diff --git a/books/bookvolbib.pamphlet b/books/bookvolbib.pamphlet
index eb8d61f..600363f 100644
--- a/books/bookvolbib.pamphlet
+++ b/books/bookvolbib.pamphlet
@@ -10,11351 +10,12439 @@ initially derived with permission from Nelson Beebe's collection.
The second section contains references from Axiom to the literature.
The third section sorts papers by topic.
\chapter{The Bibliography}
-\section{Special Topics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Solving Systems of Equations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Linear Algebra} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{axiom.bib}
-@inproceedings{Bro86,
- author = "Bronstein, Manuel",
- title = "Gsolve: a faster algorithm for solving systems of algebraic
- equations",
- booktitle = "Proc of 5th ACM SYMSAC",
- year = "1986",
- pages = "247-249",
- isbn = "0-89791-199-7",
- abstract = "
- We apply the elimination property of Gr{\"o}bner bases with respect to
- pure lexicographic ordering to solve systems of algebraic equations.
- We suggest reasons for this approach to be faster than the resultant
- technique, and give examples and timings that show that it is indeed
- faster and more correct, than MACSYMA's solve."
+@Unpublished{Kalt01,
+ author = "Kaltofen, E.",
+ title = "Algorithms for sparse and black box matrices
+ over finite fields (Invited talk)",
+ year = "2001",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/01/Ka01_Fq6.pdf",
+ paper = "Kalt01.pdf"
}
\end{chunk}
-\subsection{Numerical Algorithms} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{chunk}{ignore}
-{Bro99,
- author = "Bronstein, Manuel",
- title = "Fast Deterministic Computation of Determinants of Dense Matrices",
- url = "http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html",
- paper = "Bro99.pdf",
- abstract = "
- In this paper we consider deterministic computation of the exact
- determinant of a dense matrix $M$ of integers. We present a new
- algorithm with worst case complexity
- \[O(n^4(log n+ log \verb?||M||?)+x^3 log^2 \verb?||M||?)\],
- where $n$ is the dimension of the matrix
- and \verb?||M||? is a bound on the entries in $M$, but with
- average expected complexity
- \[O(n^4+m^3(log n + log \verb?||M||?)^2)\],
- assuming some plausible properties about the distribution of $M$.
- We will also describe a practical version of the algorithm and include
- timing data to compare this algorithm with existing ones. Our result
- does not depend on ``fast'' integer or matrix techniques."
+\begin{chunk}{axiom.bib}
+@Article{Chen02,
+ author = "Chen, L. and Eberly, W. and Kaltofen, E.
+ and Saunders, B. D. and Turner, W. J. and Villard, G.",
+ title = "Efficient Matrix Preconditioners for Black Box Linear Algebra",
+ journal = "Linear Algebra and Applications",
+ year = "2002",
+ volume = "343--344",
+ pages = "119--146",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/02/CEKSTV02.pdf",
+ paper = "Chen02.pdf"
}
\end{chunk}
-\begin{chunk}{ignore}
-{Kel00,
- author = "Kelsey, Tom",
- title = "Exact Numerical Computation via Symbolic Computation",
- url = "http://tom.host.cs.st-andrews.ac.uk/pub/ccapaper.pdf",
- paper = "Kel00.pdf",
- abstract = "
- We provide a method for converting any symbolic algebraic expression
- that can be converted into a floating point number into an exact
- numeric representation. We use this method to demonstrate a suite of
- procedures for the representation of, and arithmetic over, exact real
- numbers in the Maple computer algebra system. Exact reals are
- represented by potentially infinite lists of binary digits, and
- interpreted as sums of negative powers of the golden ratio."
+\begin{chunk}{axiom.bib}
+@InCollection{Kalt11d,
+ author = "Kaltofen, Erich and Storjohann, Arne",
+ title = "The Complexity of Computational Problems in Exact Linear Algebra",
+ booktitle = "Encyclopedia of Applied and Computational Mathematics",
+ crossref = "EACM",
+ year = "2011",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/11/KS11.pdf",
+ paper = "Kalt11d.pdf"
}
\end{chunk}
-\begin{chunk}{ignore}
-{Yang14,
- author ="Yang, Xiang and Mittal, Rajat",
- title = "Acceleration of the Jacobi iterative method by factors exceeding 100
- using scheduled relation",
- url =
-"http://engineering.jhu.edu/fsag/wp-content/uploads/sites/23/2013/10/JCP_revised_WebPost.pdf",
- paper = "Yang14.pdf"
+\begin{chunk}{axiom.bib}
+@Article{Come12,
+ author = "Comer, Matthew T. and Kaltofen, Erich L.",
+ title = "On the {Berlekamp}/{Massey} Algorithm and Counting Singular {Hankel}
+ Matrices over a Finite Field",
+ year = "2012",
+ month = "April",
+ journal = "Journal of Symbolic Computation",
+ volume = "47",
+ number = "4",
+ pages = "480--491",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/10/CoKa10.pdf",
+ paper = "Come12.pdf"
}
\end{chunk}
-\subsection{Special Functions} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{chunk}{ignore}
-{Corl0,
- author = "Corless, Robert M. and Jeffrey, David J. and Watt, Stephen M.
- and Bradford, Russell and Davenport, James H.",
- title = "Reasoning about the elementary functions of complex analysis",
- url = "http://www.csd.uwo.ca/~watt/pub/reprints/2002-amai-reasoning.pdf",
- paper = "Corl05.pdf",
- abstract = "
- There are many problems with the simplification of elementary
- functions, particularly over the complex plane. Systems tend to make
- ``howlers'' or not to simplify enough. In this paper we outline the
- ``unwinding number'' approach to such problems, and show how it can be
- used to prevent errors and to systematise such simplification, even
- though we have not yet reduced the simplification process to a
- complete algorithm. The unsolved problems are probably more amenable
- to the techniques of artificial intelligence and theorem proving than
- the original problem of complex-variable analysis."
+\begin{chunk}{axiom.bib}
+@Article{Kalt13a,
+ author = "Kaltofen, Erich and Yuhasz, George",
+ title = "A Fraction Free Matrix {Berlekamp}/{Massey} Algorithm",
+ journal = "Linear Algebra and Applications",
+ year = "2013",
+ volume = "439",
+ number = "9",
+ month = "November",
+ pages = "2515--2526",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/08/KaYu08.pdf",
+ paper = "Kalt13a.pdf"
}
\end{chunk}
-\begin{chunk}{ignore}
-{Ng68,
- author = "Ng, Edward W. and Geller, Murray",
- title = "A Table of Integrals of the Error functions",
- url = "http://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn1p1_A1b.pdf",
- paper = "Ng68.pdf",
- abstract = "
- This is a compendium of indefinite and definite integrals of products
- of the Error functions with elementary and transcendental functions."
+\begin{chunk}{axiom.bib}
+@Article{Kalt13,
+ author = "Kaltofen, Erich and Yuhasz, George",
+ title = "On The Matrix {Berlekamp}-{Massey} Algorithm",
+ year = "2013",
+ volume = "9",
+ number = "4",
+ month = "September",
+ journal = "ACM Trans. Algorithms",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/06/KaYu06.pdf",
+ paper = "Kalt13.pdf"
}
\end{chunk}
-\subsection{Exponential Integral $E_1(x)$} %%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{chunk}{ignore}
-{Gell69,
- author = "Geller, Murray and Ng, Edward W.",
- title = "A Table of Integrals of the Exponential Integral",
- url = "http://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn3p191_A1b.pdf",
- paper = "Gell69.pdf",
- abstract = "
- This is a compendium of indefinite and definite integrals of products
- of the Exponential Integral with elementary or transcendental functions."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt02a,
+ author = "Kaltofen, Erich",
+ title = "An output-sensitive variant of the baby steps/\allowbreak
+ giant steps determinant algorithm",
+ booktitle = "Proc. 2002 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC02",
+ pages = "138--144",
+ year = "2002",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/02/Ka02.pdf",
+ paper = "Kalt02a.pdf"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@techreport{Segl98,
- author = "Segletes, S.B.",
- title = "A compact analytical fit to the exponential integral $E_1(x)$",
- year = "1998",
- institution = "U.S. Army Ballistic Research Laboratory,
- Aberdeen Proving Ground, MD",
- type = "Technical Report",
- number = "ARL-TR-1758",
- paper = "Segl98.pdf",
- abstract = "
- A four-parameter fit is developed for the class of integrals known as
- the exponential integral (real branch). Unlike other fits that are
- piecewise in nature, the current fit to the exponential integral is
- valid over the complete domain of the function (compact) and is
- everywhere accurate to within $\pm 0.0052\%$ when evaluating the first
- exponential integral, $E_1$. To achieve this result, a methodology
- that makes use of analytically known limiting behaviors at either
- extreme of the domain is employed. Because the fit accurately captures
- limiting behaviors of the $E_1$ function, more accuracy is retained
- when the fit is used as part of the scheme to evaluate higher-order
- exponential integrals, $E_n$, as compared with the use of brute-force
- fits to $E_1$, which fail to accurately model limiting
- behaviors. Furthermore, because the fit is compact, no special
- accommodations are required (as in the case of spliced piecewise fits)
- to smooth the value, slope, and higher derivatives in the transition
- region between two piecewise domains. The general methodology employed
- to develop this fit is outlined, since it may be used for other
- problems as well."
+@InProceedings{Kalt01a,
+ author = "Kaltofen, E. and Villard, G.",
+ title = "On the complexity of computing determinants",
+ booktitle = "Proc. Fifth Asian Symposium on Computer Mathematics
+ (ASCM 2001)",
+ crossref = "ASCM01",
+ pages = "13--27",
+ isbn = "981-02-4763-X",
+ year = "2001",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/01/KaVi01.pdf",
+ paper = "Kalt01a.pdf"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@techreport{Se09,
- author = "Segletes, S.B.",
- title = "Improved fits for $E_1(x)$ {\sl vis-\'a-vis} those presented
- in ARL-TR-1758",
- type = "Technical Report",
- number = "ARL-TR-1758",
- institution ="U.S. Army Ballistic Research Laboratory,
- Aberdeen Proving Ground, MD",
- year = "1998",
- month = "September",
- paper = "Se09.pdf",
- abstract = "
- This is a writeup detailing the more accurate fits to $E_1(x)$,
- relative to those presented in ARL-TR-1758. My actual fits are to
- \[F1 =[x\ exp(x) E_1(x)]\] which spans a functional range from 0 to 1.
- The best accuracy I have been yet able to achieve, defined by limiting
- the value of \[[(F1)_{fit} - F1]/F1\] over the domain, is
- approximately 3.1E-07 with a 12-parameter fit, which unfortunately
- isn't quite to 32-bit floating-point accuracy. Nonetheless, the fit
- is not a piecewise fit, but rather a single continuous function over
- the domain of nonnegative x, which avoids some of the problems
- associated with piecewise domain splicing."
+@Article{Kalt04a,
+ author = "Kaltofen, Erich and Villard, Gilles",
+ title = "On the Complexity of Computing Determinants",
+ journal = "Computational Complexity",
+ volume = "13",
+ number = "3-4",
+ year = "2004",
+ pages = "91--130",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/04/KaVi04_2697263.pdf",
+ paper = "Kalt04a.pdf"
}
\end{chunk}
-\subsection{Polynomial GCD} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt97b,
+ author = "Eberly, W. and Kaltofen, E.",
+ title = "On Randomized {Lanczos} Algorithms",
+ booktitle = "Proc. 1997 Internat. Symp. Symbolic Algebraic Comput.",
+ year = "1997",
+ crossref = "ISSAC97",
+ pages = "176--183",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/EbKa97.pdf",
+ paper = "Kalt97b.pdf"
-\begin{chunk}{ignore}
-\bibitem[Knuth 71]{ST-PGCD-Knu71} Knuth, Donald
-``The Art of Computer Programming''
-2nd edition Vol. 2 (Seminumerical Algorithms) 1st edition, 2nd printing,
-Addison-Wesley 1971, section 4.6 pp399-505
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Ma 90]{ST-PGCD-Ma90} Ma, Keju; Gathen, Joachim von zur
-``Analysis of Euclidean Algorithms for Polynomials over Finite Fields''
-J. Symbolic Computation (1990) Vol 9 pp429-455\hfill{}
-\verb|www.researchgate.net/publication/220161718_Analysis_of_Euclidean_|
-\verb|Algorithms_for_Polynomials_over_Finite_Fields/file/|
-\verb|60b7d52b326a1058e4.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/ST-PGCD-Ma90.pdf|
- abstract = "
- This paper analyzes the Euclidean algorithm and some variants of it
- for computing the greatest common divisor of two univariate polynomials
- over a finite field. The minimum, maximum, and average number of
- arithmetic operations both on polynomials and in the ground field
- are derived."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt94b,
+ author = "Kaltofen, E.",
+ title = "Asymptotically fast solution of {Toeplitz}-like singular
+ linear systems",
+ booktitle = "Proc. 1994 Internat. Symp. Symbolic Algebraic Comput.",
+ pages = "297--304",
+ crossref = "ISSAC94",
+ year = "1994",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/Ka94_issac.pdf",
+ paper = "Kalt94b.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Naylor 00a]{N00} Naylor, Bill
-``Polynomial GCD Using Straight Line Program Representation''
-PhD. Thesis, University of Bath, 2000
-\verb|www.sci.csd.uwo.ca/~bill/thesis.ps|
-%\verb|axiom-developer.org/axiom-website/papers/N00.pdf|
- abstract = "
- This thesis is concerned with calculating polynomial greatest common
- divisors using straight line program representation.
-
- In the Introduction chapter, we introduce the problem and describe
- some of the traditional representations for polynomials, we then talk
- about some of the general subjects central to the thesis, terminating
- with a synopsis of the category theory which is central to the Axiom
- computer algebra system used during this research.
-
- The second chapter is devoted to describing category theory. We follow
- with a chapter detailing the important sections of computer code
- written in order to investigate the straight line program subject.
- The following chapter on evalution strategies and algorithms which are
- dependant on these follows, the major algorith which is dependant on
- evaluation and which is central to our theis being that of equality
- checking. This is indeed central to many mathematical problems.
- Interpolation, that is the determination of coefficients of a
- polynomial is the subject of the next chapter. This is very important
- for many straight line program algorithms, as their non-canonical
- structure implies that it is relatively difficult to determine
- coefficients, these being the basic objects that many algorithms work
- on. We talk about three separate interpolation techniques and compare
- their advantages and disadvantages. The final two chapters describe
- some of the results we have obtained from this research and finally
- conclusions we have drawn as to the viability of the straight line
- program approach and possible extensions.
-
- Finally we terminate with a number of appendices discussing side
- subjects encountered during the thesis."
+\begin{chunk}{axiom.bib}
+@Article{Kalt99,
+ author = "Kaltofen, E. and Lobo, A",
+ title = "Distributed matrix-free solution of large sparse linear systems over
+ finite fields",
+ journal = "Algorithmica",
+ year = "1999",
+ pages = "331--348",
+ month = "July--Aug.",
+ volume = "24",
+ number = "3--4",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/99/KaLo99.pdf",
+ paper = "Kalt99.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Shoup 93]{ST-PGCD-Sh93} Shoup, Victor
-``Factoring Polynomials over Finite Fields: Asymptotic Complexity vs
-Reality*''
-Proc. IMACS Symposium, Lille, France, (1993)
-\verb|www.shoup.net/papers/lille.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/ST-PGCD-Sh93.pdf|
- abstract = "
- This paper compares the algorithms by Berlekamp, Cantor and
- Zassenhaus, and Gathen and Shoup to conclude that (a) if large
- polynomials are factored the FFT should be used for polynomial
- multiplication and division, (b) Gathen and Shoup should be used if
- the number of irreducible factors of $f$ is small. (c) if nothing is
- know about the degrees of the factors then Berlekamp's algorithm
- should be used."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt96a,
+ author = "Kaltofen, E. and Lobo, A.",
+ title = "Distributed matrix-free solution of large sparse linear systems
+ over finite fields",
+ booktitle = "Proc. High Performance Computing '96",
+ year = "1996",
+ editor = "A. M. Tentner",
+ pages = "244--247",
+ organization = "Society for Computer Simulation",
+ publisher = "Simulation Councils, Inc.",
+ address = "San Diego, CA",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/96/KaLo96_hpc.pdf",
+ paper = "Kalt96a.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Gathen 01]{ST-PGCD-Ga01} Gathen, Joachim von zur; Panario, Daniel
-``Factoring Polynomials Over Finite Fields: A Survey''
-J. Symbolic Computation (2001) Vol 31, pp3-17\hfill{}
-\verb|people.csail.mit.edu/dmoshdov/courses/codes/poly-factorization.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/ST-PGCD-Ga01.pdf|
- keywords = "survey",
- abstract = "
- This survey reviews several algorithms for the factorization of
- univariate polynomials over finite fields. We emphasize the main ideas
- of the methods and provide and up-to-date bibliography of the problem.
- This paper gives algorithms for {\sl squarefree factorization},
- {\sl distinct-degree factorization}, and {\sl equal-degree factorization}.
- The first and second algorithms are deterministic, the third is
- probabilistic."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt94a,
+ author = "Kaltofen, E. and Lobo, A.",
+ title = "Factoring high-degree polynomials by the black box
+ Berlekamp algorithm",
+ booktitle = "Proc. 1994 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC94",
+ pages = "90--98",
+ year = "1994",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/KaLo94.ps.gz",
+ paper = "Kalt94a.ps"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[van Hoeij]{Hoeij04} Hoeij, Mark van; Monagen, Michael
-``Algorithms for Polynomial GCD Computation over Algebraic Function Fields''
-\verb|www.cecm.sfu.ca/personal/mmonagan/papers/AFGCD.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Hoeij04.pdf|
- abstract = "
- Let $L$ be an algebraic function field in $k \ge 0$ parameters
- $t_1,\ldots,t)k$. Let $f_1$, $f_2$ be non-zero polynomials in
- $L[x]$. We give two algorithms for computing their gcd. The first, a
- modular GCD algorithm, is an extension of the modular GCD algorithm
- for Brown for {\bf Z}$[x_1,\ldots,x_n]$ and Encarnacion for {\bf
- Q}$(\alpha[x])$ to function fields. The second, a fraction-free
- algorithm, is a modification of the Moreno Maza and Rioboo algorithm
- for computing gcds over triangular sets. The modification reduces
- coefficient grownth in $L$ to be linear. We give an empirical
- comparison of the two algorithms using implementations in Maple."
+\begin{chunk}{axiom.bib}
+@Article{Kalt95,
+ author = "Kaltofen, E.",
+ title = "Analysis of {Coppersmith}'s block {Wiedemann} algorithm for the
+ parallel solution of sparse linear systems",
+ journal = "Math. Comput.",
+ year = "1995",
+ volume = "64",
+ number = "210",
+ pages = "777--806",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/Ka95_mathcomp.pdf",
+ paper = "Kalt95.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Wang 78]{Wang78} Wang, Paul S.
-``An Improved Multivariate Polynomial Factoring Algorithm''
-Mathematics of Computation, Vol 32, No 144 Oct 1978, pp1215-1231
-\verb|www.ams.org/journals/mcom/1978-32-144/S0025-5718-1978-0568284-3/|
-\verb|S0025-5718-1978-0568284-3.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Wang78.pdf|
- abstract = "
- A new algorithm for factoring multivariate polynomials over the
- integers based on an algorithm by Wang and Rothschild is described.
- The new algorithm has improved strategies for dealing with the known
- problems of the original algorithm, namely, the leading coefficient
- problem, the bad-zero problem and the occurence of extraneous factors.
- It has an algorithm for correctly predetermining leading coefficients
- of the factors. A new and efficient p-adic algorith named EEZ is
- described. Basically it is a linearly convergent variable-by-variable
- parallel construction. The improved algorithm is generally faster and
- requires less store than the original algorithm. Machine examples with
- comparative timing are included."
+\begin{chunk}{axiom.bib}
+@Article{Kalt90a,
+ author = "Kaltofen, E. and Krishnamoorthy, M.S. and Saunders, B.D.",
+ title = "Parallel algorithms for matrix normal forms",
+ journal = "Linear Algebra and Applications",
+ year = "1990",
+ volume = "136",
+ pages = "189--208",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/KKS90.pdf",
+ paper = "Kalt90a.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Wiki 4]{Wiki4}.
-``Polynomial greatest common divisor''
-\verb|en.wikipedia.org/wiki/Polynomial_greatest_common_divisor|
+\begin{chunk}{axiom.bib}
+@Article{Kalt87,
+ author = "Kaltofen, E. and Krishnamoorthy, M.S. and Saunders, B.D.",
+ title = "Fast parallel computation of Hermite and Smith forms of
+ polynomial matrices",
+ journal = "SIAM J. Alg. Discrete Math.",
+ year = "1987",
+ volume = "8",
+ pages = "683--690",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/KKS87.pdf",
+ paper = "Kalt87.pdf"
+}
\end{chunk}
-\subsection{Category Theory} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{chunk}{ignore}
-\bibitem[Baez 09]{Baez09} Baez, John C.; Stay, Mike
-``Physics, Topology, Logic and Computation: A Rosetta Stone''
-\verb|arxiv.org/pdf/0903.0340v3.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Baez09.pdf|
- abstract = "
- In physics, Feynman diagrams are used to reason about quantum
- processes. In the 1980s, it became clear that underlying these
- diagrams is a powerful analogy between quantum physics and
- topology. Namely, a linear operator behaves very much like a
- ``cobordism'': a manifold representing spacetime, going between two
- manifolds representing space. But this was just the beginning: simiar
- diagrams can be used to reason about logic, where they represent
- proofs, and computation, where they represent programs. With the rise
- of interest in quantum cryptography and quantum computation, it became
- clear that there is an extensive network of analogies between physics,
- topology, logic and computation. In this expository paper, we make
- some of these analogies precise using the concept of ``closed
- symmetric monodial category''. We assume no prior knowledge of
- category theory, proof theory or computer science."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt92,
+ author = "Kaltofen, E. and Pan, V.",
+ title = "Processor-efficient parallel solution of linear systems {II}:
+ the positive characteristic and singular cases",
+ booktitle = "Proc. 33rd Annual Symp. Foundations of Comp. Sci.",
+ year = "1992",
+ pages = "714--723",
+ publisher = "IEEE Computer Society Press",
+ address = "Los Alamitos, California",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/92/KaPa92.pdf",
+ paper = "Kalt92.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Meijer 91]{Meij91} Meijer, Erik; Fokkinga, Maarten; Paterson, Ross
-``Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire''
-\verb|eprints.eemcs.utwente.nl/7281/01/db-utwente-40501F46.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Meij91.pdf|
- abstract = "
- We develop a calculus for lazy functional programming based on
- recursion operators associated with data type definitions. For these
- operators we derive various algebraic laws that are useful in deriving
- and manipulating programs. We shall show that all example functions in
- Bird and Wadler's ``Introduction to Functional Programming'' can be
- expressed using these operators."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt91c,
+ author = "Kaltofen, E. and Pan, V.",
+ title = "Processor efficient parallel solution of linear systems over
+ an abstract field",
+ booktitle = "Proc. SPAA '91 3rd Ann. ACM Symp. Parallel Algor. Architecture",
+ pages = "180--191",
+ publisher = "ACM Press",
+ year = "1991",
+ address = "New York, N.Y.",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaPa91.pdf",
+ paper = "Kalt91c.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Youssef 04]{You04} Youssef, Saul
-``Prospects for Category Theory in Aldor''
-October 2004
-%\verb|axiom-developer.org/axiom-website/papers/You04.pdf|
- abstract = "
- Ways of encorporating category theory constructions and results into
- the Aldor language are discussed. The main features of Aldor which
- make this possible are identified, examples of categorical
- constructions are provided and a suggestion is made for a foundation
- for rigorous results."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt91,
+ author = "Kaltofen, E. and Saunders, B.D.",
+ editor = "H. F. Mattson and T. Mora and T. R. N. Rao",
+ title = "On {Wiedemann's} method of solving sparse linear systems",
+ booktitle = "Proc. AAECC-9",
+ series = "Lect. Notes Comput. Sci.",
+ volume = "539",
+ pages = "29--38",
+ publisher = "Springer-Verlag",
+ year = "1991",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaSa91.pdf",
+ paper = "Kalt91.pdf"
+}
\end{chunk}
-\subsection{Proving Axiom Correct} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Algebraic Algorithms} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{chunk}{ignore}
-\bibitem[Adams 99]{Adam99} Adams, A.A.; Gottlieben, H.; Linton, S.A.;
-Martin, U.
-``Automated theorem proving in support of computer algebra:''
-`` symbolic definite integration as a case study''
-%\verb|axiom-developer.org/axiom-website/papers/Adam99.pdf|
- abstract = "
- We assess the current state of research in the application of computer
- aided formal reasoning to computer algebra, and argue that embedded
- verification support allows users to enjoy its benefits without
- wrestling with technicalities. We illustrate this claim by considering
- symbolic definite integration, and present a verifiable symbolic
- definite integral table look up: a system which matches a query
- comprising a definite integral with parameters and side conditions,
- against an entry in a verifiable table and uses a call to a library of
- lemmas about the reals in the theorem prover PVS to aid in the
- transformation of the table entry into an answer. We present the full
- model of such a system as well as a description of our prototype
- implementation showing the efficacy of such a system: for example, the
- prototype is able to obtain correct answers in cases where computer
- algebra systems [CAS] do not. We extend upon Fateman's web-based table
- by including parametric limits of integration and queries with side
- conditions."
+\begin{chunk}{axiom.bib}
+@InCollection{Diaz97,
+ author = "Diaz, A. and Kaltofen, E. and Pan, V.",
+ title = "Algebraic Algorithms",
+ booktitle = "The Computer Science and Engineering Handbook",
+ publisher = "CRC Press",
+ year = "1997",
+ editor = "A. B. Tucker",
+ pages = "226--248",
+ address = "Boca Raton, Florida",
+ chapter = "10",
+ keywords = "survey",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/DKP97.ps.gz",
+ paper = "Diaz97.ps"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Adams 01]{Adam01} Adams, Andrew; Dunstan, Martin; Gottliebsen, Hanne;
-Kelsey, Tom; Martin, Ursula; Owre, Sam
-``Computer Algebra Meets Automated Theorem Proving: Integrating Maple and PVS''
-\verb|www.csl.sri.com/~owre/papers/tphols01/tphols01.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Adam01.pdf|
- abstract = "
- We describe an interface between version 6 of the Maple computer
- algebra system with the PVS automated theorem prover. The interface is
- designed to allow Maple users access to the robust and checkable proof
- environment of PVS. We also extend this environment by the provision
- of a library of proof strategies for use in real analysis. We
- demonstrate examples using the interface and the real analysis
- library. These examples provide proofs which are both illustrative and
- applicable to genuine symbolic computation problems."
+\begin{chunk}{axiom.bib}
+@InCollection{Diaz99,
+ author = "Diaz, A. and Emiris, I. and Kaltofen, E. and Pan, V.",
+ title = "Algebraic Algorithms",
+ booktitle = "Algorithms \& Theory of Computation Handbook",
+ publisher = "CRC Press",
+ year = "1999",
+ editor = "M. J. Atallah",
+ address = "Boca Raton, Florida",
+ pages = "16.1--16.27",
+ isbn = "0-8493-2649-4",
+ keywords = "survey",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/99/DEKP99.ps.gz",
+ paper = "Diaz99.ps"
+}
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Mahb06,
- author = "Mahboubi, Assia",
- title = "Proving Formally the Implementation of an Efficient gcd
- Algorithm for Polynomials",
- journal = "Lecture Notes in Computer Science",
- volume = "4130",
- year = "2006",
- pages = "438-452",
- paper = "Mahb06.pdf",
- abstract = "
- We describe here a formal proof in the Coq system of the structure
- theorem for subresultants which allows to prove formally the
- correctness of our implementation of the subresultants algorithm.
- Up to our knowledge it is the first mechanized proof of this result."
+@InCollection{Kalt87a,
+ author = "Kaltofen, E.",
+ editor = "J. F. Traub",
+ title = "Computer algebra algorithms",
+ booktitle = "Annual Review in Computer Science",
+ pages = "91--118",
+ publisher = "Annual Reviews Inc.",
+ year = "1987",
+ volume = "2",
+ address = "Palo Alto, California",
+ keywords = "survey",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_annrev.pdf",
+ paper = "Kalt87a.pdf"
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Ballarin 99]{Ball99} Ballarin, Clemens; Paulson, Lawrence C.
-``A Pragmatic Approach to Extending Provers by Computer Algebra --
- with Applications to Coding Theory''
-\verb|www.cl.cam.ac.uk/~lp15/papers/Isabelle/coding.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Ball99.pdf|
- abstract = "
- The use of computer algebra is usually considered beneficial for
- mechanised reasoning in mathematical domains. We present a case study,
- in the application domain of coding theory, that supports this claim:
- the mechanised proofs depend on non-trivial algorithms from computer
- algebra and increase the reasoning power of the theorem prover.
+\section{Sparse Linear Systems} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- The unsoundness of computer algebra systems is a major problem in
- interfacing them to theorem provers. Our approach to obtaining a sound
- overall system is not blanket distrust but based on the distinction
- between algorithms we call sound and {\sl ad hoc} respectively. This
- distinction is blurred in most computer algebra systems. Our
- experimental interface therefore uses a computer algebra library. It
- is based on formal specifications for the algorithms, and links the
- computer algebra library Sumit to the prover Isabelle.
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt96b,
+ author = "Kaltofen, E.",
+ title = "Blocked iterative sparse linear system solvers for finite fields",
+ booktitle = "Proc. Symp. Parallel Comput. Solving Large Scale Irregular
+ Applic. (Stratagem '96)",
+ editor = "C. Roucairol",
+ publisher = "INRIA",
+ address = "Sophia Antipolis, France",
+ pages = "91--95",
+ year = "1996",
+ keywords = "survey",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/96/Ka96_stratagem.ps.gz",
+ paper = "Kalt96b.ps"
+}
- We give details of the interface, the use of the computer algebra
- system on the tactic-level of Isabelle and its integration into proof
- procedures."
+\end{chunk}
+
+\section{Matrix Determinants} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@Article{Kalt04,
+ author = "Kaltofen, E. and Villard, G.",
+ title = "Computing the sign or the value of the determinant of an integer
+ matrix, a complexity survey",
+ journal = "J. Computational Applied Math.",
+ volume = "162",
+ number = "1",
+ month = "January",
+ pages = "133--146",
+ year = "2004",
+ keywords = "survey",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/02/KaVi02.pdf",
+ paper = "Kalt04.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Bertot 04]{Bert04} Bertot, Yves; Cast\'eran, Pierre
-``Interactive Theorem Proving and Program Development''
-Springer ISBN 3-540-20854-2
- abstract = "
- Coq is an interactive proof assistant for the development of
- mathematical theories and formally certified software. It is based on
- a theory called the calculus of inductive constructions, a variant of
- type theory.
- This book provides a pragmatic introduction to the development of
- proofs and certified programs using Coq. With its large collection of
- examples and exercies it is an invaluable tool for researchers,
- students, and engineers interested in formal methods and the
- development of zero-fault software."
+\section{Open Problems} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@Article{Kalt00,
+ author = "Kaltofen, E.",
+ title = "Challenges of Symbolic Computation My Favorite Open Problems",
+ journal = "Journal of Symbolic Computation",
+ volume = "29",
+ number = "6",
+ pages = "891--919",
+ year = "2000",
+ keywords = "survey",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/2K/Ka2K.pdf",
+ paper = "Kalt00.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Boulme 00]{BHR00} Boulm\'e, S.; Hardin, T.; Rioboo, R.
-``Polymorphic Data Types, Objects, Modules and Functors,: is it too much?''
-%\verb|axiom-developer.org/axiom-website/papers/BHR00.pdf|
- abstract = "
- Abstraction is a powerful tool for developers and it is offered by
- numerous features such as polymorphism, classes, modules, and
- functors, $\ldots$ A working programmer may be confused by this
- abundance. We develop a computer algebra library which is being
- certificed. Reporting this experience made with a language (Ocaml)
- offering all these features, we argue that the are all needed
- together. We compare several ways of using classes to represent
- algebraic concepts, trying to follow as close as possible mathematical
- specification. Thenwe show how to combine classes and modules to
- produce code having very strong typing properties. Currently, this
- library is made of one hundred units of functional code and behaves
- faster than analogous ones such as Axiom."
+\section{Parallel Evaluation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@InCollection{Kalt93a,
+ author = "Kaltofen, E.",
+ editor = "J. Reif",
+ title = "Dynamic parallel evaluation of computation {DAG}s",
+ booktitle = "Synthesis of Parallel Algorithms",
+ pages = "723--758",
+ publisher = "Morgan Kaufmann Publ.",
+ year = "1993",
+ address = "San Mateo, California",
+ keywords = "survey",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_synthesis.ps.gz",
+ paper = "Kalt93a.ps"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Boulme 01]{BHHMR01}
-Boulm\'e, S.; Hardin, T.; Hirschkoff, D.; M\'enissier-Morain, V.; Rioboo, R.
-``On the way to certify Computer Algebra Systems''
-Calculemus-2001
-%\verb|axiom-developer.org/axiom-website/papers/BHHMR01.pdf|
- abstract = "
- The FOC project aims at supporting, within a coherent software system,
- the entire process of mathematical computation, starting with proved
- theories, ending with certified implementations of algorithms. In this
- paper, we explain our design requirements for the implementation,
- using polynomials as a running example. Indeed, proving correctness of
- implementations depends heavily on the way this design allows
- mathematical properties to be truly handled at the programming level.
+\section{Hybrid Symbolic/Numeric} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- The FOC project, started at the fall of 1997, is aimed to build a
- programming environment for the development of certified symbolic
- computation. The working languages are Coq and Ocaml. In this paper,
- we present first the motivations of the project. We then explain why
- and how our concern for proving properties of programs has led us to
- certain implementation choices in Ocaml. This way, the sources express
- exactly the mathematical dependencies between different structures.
- This may ease the achievement of proofs."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt06,
+ author = "Kaltofen, Erich and Zhi, Lihong",
+ title = "Hybrid Symbolic-Numeric Computation",
+ year = "2006",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'06",
+ crossref = "ISSAC06",
+ pages = "7",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/06/KaZhi06.pdf",
+ paper = "Kalt06.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Daly 10]{Daly10} Daly, Timothy
-``Intel Instruction Semantics Generator''
-\verb|daly.axiom-developer.org/TimothyDaly_files/publications/sei/intel/intel.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Daly10.pdf|
- abstract = "
- Given an Intel x86 binary, extract the semantics of the instruction
- stream as Conditional Concurrent Assignments (CCAs). These CCAs
- represent the semantics of each individual instruction. They can be
- composed to represent higher level semantics."
+\begin{chunk}{axiom.bib}
+@InProceedings{Hutt10,
+ author = "Hutton, Sharon E. and Kaltofen, Erich L. and Zhi, Lihong",
+ title = "Computing the radius of positive semidefiniteness of a
+ multivariate real polynomial via a dual of {Seidenberg}'s method",
+ year = "2010",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'10",
+ crossref = "ISSAC10",
+ pages = "227--234",
+ month = "July",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/10/HKZ10.pdf",
+ paper = "Hutt10.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Danielsson 06]{Dani06} Danielsson, Nils Anders; Hughes, John;
-Jansson, Patrik; Gibbons, Jeremy
-``Fast and Loose Reasoning is Morally Correct''
-ACM POPL'06 January 2005, Charleston, South Carolina, USA
-%\verb|axiom-developer.org/axiom-website/papers/Dani06.pdf|
- abstract = "
- Functional programmers often reason about programs as if they were
- written in a total language, expecting the results to carry over to
- non-toal (partial) languages. We justify such reasoning.
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt09,
+ author = "Kaltofen, Erich and Yang, Zhengfeng and Zhi, Lihong",
+ title = "A Proof of the {Monotone Column Permanent (MCP) Conjecture} for
+ Dimension 4 via Sums-Of-Squares of Rational Functions",
+ year = "2009",
+ booktitle = "Proc. 2009 Internat. Workshop on Symbolic-Numeric Comput.",
+ crossref = "SNC09",
+ pages = "65--69",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/09/KYZ09.pdf",
+ paper = "Kalt09.pdf"
+}
- Two languages are defined, one total and one partial, with identical
- syntax. The semantics of the partial language includes partial and
- infinite values, and all types are lifted, including the function
- spaces. A partial equivalence relation (PER) is then defined, the
- domain of which is the total subset of the partial language. For types
- not containing function spaces the PER relates equal values, and
- functions are related if they map related values to related values.
+\end{chunk}
- It is proved that if two closed terms have the same semantics in the
- total language, then they have related semantics in the partial
- language. It is also shown that the PER gives rise to a bicartesian
- closed category which can be used to reason about values in the domain
- of the relation."
+\begin{chunk}{axiom.bib}
+@Article{Kalt12,
+ author = "Kaltofen, Erich L. and Li, Bin and Yang, Zhengfeng and
+ Zhi, Lihong",
+ title = "Exact Certification in Global Polynomial Optimization
+ Via Sums-Of-Squares of Rational Functions
+ with Rational Coefficients",
+ year = "2012",
+ month = "January",
+ journal = "Journal of Symbolic Computation",
+ volume = "47",
+ number = "1",
+ pages = "1--15",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/09/KLYZ09.pdf",
+ paper = "Kalt12.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Davenport 12]{Davenp12} Davenport, James H.; Bradford, Russell;
-England, Matthew; Wilson, David
-``Program Verification in the presence of complex numbers, functions with
-branch cuts etc.''
-\verb|arxiv.org/pdf/1212.5417.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Davenp12.pdf|
- abstract = "
- In considering the reliability of numerical programs, it is normal to
- ``limit our study to the semantics dealing with numerical precision''.
- On the other hand, there is a great deal of work on the reliability of
- programs that essentially ignores the numerics. The thesis of this
- paper is that there is a class of problems that fall between these
- two, which could be described as ``does the low-level arithmetic
- implement the high-level mathematics''. Many of these problems arise
- because mathematics, particularly the mathematics of the complex
- numbers, is more difficult than expected: for example the complex
- function log is not continuous, writing down a program to compute an
- inverse function is more complicated than just solving an equation,
- and many algebraic simplification rules are not universally valid.
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt08b,
+ author = "Kaltofen, Erich and Li, Bin and Yang, Zhengfeng and Zhi, Lihong",
+ title = "Exact Certification of Global Optimality of Approximate
+ Factorizations Via Rationalizing Sums-Of-Squares
+ with Floating Point Scalars",
+ year = "2008",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'08",
+ crossref = "ISSAC08",
+ pages = "155--163",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/08/KLYZ08.pdf",
+ paper = "Kalt08b.pdf"
+}
- The good news is that these problems are {\sl theoretically} capable
- of being solved, and are {\sl practically} close to being solved, but
- not yet solved, in several real-world examples. However, there is
- still a long way to go before implementations match the theoretical
- possibilities."
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt06b,
+ author = "Kaltofen, Erich and Yang, Zhengfeng and Zhi, Lihong",
+ title = "Approximate greatest common divisors of several polynomials
+ with linearly constrained coefficients and singular polynomials",
+ year = "2006",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'06",
+ crossref = "ISSAC06",
+ pages = "169--176",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/06/KYZ06.pdf",
+ paper = "Kalt06b.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Dolzmann 97]{Dolz97} Dolzmann, Andreas; Sturm, Thomas
-``Guarded Expressions in Practice''
-\verb|redlog.dolzmann.de/papers/pdf/MIP-9702.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Dolz97.pdf|
- abstract = "
- Computer algebra systems typically drop some degenerate cases when
- evaluating expressions, e.g. $x/x$ becomes 1 dropping the case
- $x=0$. We claim that it is feasible in practice to compute also the
- degenerate cases yielding {\sl guarded expressions}. We work over real
- closed fields but our ideas about handling guarded expressions can be
- easily transferred to other situations. Using formulas as guards
- provides a powerful tool for heuristically reducing the combinatorial
- explosion of cases: equivalent, redundant, tautological, and
- contradictive cases can be detected by simplification and quantifier
- elimination. Our approach allows to simplify the expressions on the
- basis of simplification knowledge on the logical side. The method
- described in this paper is implemented in the REDUCE package GUARDIAN,
- which is freely available on the WWW."
+\begin{chunk}{axiom.bib}
+@InCollection{Kalt05,
+ author = "Kaltofen, Erich and Yang, Zhengfeng and Zhi, Lihong",
+ title = "Structured Low Rank Approximation of a {Sylvester} Matrix",
+ booktitle = "Symbolic-Numeric Computation",
+ crossref = "SNC06",
+ pages = "69--83",
+ year = "2005",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/05/KYZ05.pdf",
+ paper = "Kalt05.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Dos Reis 11]{DR11} Dos Reis, Gabriel; Matthews, David; Li, Yue
-``Retargeting OpenAxiom to Poly/ML: Towards an Integrated Proof Assistants
-and Computer Algebra System Framework''
-Calculemus (2011) Springer
-\verb|paradise.caltech.edu/~yli/paper/oa-polyml.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/DR11.pdf|
- abstract = "
- This paper presents an ongoing effort to integrate the Axiom family of
- computer algebra systems with Poly/ML-based proof assistants in the
- same framework. A long term goal is to make a large set of efficient
- implementations of algebraic algorithms available to popular proof
- assistants, and also to bring the power of mechanized formal
- verification to a family of strongly typed computer algebra systems at
- a modest cost. Our approach is based on retargeting the code generator
- of the OpenAxiom compiler to the Poly/ML abstract machine."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt03a,
+ author = "Kaltofen, Erich and May, John",
+ title = "On Approximate Irreducibility of Polynomials in Several Variables",
+ year = "2003",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'03",
+ crossref = "ISSAC03",
+ pages = "161--168",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/03/KM03.pdf",
+ paper = "Kalt03a.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Dunstan 00a]{Dun00a} Dunstan, Martin N.
-``Adding Larch/Aldor Specifications to Aldor''
-%\verb|axiom-developer.org/axiom-website/papers/Dunxx.pdf|
- abstract = "
- We describe a proposal to add Larch-style annotations to the Aldor
- programming language, based on our PhD research. The annotations
- are intended to be machine-checkable and may be used for a variety
- of purposes ranging from compiler optimizations to verification
- condition (VC) generation. In this report we highlight the options
- available and describe the changes which would need to be made to
- the compiler to make use of this technology."
+\begin{chunk}{axiom.bib}
+@InProceedings{Gao04a,
+ author = "Shuhong, Gao and Kaltofen, Erich and May, John P. and
+ Yang, Zhengfeng and Zhi, Lihong",
+ title = "Approximate factorization of multivariate polynomials via
+ differential equations",
+ year = "2004",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'04",
+ crossref = "ISSAC04",
+ pages = "167--174",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/04/GKMYZ04.pdf",
+ paper = "Gao04a.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Dunstan 98]{Dun98} Dunstan, Martin; Kelsey, Tom; Linton, Steve;
-Martin, Ursula
-``Lightweight Formal Methods For Computer Algebra Systems''
-\verb|www.cs.st-andrews.ac.uk/~tom/pub/issac98.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Dun98.pdf|
- abstract = "
- Demonstrates the use of formal methods tools to provide a semantics
- for the type hierarchy of the Axiom computer algebra system, and a
- methodology for Aldor program analysis and verification. There are
- examples of abstract specifications of Axiom primitives."
+\begin{chunk}{axiom.bib}
+@Article{Kalt08,
+ author = "Kaltofen, Erich and May, John and Yang, Zhengfeng and Zhi, Lihong",
+ title = "Approximate Factorization of Multivariate Polynomials Using
+ Singular Value Decomposition",
+ year = "2008",
+ journal = "Journal of Symbolic Computation",
+ volume = "43",
+ number = "5",
+ pages = "359--376",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/07/KMYZ07.pdf",
+ paper = "Kalt08.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Dunstan 99a]{Dun99a} Dunstan, MN
-``Larch/Aldor - A Larch BISL for AXIOM and Aldor''
-PhD Thesis, 1999
-\verb|www.cs.st-andrews.uk/files/publications/Dun99.php|
-%\verb|axiom-developer.org/axiom-website/papers/Dun99a.pdf|
- abstract = "
- In this thesis we investigate the use of lightweight formal methods
- and verification conditions (VCs) to help improve the reliability of
- components constructed within a computer algebra system. We follow the
- Larch approach to formal methods and have designed a new behavioural
- interface specification language (BISL) for use with Aldor: the
- compiled extension language of Axiom and a fully-featured programming
- language in its own right. We describe our idea of lightweight formal
- methods, present a design for a lightweight verification condition
- generator and review our implementation of a prototype verification
- condition generator for Larch/Aldor."
+\begin{chunk}{axiom.bib}
+@InProceedings{Hitz99,
+ author = "Hitz, M.A. and Kaltofen, E. and Lakshman, Y.N.",
+ title = "Efficient Algorithms for Computing the Nearest Polynomial
+ With A Real Root and Related Problems",
+ booktitle = "Proc. 1999 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC99",
+ pages = "205--212",
+ year = "1999",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/99/HKL99.pdf",
+ paper = "Hitz99.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Dunstan 00]{Dun00} Dunstan, Martin; Kelsey, Tom; Martin, Ursula;
-Linton, Steve
-``Formal Methods for Extensions to CAS''
-FME'99, Toulouse, France, Sept 20-24, 1999, pp 1758-1777
-\verb|tom.host.cs.st-andrews.ac.uk/pub/fm99.ps|
-%\verb|axiom-developer.org/axiom-website/papers/Dun00.pdf|
- abstract = "
- We demonstrate the use of formal methods tools to provide a semantics
- for the type hierarchy of the AXIOM computer algebra system, and a
- methodology for Aldor program analysis and verification. We give a
- case study of abstract specifications of AXIOM primitives, and provide
- an interface between these abstractions and Aldor code."
+\begin{chunk}{axiom.bib}
+@InProceedings{Hitz98,
+ author = "Hitz, M. A. and Kaltofen, E.",
+ title = "Efficient Algorithms for Computing the Nearest Polynomial
+ with Constrained Roots",
+ booktitle = "Proc. 1998 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC98",
+ year = "1998",
+ pages = "236--243",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/98/HiKa98.pdf",
+ paper = "Hitz98.pdf"
+}
\end{chunk}
+\section{Software Systems} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{axiom.bib}
-@misc{Hard13,
- author = "Hardin, David S. and McClurg, Jedidiah R. and Davis, Jennifer A.",
- title = "Creating Formally Verified Components for Layered Assurance with an LLVM to ACL2 Translator",
- url = "http://www.jrmcclurg.com/papers/law_2013_paper.pdf",
- paper = "Hard13.pdf",
- abstract = "
- This paper describes an effort to create a library of formally
- verified software component models from code that have been compiled
- using the Low-Level Virtual Machine (LLVM) intermediate form. The idea
- is to build a translator from LLVM to the applicative subset of Common
- Lisp accepted by the ACL2 theorem prover. They perform verification of
- the component model using ACL2's automated reasoning capabilities."
+@InProceedings{Diaz91,
+ author = "Diaz, A.; Kaltofen,E.; Schmitz, K.; Valente, T.",
+ title = "DSC A System for Distributed Symbolic Computation",
+ booktitle = "Proc. 1991 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC91",
+ pages = "323--332",
+ year = "1991",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/DKSV91.pdf",
+ paper = "Diaz91.pdf"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@misc{Hard14,
- author = "Hardin, David S. and Davis, Jennifer A. and Greve, David A. and
- McClurg, Jedidiah R.",
- title = "Development of a Translator from LLVM to ACL2",
- url = "http://arxiv.org/pdf/1406.1566",
- paper = "Hard14.pdf",
- abstract = "
- In our current work a library of formally verified software components
- is to be created, and assembled, using the Low-Level Virtual Machine
- (LLVM) intermediate form, into subsystems whose top-level assurance
- relies on the assurance of the individual components. We have thus
- undertaken a project to build a translator from LLVM to the
- applicative subset of Common Lisp accepted by the ACL2 theorem
- prover. Our translator produces executable ACL2 formal models,
- allowing us to both prove theorems about the translated models as well
- as validate those models by testing. The resulting models can be
- translated and certified without user intervention, even for code with
- loops, thanks to the use of the def::ung macro which allows us to
- defer the question of termination. Initial measurements of concrete
- execution for translated LLVM functions indicate that performance is
- nearly 2.4 million LLVM instructions per second on a typical laptop
- computer. In this paper we overview the translation process and
- illustrate the translator's capabilities by way of a concrete example,
- including both a functional correctness theorem as well as a
- validation test for that example."
+@InProceedings{Chan94,
+ author = "Chan, K.C. and Diaz, A. and Kaltofen, E.",
+ editor = "R. J. Lopez",
+ title = "A distributed approach to problem solving in Maple",
+ booktitle = "Maple V: Mathematics and its Application",
+ pages = "13--21",
+ publisher = {Birkh\"auser},
+ year = "1994",
+ series = "Proceedings of the Maple Summer Workshop and Symposium (MSWS'94)",
+ address = "Boston",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/CDK94.ps.gz",
+ paper = "Chan94.ps"
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Lamport 02]{Lamp02} Lamport, Leslie
-``Specifying Systems''
-\verb|research.microsoft.com/en-us/um/people/lamport/tla/book-02-08-08.pdf|
-Addison-Wesley ISBN 0-321-14306-X
-%\verb|axiom-developer.org/axiom-website/papers/Lamp02.pdf|
+\begin{chunk}{axiom.bib}
+@InProceedings{Duma02,
+ author = "Dumas, J.-G. and Gautier, T. and Giesbrecht, M. and Giorgi, P.
+ and Hovinen, B. and Kaltofen, E. and Saunders, B.D. and
+ Turner, W.J. and Villard, G.",
+ title = "{LinBox}: A Generic Library for Exact Linear Algebra",
+ booktitle = "Proc. First Internat. Congress Math. Software ICMS 2002,
+ Beijing, China",
+ crossref = "ICMS02",
+ pages = "40--50",
+ year = "2002",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/02/Detal02.pdf",
+ paper = "Duma02.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Martin 97]{Mart97} Martin, U.; Shand, D.
-``Investigating some Embedded Verification Techniques for Computer
- Algebra Systems''
-\verb|www.risc.jku.at/conferences/Theorema/papers/shand.ps.gz|
-%\verb|axiom-developer.org/axiom-website/papers/Mart97.ps|
- abstract = "
- This paper reports some preliminary ideas on a collaborative project
- between St. Andrews University in the UK and NAG Ltd. The project aims
- to use embedded verification techniques to improve the reliability and
- mathematical soundness of computer algebra systems. We give some
- history of attempts to integrate computer algebra systems and
- automated theorem provers and discuss possible advantages and
- disadvantages of these approaches. We also discuss some possible case
- studies."
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt05a,
+ author = "Kaltofen, Erich and Morozov, Dmitriy and Yuhasz, George",
+ title = "Generic Matrix Multiplication and Memory Management in {LinBox}",
+ year = "2005",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'05",
+ crossref = "ISSAC05",
+ pages = "216--223",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/05/KMY05.pdf",
+ paper = "Kalt05a.pdf"
+}
\end{chunk}
\begin{chunk}{axiom.bib}
-@book{Maso86,
- author = "Mason, Ian A.",
- title = "The Semantics of Destructive Lisp",
- publisher = "Center for the Study of Language and Information",
- year = "1986",
- isbn = "0-937073-06-7",
- abstract = "
- Our basic premise is that the ability to construct and modify programs
- will not improve without a new and comprehensive look at the entire
- programming process. Past theoretical research, say, in the logic of
- programs, has tended to focus on methods for reasoning about
- individual programs; little has been done, it seems to us, to develop
- a sound understanding of the process of programming -- the process by
- which programs evolve in concept and in practice. At present, we lack
- the means to describe the techniques of program construction and
- improvement in ways that properly link verification, documentation and
- adaptability."
+@InProceedings{Diaz98,
+ author = "Diaz, A. and Kaltofen, E.",
+ title = "{FoxBox}, a System for Manipulating Symbolic Objects in Black Box
+ Representation",
+ booktitle = "Proc. 1998 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC98",
+ year = "1998",
+ pages = "30--37",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/98/DiKa98.pdf",
+ paper = "Diaz98.pdf"
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Newcombe 13]{Newc13} Newcombe, Chris; Rath, Tim; Zhang, Fan;
-Munteanu, Bogdan; Brooker, Marc; Deardeuff, Michael
-``Use of Formal Methods at Amazon Web Services''
-\verb|research.microsoft.com/en-us/um/people/lamport/tla/|
-\verb|formal-methods-amazon.pdf|
- abstract = "
- In order to find subtle bugs in a system design, it is necessary to
- have a precise description of that design. There are at least two
- major benefits to writing a precise design; the author is forced to
- think more clearly, which helps eliminate ``plausible hand-waving'',
- and tools can be applied to check for errors in the design, even while
- it is being written. In contrast, conventional design documents
- consist of prose, static diagrams, and perhaps pseudo-code in an ad
- hoc untestable language. Such descriptions are far from precise; they
- are often ambiguous, or omit critical aspects such as partial failure
- or the granularity of concurrency (i.e. which constructs are assumed
- to be atomic). At the other end of the spectrum, the final executable
- code is unambiguous, but contains an overwhelming amount of detail. We
- needed to be able to capture the essence of a design in a few hundred
- lines of precise description. As our designs are unavoidably complex,
- we need a highly-expressive language, far above the level of code, but
- with precise semantics. That expressivity must cover real-world
- concurrency and fault-tolerance. And, as we wish to build services
- quickly, we wanted a language that is simple to learn and apply,
- avoiding esoteric concepts. We also very much wanted an existing
- ecosystem of tools. We found what we were looking for in TLA+, a
- formal specification language."
+\begin{chunk}{axiom.bib}
+@InProceedings{Diaz93,
+ author = "Diaz, A. and Kaltofen, E. and Lobo, A. and Valente, T.",
+ editor = "A. Miola",
+ title = "Process scheduling in {DSC} and the large sparse linear
+ systems challenge",
+ booktitle = "Proc. DISCO '93",
+ series = "Lect. Notes Comput. Sci.",
+ pages = "66--80",
+ year = "1993",
+ volume = "722",
+ publisher = "Springer-Verlag",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/93/DHKLV93.pdf",
+ paper = "Diaz93.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Poll 99a]{P99a} Poll, Erik
-``The Type System of Axiom''
-\verb|www.cs.ru.nl/E.Poll/talks/axiom.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/P99a.pdf|
- abstract = "
- This is a slide deck from a talk on the correspondence between
- Axiom/Aldor types and Logic."
+\begin{chunk}{axiom.bib}
+@Article{Diaz95a,
+ author = "Diaz, A. and Hitz, M. and Kaltofen, E. and Lobo, A. and
+ Valtente, T.",
+ title = "Process scheduling in {DSC} and the large sparse linear
+ systems challenge",
+ journal = "Journal of Symbolic Computing",
+ year = "1995",
+ volume = "19",
+ number = "1--3",
+ pages = "269--282",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/DHKLV95.pdf",
+ paper = "Diaz95a.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Poll 99]{PT99} Poll, Erik; Thompson, Simon
-``The Type System of Aldor''
-\verb|www.cs.kent.ac.uk/pubs/1999/874/content.ps|
-%\verb|axiom-developer.org/axiom-website/papers/PT99.pdf|
- abstract = "
- This paper gives a formal description of -- at least a part of --
- the type system of Aldor, the extension language of the Axiom.
- In the process of doing this a critique of the design of the system
- emerges."
+\begin{chunk}{axiom.bib}
+@Article{Free88,
+ author = "Freeman, T.S. and Imirzian, G. and Kaltofen, E. and
+ Yagati, Lakshman",
+ title = "DAGWOOD: A system for manipulating polynomials given by
+ straight-line programs",
+ journal = "ACM Trans. Math. Software",
+ year = "1988",
+ volume = "14",
+ number = "3",
+ pages = "218--240",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/FIKY88.pdf",
+ paper = "Free88.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Poll (a)]{PTxx} Poll, Erik; Thompson, Simon
-``Adding the axioms to Axiom. Toward a system of automated reasoning in
-Aldor''
-\verb|citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.7.1457&rep=rep1&type=ps|
-%\verb|axiom-developer.org/axiom-website/papers/PTxx.pdf|
- abstract = "
- This paper examines the proposal of using the type system of Axiom to
- represent a logic, and thus to use the constructions of Axiom to
- handle the logic and represent proofs and propositions, in the same
- way as is done in theorem provers based on type theory such as Nuprl
- or Coq.
-
- The paper shows an interesting way to decorate Axiom with pre- and
- post-conditions.
+\section{The Seven Dwarfs} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- The Curry-Howard correspondence used is
- \begin{verbatim}
- PROGRAMMING LOGIC
- Type Formula
- Program Proof
- Product/record type (...,...) Conjunction
- Sum/union type \/ Disjunction
- Function type -> Implication
- Dependent function type (x:A) -> B(x) Universal quantifier
- Dependent product type (x:A,B(x)) Existential quantifier
- Empty type Exit Contradictory proposition
- One element type Triv True proposition
- \end{verbatim}"
+\begin{chunk}{axiom.bib}
+@InCollection{Kalt10a,
+ author = "Kaltofen, Erich L.",
+ title = "The ``{Seven} {Dwarfs}'' of Symbolic Computation",
+ booktitle = "Numeric and Symbolic Scientific Computing
+ Progress and Prospects",
+ crossref = "LaPau12",
+ pages = "95--104",
+ year = "2010",
+ keywords = "survey",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/10/Ka10_7dwarfs.pdf",
+ paper = "Kalt10a.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Poll 00]{PT00} Poll, Erik; Thompson, Simon
-``Integrating Computer Algebra and Reasoning through the Type System
-of Aldor''
-%\verb|axiom-developer.org/axiom-website/papers/PT00.pdf|
+\section{Solving Systems of Equations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@inproceedings{Bro86,
+ author = "Bronstein, Manuel",
+ title = "Gsolve: a faster algorithm for solving systems of algebraic
+ equations",
+ booktitle = "Proc of 5th ACM SYMSAC",
+ year = "1986",
+ pages = "247-249",
+ isbn = "0-89791-199-7",
abstract = "
- A number of combinations of reasoning and computer algebra systems
- have been proposed; in this paper we describe another, namely a way to
- incorporate a logic in the computer algebra system Axiom. We examine
- the type system of Aldor -- the Axiom Library Compiler -- and show
- that with some modifications we can use the dependent types of the
- system to model a logic, under the Curry-Howeard isomorphism. We give
- a number of example applications of the logi we construct and explain
- a prototype implementation of a modified type-checking system written
- in Haskell."
+ We apply the elimination property of Gr{\"o}bner bases with respect to
+ pure lexicographic ordering to solve systems of algebraic equations.
+ We suggest reasons for this approach to be faster than the resultant
+ technique, and give examples and timings that show that it is indeed
+ faster and more correct, than MACSYMA's solve."
+}
\end{chunk}
-\subsection{Interval Arithmetic} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{chunk}{ignore}
-\bibitem[Boehm 86]{Boe86} Boehm, Hans-J.; Cartwright, Robert; Riggle, Mark;
-O'Donnell, Michael J.
-``Exact Real Arithmetic: A Case Study in Higher Order Programming''
-\verb|dev.acm.org/pubs/citations/proceedings/lfp/319838/p162-boehm|
-%\verb|axiom-developer.org/axiom-website/papers/Boe86.pdf|
-
-\end{chunk}
+\section{Numerical Algorithms} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
-\bibitem[Briggs 04]{Bri04} Briggs, Keith
-``Exact real arithmetic''
-\verb|keithbriggs.info/documents/xr-kent-talk-pp.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Bri04.pdf|
+{Bro99,
+ author = "Bronstein, Manuel",
+ title = "Fast Deterministic Computation of Determinants of Dense Matrices",
+ url = "http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html",
+ paper = "Bro99.pdf",
+ abstract = "
+ In this paper we consider deterministic computation of the exact
+ determinant of a dense matrix $M$ of integers. We present a new
+ algorithm with worst case complexity
+ \[O(n^4(log n+ log \verb?||M||?)+x^3 log^2 \verb?||M||?)\],
+ where $n$ is the dimension of the matrix
+ and \verb?||M||? is a bound on the entries in $M$, but with
+ average expected complexity
+ \[O(n^4+m^3(log n + log \verb?||M||?)^2)\],
+ assuming some plausible properties about the distribution of $M$.
+ We will also describe a practical version of the algorithm and include
+ timing data to compare this algorithm with existing ones. Our result
+ does not depend on ``fast'' integer or matrix techniques."
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Fateman 94]{Fat94} Fateman, Richard J.; Yan, Tak W.
-``Computation with the Extended Rational Numbers and an Application to
-Interval Arithmetic''
-\verb|www.cs.berkeley.edu/~fateman/papers/extrat.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Fat94.pdf|
+{Kel00,
+ author = "Kelsey, Tom",
+ title = "Exact Numerical Computation via Symbolic Computation",
+ url = "http://tom.host.cs.st-andrews.ac.uk/pub/ccapaper.pdf",
+ paper = "Kel00.pdf",
abstract = "
- Programming languages such as Common Lisp, and virtually every
- computer algebra system (CAS), support exact arbitrary-precision
- integer arithmetic as well as exect rational number computation.
- Several CAS include interval arithmetic directly, but not in the
- extended form indicated here. We explain why changes to the usual
- rational number system to include infinity and ``not-a-number'' may be
- useful, especially to support robust interval computation. We describe
- techniques for implementing these changes."
+ We provide a method for converting any symbolic algebraic expression
+ that can be converted into a floating point number into an exact
+ numeric representation. We use this method to demonstrate a suite of
+ procedures for the representation of, and arithmetic over, exact real
+ numbers in the Maple computer algebra system. Exact reals are
+ represented by potentially infinite lists of binary digits, and
+ interpreted as sums of negative powers of the golden ratio."
+}
\end{chunk}
-\begin{chunk}{axiom.bib}
-@incollection{Lamb06,
- author = "Lambov, Branimir",
- title = "Interval Arithmetic Using SSE-2",
- booktitle = "Lecture Notes in Computer Science",
- publisher = "Springer-Verlag",
- year = "2006",
- isbn = "978-3-540-85520-0",
- pages = "102-113"
+\begin{chunk}{ignore}
+{Yang14,
+ author ="Yang, Xiang and Mittal, Rajat",
+ title = "Acceleration of the Jacobi iterative method by factors exceeding 100
+ using scheduled relation",
+ url =
+"http://engineering.jhu.edu/fsag/wp-content/uploads/sites/23/2013/10/JCP_revised_WebPost.pdf",
+ paper = "Yang14.pdf"
}
\end{chunk}
-\subsection{Numerics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Special Functions} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
-\bibitem[Atkinson 09]{Atk09} Atkinson, Kendall; Han, Welmin; Stewear, David
-``Numerical Solution of Ordinary Differential Equations''
-\verb|homepage.math.uiowa.edu/~atkinson/papers/NAODE_Book.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Atk09.pdf|
+{Corl0,
+ author = "Corless, Robert M. and Jeffrey, David J. and Watt, Stephen M.
+ and Bradford, Russell and Davenport, James H.",
+ title = "Reasoning about the elementary functions of complex analysis",
+ url = "http://www.csd.uwo.ca/~watt/pub/reprints/2002-amai-reasoning.pdf",
+ paper = "Corl05.pdf",
abstract = "
- This book is an expanded version of supplementary notes that we used
- for a course on ordinary differential equations for upper-division
- undergraduate students and beginning graduate students in mathematics,
- engineering, and sciences. The book introduces the numerical analysis
- of differential equations, describing the mathematical background for
- understanding numerical methods and giving information on what to
- expect when using them. As a reason for studying numerical methods as
- a part of a more general course on differential equations, many of the
- basic ideas of the numerical analysis of differential equations are
- tied closely to theoretical behavior associated with the problem being
- solved. For example, the criteria for the stability of a numerical
- method is closely connected to the stability of the differential
- equation problem being solved."
-
-\end{chunk}
-
-\begin{chunk}{ignore}
-\bibitem[Crank 96]{Cran96} Crank, J.; Nicolson, P.
-``A practical method for numerical evaluations of solutions of partial
- differential equations of heat-conduction type''
-Advances in Computational Mathematics Vol 6 pp207-226 (1996)
-\verb|www.acms.arizona.edu/FemtoTheory/MK_personal/opti547/literature/|
-\verb|CNMethod-original.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Cran96.pdf|
+ There are many problems with the simplification of elementary
+ functions, particularly over the complex plane. Systems tend to make
+ ``howlers'' or not to simplify enough. In this paper we outline the
+ ``unwinding number'' approach to such problems, and show how it can be
+ used to prevent errors and to systematise such simplification, even
+ though we have not yet reduced the simplification process to a
+ complete algorithm. The unsolved problems are probably more amenable
+ to the techniques of artificial intelligence and theorem proving than
+ the original problem of complex-variable analysis."
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lef\'evre 06]{Lef06} Lef\'evre, Vincent; Stehl\'e, Damien;
-Zimmermann, Paul
-``Worst Cases for the Exponential Function
-in the IEEE-754r decimal64 Format''
-in Lecture Notes in Computer Science, Springer ISBN 978-3-540-85520-0
-(2006) pp114-125
+{Ng68,
+ author = "Ng, Edward W. and Geller, Murray",
+ title = "A Table of Integrals of the Error functions",
+ url = "http://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn1p1_A1b.pdf",
+ paper = "Ng68.pdf",
abstract = "
- We searched for the worst cases for correct rounding of the
- exponential function in the IEEE 754r decimal64 format, and computed
- all the bad cases whose distance from a breakpoint (for all rounding
- modes) is less than $10^{-15}$ ulp, and we give the worst ones. In
- particular, the worst case for
- $\vert{}x\vert{} \ge 3 x 10^{-11}$ is
- \[
- exp(9.407822313572878x10^{-2} =
- 1.09864568206633850000000000000000278\ldots
- \]
- This work can be extended to other elementary functions in the decimal64
- format and allows the design of reasonably fast routines that will
- evaluate these functions with correct rounding, at least in some
- situations."
-
-\end{chunk}
-
-\begin{chunk}{axiom.bib}
-@book{Hamm62,
- author = "Hamming R W.",
- title = "Numerical Methods for Scientists and Engineers",
- publisher = "Dover",
- year = "1973",
- isbn = "0-486-65241-6"
+ This is a compendium of indefinite and definite integrals of products
+ of the Error functions with elementary and transcendental functions."
}
\end{chunk}
-\subsection{Advanced Documentation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Exponential Integral $E_1(x)$} %%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
-\bibitem [Bostock 14]{Bos14} Bostock, Mike
-``Visualizing Algorithms''
-\verb|bost.ocks.org/mike/algorithms|
+{Gell69,
+ author = "Geller, Murray and Ng, Edward W.",
+ title = "A Table of Integrals of the Exponential Integral",
+ url = "http://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn3p191_A1b.pdf",
+ paper = "Gell69.pdf",
abstract = "
- This website hosts various ways of visualizing algorithms. The hope is
- that these kind of techniques can be applied to Axiom."
+ This is a compendium of indefinite and definite integrals of products
+ of the Exponential Integral with elementary or transcendental functions."
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Leeuwen]{Leexx} van Leeuwen, Andr\'e M.A.
-``Representation of mathematical object in interactive books''
-%\verb|axiom-developer.org/axiom-website/papers/Leexx.pdf|
- abstract = "
- We present a model for the representation of mathematical objects in
- structured electronic documents, in a way that allows for interaction
- with applications such as computer algebra systems and proof checkers.
- Using a representation that reflects only the intrinsic information of
- an object, and storing application-dependent information in so-called
- {\sl application descriptions}, it is shown how the translation from
- the internal to an external representation and {\sl vice versa} can be
- achieved. Hereby a formalisation of the concept of {\sl context} is
- introduced. The proposed scheme allows for a high degree of
- application integration, e.g., parallel evaluation of subexpressions
- (by different computer algebra systems), or a proof checker using a
- computer algebra system to verify an equation involving a symbolic
- computation."
-
-\end{chunk}
-
-\begin{chunk}{ignore}
-\bibitem[Soiffer 91]{Soif91} Soiffer, Neil Morrell
-``The Design of a User Interface for Computer Algebra Systems''
-\verb|www.eecs.berkeley.edu/Pubs/TechRpts/1991/CSD-91-626.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Soif91.pdf|
+\begin{chunk}{axiom.bib}
+@techreport{Segl98,
+ author = "Segletes, S.B.",
+ title = "A compact analytical fit to the exponential integral $E_1(x)$",
+ year = "1998",
+ institution = "U.S. Army Ballistic Research Laboratory,
+ Aberdeen Proving Ground, MD",
+ type = "Technical Report",
+ number = "ARL-TR-1758",
+ paper = "Segl98.pdf",
abstract = "
- This thesis discusses the design and implementation of natural user
- interfaces for Computer Algebra Systems. Such an interface must not
- only display expressions generated by the Computer Algebra System in
- standard mathematical notation, but must also allow easy manipulation
- and entry of expressions in that notation. The user interface should
- also assist in understanding of large expressions that are generated
- by Computer Algebra Systems and should be able to accommodate new
- notational forms."
+ A four-parameter fit is developed for the class of integrals known as
+ the exponential integral (real branch). Unlike other fits that are
+ piecewise in nature, the current fit to the exponential integral is
+ valid over the complete domain of the function (compact) and is
+ everywhere accurate to within $\pm 0.0052\%$ when evaluating the first
+ exponential integral, $E_1$. To achieve this result, a methodology
+ that makes use of analytically known limiting behaviors at either
+ extreme of the domain is employed. Because the fit accurately captures
+ limiting behaviors of the $E_1$ function, more accuracy is retained
+ when the fit is used as part of the scheme to evaluate higher-order
+ exponential integrals, $E_n$, as compared with the use of brute-force
+ fits to $E_1$, which fail to accurately model limiting
+ behaviors. Furthermore, because the fit is compact, no special
+ accommodations are required (as in the case of spliced piecewise fits)
+ to smooth the value, slope, and higher derivatives in the transition
+ region between two piecewise domains. The general methodology employed
+ to develop this fit is outlined, since it may be used for other
+ problems as well."
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Victor 11]{Vict11} Victor, Bret
-``Up and Down the Ladder of Abstraction''
-\verb|worrydream.com/LadderOfAbstraction|
+\begin{chunk}{axiom.bib}
+@techreport{Se09,
+ author = "Segletes, S.B.",
+ title = "Improved fits for $E_1(x)$ {\sl vis-\'a-vis} those presented
+ in ARL-TR-1758",
+ type = "Technical Report",
+ number = "ARL-TR-1758",
+ institution ="U.S. Army Ballistic Research Laboratory,
+ Aberdeen Proving Ground, MD",
+ year = "1998",
+ month = "September",
+ paper = "Se09.pdf",
abstract = "
- This interactive essay presents the ladder of abstraction, a technique for
- thinking explicitly about these levels, so a designer can move among
- them consciously and confidently. "
+ This is a writeup detailing the more accurate fits to $E_1(x)$,
+ relative to those presented in ARL-TR-1758. My actual fits are to
+ \[F1 =[x\ exp(x) E_1(x)]\] which spans a functional range from 0 to 1.
+ The best accuracy I have been yet able to achieve, defined by limiting
+ the value of \[[(F1)_{fit} - F1]/F1\] over the domain, is
+ approximately 3.1E-07 with a 12-parameter fit, which unfortunately
+ isn't quite to 32-bit floating-point accuracy. Nonetheless, the fit
+ is not a piecewise fit, but rather a single continuous function over
+ the domain of nonnegative x, which avoids some of the problems
+ associated with piecewise domain splicing."
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Victor 12]{Vict12} Victor, Bret
-``Inventing on Principle''
-\verb|www.youtube.com/watch?v=PUv66718DII|
- abstract = "
- This video raises the level of discussion about human-computer
- interaction from a technical question to a question of effectively
- capturing ideas. In particular, this applies well to Axiom's focus on
- literate programming."
+\section{Polynomial GCD} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\end{chunk}
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt99a,
+ author = "Kaltofen, E. and Monagan, M.",
+ title = "On the Genericity of the Modular Polynomial {GCD} Algorithm",
+ booktitle = "Proc. 1999 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC99",
+ year = "1999",
+ pages = "59--66",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/99/KaMo99.pdf",
+ paper = "Kalt99a.pdf"
+}
-\subsection{Differential Equations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Abramov 95]{Abra95} Abramov, Sergei A.; Bronstein, Manuel;
-Petkovsek, Marko
-``On Polynomial Solutions of Linear Operator Equations''
-\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html|
-%\verb|axiom-developer.org/axiom-website/papers/Abra95.pdf|
+\bibitem[Knuth 71]{ST-PGCD-Knu71} Knuth, Donald
+``The Art of Computer Programming''
+2nd edition Vol. 2 (Seminumerical Algorithms) 1st edition, 2nd printing,
+Addison-Wesley 1971, section 4.6 pp399-505
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Abramov 01]{Abra01} Abramov, Sergei; Bronstein, Manuel
-``On Solutions of Linear Functional Systems''
-\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html|
-%\verb|axiom-developer.org/axiom-website/papers/Abra01.pdf|
+\bibitem[Ma 90]{ST-PGCD-Ma90} Ma, Keju; Gathen, Joachim von zur
+``Analysis of Euclidean Algorithms for Polynomials over Finite Fields''
+J. Symbolic Computation (1990) Vol 9 pp429-455\hfill{}
+\verb|www.researchgate.net/publication/220161718_Analysis_of_Euclidean_|
+\verb|Algorithms_for_Polynomials_over_Finite_Fields/file/|
+\verb|60b7d52b326a1058e4.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/ST-PGCD-Ma90.pdf|
abstract = "
- We describe a new direct algorithm for transforming a linear system of
- recurrences into an equivalent one with nonsingular leading or
- trailing matrix. Our algorithm, which is an improvement to the EG
- elimination method, uses only elementary linear algebra operations
- (ranks, kernels, and determinants) to produce an equation satisfied by
- the degress of the solutions with finite support. As a consequence, we
- can boudn and compute the polynomial and rational solutions of very
- general linear functional systems such as systems of differential or
- ($q$)-difference equations."
+ This paper analyzes the Euclidean algorithm and some variants of it
+ for computing the greatest common divisor of two univariate polynomials
+ over a finite field. The minimum, maximum, and average number of
+ arithmetic operations both on polynomials and in the ground field
+ are derived."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Bronstein 96b]{Bro96b} Bronstein, Manuel
-``On the Factorization of Linear Ordinary Differential Operators''
-Mathematics and Computers in Simulation 42 pp 387-389 (1996)
-%\verb|axiom-developer.org/axiom-website/papers/Bro96b.pdf|
+\bibitem[Naylor 00a]{N00} Naylor, Bill
+``Polynomial GCD Using Straight Line Program Representation''
+PhD. Thesis, University of Bath, 2000
+\verb|www.sci.csd.uwo.ca/~bill/thesis.ps|
+%\verb|axiom-developer.org/axiom-website/papers/N00.pdf|
abstract = "
- After reviewing the arithmetic of linear ordinary differential
- operators, we describe the current status of the factorisation
- algorithm, specially with respect to factoring over non-algebraically
- closed constant fields. We also describe recent results from Singer
- and Ulmer that reduce determining the differential Galois group of an
- operator to factoring."
+ This thesis is concerned with calculating polynomial greatest common
+ divisors using straight line program representation.
+
+ In the Introduction chapter, we introduce the problem and describe
+ some of the traditional representations for polynomials, we then talk
+ about some of the general subjects central to the thesis, terminating
+ with a synopsis of the category theory which is central to the Axiom
+ computer algebra system used during this research.
+
+ The second chapter is devoted to describing category theory. We follow
+ with a chapter detailing the important sections of computer code
+ written in order to investigate the straight line program subject.
+ The following chapter on evalution strategies and algorithms which are
+ dependant on these follows, the major algorith which is dependant on
+ evaluation and which is central to our theis being that of equality
+ checking. This is indeed central to many mathematical problems.
+ Interpolation, that is the determination of coefficients of a
+ polynomial is the subject of the next chapter. This is very important
+ for many straight line program algorithms, as their non-canonical
+ structure implies that it is relatively difficult to determine
+ coefficients, these being the basic objects that many algorithms work
+ on. We talk about three separate interpolation techniques and compare
+ their advantages and disadvantages. The final two chapters describe
+ some of the results we have obtained from this research and finally
+ conclusions we have drawn as to the viability of the straight line
+ program approach and possible extensions.
+
+ Finally we terminate with a number of appendices discussing side
+ subjects encountered during the thesis."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Bronstein 96a]{Bro96a} Bronstein, Manuel; Petkovsek, Marko
-``An introduction to pseudo-linear algebra''
-Theoretical Computer Science V157 pp3-33 (1966)
-\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html|
-%\verb|axiom-developer.org/axiom-website/papers/Bro96a.pdf|
+\bibitem[Shoup 93]{ST-PGCD-Sh93} Shoup, Victor
+``Factoring Polynomials over Finite Fields: Asymptotic Complexity vs
+Reality*''
+Proc. IMACS Symposium, Lille, France, (1993)
+\verb|www.shoup.net/papers/lille.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/ST-PGCD-Sh93.pdf|
abstract = "
- Pseudo-linear algebra is the study of common properties of linear
- differential and difference operators. We introduce in this paper its
- basic objects (pseudo-derivations, skew polynomials, and pseudo-linear
- operators) and describe several recent algorithms on them, which, when
- applied in the differential and difference cases, yield algorithms for
- uncoupling and solving systems of linear differential and difference
- equations in closed form."
+ This paper compares the algorithms by Berlekamp, Cantor and
+ Zassenhaus, and Gathen and Shoup to conclude that (a) if large
+ polynomials are factored the FFT should be used for polynomial
+ multiplication and division, (b) Gathen and Shoup should be used if
+ the number of irreducible factors of $f$ is small. (c) if nothing is
+ know about the degrees of the factors then Berlekamp's algorithm
+ should be used."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Bronstein xb]{Broxb} Bronstein, Manuel
-``Computer Algebra Algorithms for Linear Ordinary Differential and
-Difference equations''
-\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/ecm3.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Broxb.pdf|
+\bibitem[Gathen 01]{ST-PGCD-Ga01} Gathen, Joachim von zur; Panario, Daniel
+``Factoring Polynomials Over Finite Fields: A Survey''
+J. Symbolic Computation (2001) Vol 31, pp3-17\hfill{}
+\verb|people.csail.mit.edu/dmoshdov/courses/codes/poly-factorization.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/ST-PGCD-Ga01.pdf|
+ keywords = "survey",
abstract = "
- Galois theory has now produced algorithms for solving linear ordinary
- differential and difference equations in closed form. In addition,
- recent algorithmic advances have made those algorithms effective and
- implementable in computer algebra systems. After introducing the
- relevant parts of the theory, we describe the latest algorithms for
- solving such equations."
+ This survey reviews several algorithms for the factorization of
+ univariate polynomials over finite fields. We emphasize the main ideas
+ of the methods and provide and up-to-date bibliography of the problem.
+ This paper gives algorithms for {\sl squarefree factorization},
+ {\sl distinct-degree factorization}, and {\sl equal-degree factorization}.
+ The first and second algorithms are deterministic, the third is
+ probabilistic."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Bronstein 94]{Bro94} Bronstein, Manuel
-``An improved algorithm for factoring linear ordinary differential
-operators''
-\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html|
+\bibitem[van Hoeij]{Hoeij04} Hoeij, Mark van; Monagen, Michael
+``Algorithms for Polynomial GCD Computation over Algebraic Function Fields''
+\verb|www.cecm.sfu.ca/personal/mmonagan/papers/AFGCD.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Hoeij04.pdf|
abstract = "
- We describe an efficient algorithm for computing the associated
- equations appearing in the Beke-Schlesinger factorisation method for
- linear ordinary differential operators. This algorithm, which is based
- on elementary operations with sets of integers, can be easily
- implemented for operators of any order, produces several possible
- associated equations, of which only the simplest can be selected for
- solving, and often avoids the degenerate case, where the order of the
- associated equation is less than in the generic case. We conclude with
- some fast heuristics that can produce some factorizations while using
- only linear computations."
+ Let $L$ be an algebraic function field in $k \ge 0$ parameters
+ $t_1,\ldots,t)k$. Let $f_1$, $f_2$ be non-zero polynomials in
+ $L[x]$. We give two algorithms for computing their gcd. The first, a
+ modular GCD algorithm, is an extension of the modular GCD algorithm
+ for Brown for {\bf Z}$[x_1,\ldots,x_n]$ and Encarnacion for {\bf
+ Q}$(\alpha[x])$ to function fields. The second, a fraction-free
+ algorithm, is a modification of the Moreno Maza and Rioboo algorithm
+ for computing gcds over triangular sets. The modification reduces
+ coefficient grownth in $L$ to be linear. We give an empirical
+ comparison of the two algorithms using implementations in Maple."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Bronstein 90]{Bro90} Bronstein, Manuel
-``On Solutions of Linear Ordinary Differential Equations in their
-Coefficient Field''
-\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html|
-%\verb|axiom-developer.org/axiom-website/papers/Bro90.pdf|
+\bibitem[Wang 78]{Wang78} Wang, Paul S.
+``An Improved Multivariate Polynomial Factoring Algorithm''
+Mathematics of Computation, Vol 32, No 144 Oct 1978, pp1215-1231
+\verb|www.ams.org/journals/mcom/1978-32-144/S0025-5718-1978-0568284-3/|
+\verb|S0025-5718-1978-0568284-3.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Wang78.pdf|
abstract = "
- We describe a rational algorithm for finding the denominator of any
- solution of a linear ordinary differential equation in its coefficient
- field. As a consequence, there is now a rational algorithm for finding
- all such solutions when the coefficients can be built up from the
- rational functions by finitely many algebraic and primitive
- adjunctions. This also eliminates one of the computational bottlenecks
- in algorithms that either factor or search for Liouvillian solutions
- of such equations with Liouvillian coefficients."
+ A new algorithm for factoring multivariate polynomials over the
+ integers based on an algorithm by Wang and Rothschild is described.
+ The new algorithm has improved strategies for dealing with the known
+ problems of the original algorithm, namely, the leading coefficient
+ problem, the bad-zero problem and the occurence of extraneous factors.
+ It has an algorithm for correctly predetermining leading coefficients
+ of the factors. A new and efficient p-adic algorith named EEZ is
+ described. Basically it is a linearly convergent variable-by-variable
+ parallel construction. The improved algorithm is generally faster and
+ requires less store than the original algorithm. Machine examples with
+ comparative timing are included."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Bronstein 96]{Bro96} Bronstein, Manuel
-``$\sum^{IT}$ -- A strongly-typed embeddable computer algebra library''
-\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html|
-%\verb|axiom-developer.org/axiom-website/papers/Bro96.pdf|
- abstract = "
- We describe the new computer algebra library $\sum^{IT}$ and its
- underlying design. The development of $\sum^{IT}$ is motivated by the
- need to provide highly efficient implementations of key algorithms for
- linear ordinary differential and ($q$)-difference equations to
- scientific programmers and to computer algebra users, regardless of
- the programming language or interactive system they use. As such,
- $\sum^{IT}$ is not a computer algebra system per se, but a library (or
- substrate) which is designed to be ``plugged'' with minimal efforts
- into different types of client applications."
+\bibitem[Wiki 4]{Wiki4}.
+``Polynomial greatest common divisor''
+\verb|en.wikipedia.org/wiki/Polynomial_greatest_common_divisor|
\end{chunk}
+\section{Category Theory} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Bronstein 99a]{Bro99a} Bronstein, Manuel
-``Solving linear ordinary differential equations over
-$C(x,e^{\int{f(x)dx}})$
-\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html|
-%\verb|axiom-developer.org/axiom-website/papers/Bro99a.pdf|
+\bibitem[Baez 09]{Baez09} Baez, John C.; Stay, Mike
+``Physics, Topology, Logic and Computation: A Rosetta Stone''
+\verb|arxiv.org/pdf/0903.0340v3.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Baez09.pdf|
abstract = "
- We describe a new algorithm for computing the solutions in
- \[F=C(x,e^{\int{f(x)dx}})\] of linear ordinary differential equations
- with coefficients in $F$. Compared to the general algorithm, our
- algorithm avoids the computation of exponential solutions of equations
- with coefficients in $C(x)$, as well as the solving of linear
- differential systems over $C(x)$. Our method is effective and has been
- implemented."
+ In physics, Feynman diagrams are used to reason about quantum
+ processes. In the 1980s, it became clear that underlying these
+ diagrams is a powerful analogy between quantum physics and
+ topology. Namely, a linear operator behaves very much like a
+ ``cobordism'': a manifold representing spacetime, going between two
+ manifolds representing space. But this was just the beginning: simiar
+ diagrams can be used to reason about logic, where they represent
+ proofs, and computation, where they represent programs. With the rise
+ of interest in quantum cryptography and quantum computation, it became
+ clear that there is an extensive network of analogies between physics,
+ topology, logic and computation. In this expository paper, we make
+ some of these analogies precise using the concept of ``closed
+ symmetric monodial category''. We assume no prior knowledge of
+ category theory, proof theory or computer science."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Bronstein 00]{Bro00} Bronstein, Manuel
-``On Solutions of Linear Ordinary Differential Equations in their
- Coefficient Field''
-\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html|
-%\verb|axiom-developer.org/axiom-website/papers/Bro00.pdf|
+\bibitem[Meijer 91]{Meij91} Meijer, Erik; Fokkinga, Maarten; Paterson, Ross
+``Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire''
+\verb|eprints.eemcs.utwente.nl/7281/01/db-utwente-40501F46.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Meij91.pdf|
abstract = "
- We extend the notion of monomial extensions of differential fields,
- i.e. simple transcendental extensions in which the polynomials are
- closed under differentiation, to difference fields. The structure of
- such extensions provides an algebraic framework for solving
- generalized linear difference equations with coefficients in such
- fields. We then describe algorithms for finding the denominator of any
- solution of those equations in an important subclass of monomial
- extensions that includes transcendental indefinite sums and
- products. This reduces the general problem of finding the solutions of
- such equations in their coefficient fields to bounding their
- degrees. In the base case, this yields in particular a new algorithm
- for computing the rational solutions of $q$-difference equations with
- polynomial coefficients."
+ We develop a calculus for lazy functional programming based on
+ recursion operators associated with data type definitions. For these
+ operators we derive various algebraic laws that are useful in deriving
+ and manipulating programs. We shall show that all example functions in
+ Bird and Wadler's ``Introduction to Functional Programming'' can be
+ expressed using these operators."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Bronstein 02]{Bro02} Bronstein, Manuel; Lafaille, S\'ebastien
-``Solutions of linear ordinary differential equations in terms of
-special functions''
-\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac2002.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Bro02.pdf|
+\bibitem[Youssef 04]{You04} Youssef, Saul
+``Prospects for Category Theory in Aldor''
+October 2004
+%\verb|axiom-developer.org/axiom-website/papers/You04.pdf|
abstract = "
- We describe a new algorithm for computing special function solutions
- of the form $y(x) = m(x)F(\eta(x))$ of second order linear ordinary
- differential equations, where $m(x)$ is an arbitrary Liouvillian
- function, $\eta(x)$ is an arbitrary rational function, and $F$
- satisfies a given second order linear ordinary differential
- equations. Our algorithm, which is base on finding an appropriate
- point transformation between the equation defining $F$ and the one to
- solve, is able to find all rational transformations for a large class
- of functions $F$, in particular (but not only) the $_0F_1$ and $_1F_1$
- special functions of mathematical physics, such as Airy, Bessel,
- Kummer and Whittaker functions. It is also able to identify the values
- of the parameters entering those special functions, and can be
- generalized to equations of higher order."
+ Ways of encorporating category theory constructions and results into
+ the Aldor language are discussed. The main features of Aldor which
+ make this possible are identified, examples of categorical
+ constructions are provided and a suggestion is made for a foundation
+ for rigorous results."
\end{chunk}
+\section{Proving Axiom Correct} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Bronstein 03]{Bro03} Bronstein, Manuel; Trager, Barry M.
-``A Reduction for Regular Differential Systems''
-\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mega2003.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Bro03.pdf|
+\bibitem[Adams 99]{Adam99} Adams, A.A.; Gottlieben, H.; Linton, S.A.;
+Martin, U.
+``Automated theorem proving in support of computer algebra:''
+`` symbolic definite integration as a case study''
+%\verb|axiom-developer.org/axiom-website/papers/Adam99.pdf|
abstract = "
- We propose a definition of regularity of a linear differential system
- with coefficients in a monomial extension of a differential field, as
- well as a global and truly rational (i.e. factorisation-free)
- iteration that transforms a system with regular finite singularites
- into an equivalent one with simple finite poles. We then apply our
- iteration to systems satisfied by bases of algebraic function fields,
- obtaining algorithms for computing the number of irreducible
- components and the genus of algebraic curves."
+ We assess the current state of research in the application of computer
+ aided formal reasoning to computer algebra, and argue that embedded
+ verification support allows users to enjoy its benefits without
+ wrestling with technicalities. We illustrate this claim by considering
+ symbolic definite integration, and present a verifiable symbolic
+ definite integral table look up: a system which matches a query
+ comprising a definite integral with parameters and side conditions,
+ against an entry in a verifiable table and uses a call to a library of
+ lemmas about the reals in the theorem prover PVS to aid in the
+ transformation of the table entry into an answer. We present the full
+ model of such a system as well as a description of our prototype
+ implementation showing the efficacy of such a system: for example, the
+ prototype is able to obtain correct answers in cases where computer
+ algebra systems [CAS] do not. We extend upon Fateman's web-based table
+ by including parametric limits of integration and queries with side
+ conditions."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Bronstein 03a]{Bro03a} Bronstein, Manuel; Sol\'e, Patrick
-``Linear recurrences with polynomial coefficients''
-\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html|
-%\verb|axiom-developer.org/axiom-website/papers/Bro03a.pdf|
+\bibitem[Adams 01]{Adam01} Adams, Andrew; Dunstan, Martin; Gottliebsen, Hanne;
+Kelsey, Tom; Martin, Ursula; Owre, Sam
+``Computer Algebra Meets Automated Theorem Proving: Integrating Maple and PVS''
+\verb|www.csl.sri.com/~owre/papers/tphols01/tphols01.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Adam01.pdf|
abstract = "
- We relate sequences generated by recurrences with polynomial
- coefficients to interleaving and multiplexing of sequences generated
- by recurrences with constant coefficients. In the special case of
- finite fields, we show that such sequences are periodic and provide
- linear complexity estimates for all three constructions."
+ We describe an interface between version 6 of the Maple computer
+ algebra system with the PVS automated theorem prover. The interface is
+ designed to allow Maple users access to the robust and checkable proof
+ environment of PVS. We also extend this environment by the provision
+ of a library of proof strategies for use in real analysis. We
+ demonstrate examples using the interface and the real analysis
+ library. These examples provide proofs which are both illustrative and
+ applicable to genuine symbolic computation problems."
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Bronstein 05]{Bro05} Bronstein, Manuel; Li, Ziming; Wu, Min
-``Picard-Vessiot Extensions for Linear Functional Systems''
-\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac2005.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Bro05.pdf|
+\begin{chunk}{axiom.bib}
+@article{Mahb06,
+ author = "Mahboubi, Assia",
+ title = "Proving Formally the Implementation of an Efficient gcd
+ Algorithm for Polynomials",
+ journal = "Lecture Notes in Computer Science",
+ volume = "4130",
+ year = "2006",
+ pages = "438-452",
+ paper = "Mahb06.pdf",
abstract = "
- Picard-Vessiot extensions for ordinary differential and difference
- equations are well known and are at the core of the associated Galois
- theories. In this paper, we construct fundamental matrices and
- Picard-Vessiot extensions for systems of linear partial functional
- equations having finite linear dimension. We then use those extensions
- to show that all the solutions of a factor of such a system can be
- completed to solutions of the original system."
+ We describe here a formal proof in the Coq system of the structure
+ theorem for subresultants which allows to prove formally the
+ correctness of our implementation of the subresultants algorithm.
+ Up to our knowledge it is the first mechanized proof of this result."
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Davenport 86]{Dav86} Davenport, J.H.
-``The Risch Differential Equation Problem''
-SIAM J. COMPUT. Vol 15, No. 4 1986
-%\verb|axiom-developer.org/axiom-website/papers/Dav86.pdf|
+\bibitem[Ballarin 99]{Ball99} Ballarin, Clemens; Paulson, Lawrence C.
+``A Pragmatic Approach to Extending Provers by Computer Algebra --
+ with Applications to Coding Theory''
+\verb|www.cl.cam.ac.uk/~lp15/papers/Isabelle/coding.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Ball99.pdf|
abstract = "
- We propose a new algorithm, similar to Hermite's method for the
- integration of rational functions, for the resolution of Risch
- differential equations in closed form, or proving that they have no
- resolution. By requiring more of the presentation of our differential
- fields (in particular that the exponentials be weakly normalized), we
- can avoid the introduction of arbitrary constants which have to be
- solved for later.
+ The use of computer algebra is usually considered beneficial for
+ mechanised reasoning in mathematical domains. We present a case study,
+ in the application domain of coding theory, that supports this claim:
+ the mechanised proofs depend on non-trivial algorithms from computer
+ algebra and increase the reasoning power of the theorem prover.
- We also define a class of fields known as exponentially reduced, and
- show that solutions of Risch differential equations which arise from
- integrating in these fields satisfy the ``natural'' degree constraints
- in their main variables, and we conjecture (after Risch and Norman)
- that this is true in all variables."
+ The unsoundness of computer algebra systems is a major problem in
+ interfacing them to theorem provers. Our approach to obtaining a sound
+ overall system is not blanket distrust but based on the distinction
+ between algorithms we call sound and {\sl ad hoc} respectively. This
+ distinction is blurred in most computer algebra systems. Our
+ experimental interface therefore uses a computer algebra library. It
+ is based on formal specifications for the algorithms, and links the
+ computer algebra library Sumit to the prover Isabelle.
+
+ We give details of the interface, the use of the computer algebra
+ system on the tactic-level of Isabelle and its integration into proof
+ procedures."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Singer 9]{Sing91.pdf} singer, Michael F.
-``Liouvillian Solutions of Linear Differential Equations with Liouvillian
- Coefficients''
-J. Symbolic Computation V11 No 3 pp251-273 (1991)
-\verb|www.sciencedirect.com/science/article/pii/S074771710880048X|
-%\verb|axiom-developer.org/axiom-website/papers/Sing91.pdf|
+\bibitem[Bertot 04]{Bert04} Bertot, Yves; Cast\'eran, Pierre
+``Interactive Theorem Proving and Program Development''
+Springer ISBN 3-540-20854-2
abstract = "
- Let $L(y)=b$ be a linear differential equation with coefficients in a
- differential field $K$. We discuss the problem of deciding if such an
- equation has a non-zero solution in $K$ and give a decision procedure
- in case $K$ is an elementary extension of the field of rational
- functions or is an algebraic extension of a transcendental liouvillian
- extension of the field of rational functions We show how one can use
- this result to give a procedure to find a basis for the space of
- solutions, liouvillian over $K$, of $L(y)=0$ where $K$ is such a field
- and $L(y)$ has coefficients in $K$."
-
-\end{chunk}
+ Coq is an interactive proof assistant for the development of
+ mathematical theories and formally certified software. It is based on
+ a theory called the calculus of inductive constructions, a variant of
+ type theory.
-\begin{chunk}{ignore}
-\bibitem[Von Mohrenschildt 94]{Mohr94} Von Mohrenschildt, Martin
-``Symbolic Solutions of Discontinuous Differential Equations''
-\verb|e-collection.library.ethz.ch/eserv/eth:39463/eth-39463-01.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Mohr94.pdf|
+ This book provides a pragmatic introduction to the development of
+ proofs and certified programs using Coq. With its large collection of
+ examples and exercies it is an invaluable tool for researchers,
+ students, and engineers interested in formal methods and the
+ development of zero-fault software."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Von Mohrenschildt 98]{Mohr98} Von Mohrenschildt, Martin
-``A Normal Form for Function Rings of Piecewise Functions''
-J. Symbolic Computation (1998) Vol 26 pp607-619
-\verb|www.cas.mcmaster.ca/~mohrens/JSC.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Mohr98.pdf|
+\bibitem[Boulme 00]{BHR00} Boulm\'e, S.; Hardin, T.; Rioboo, R.
+``Polymorphic Data Types, Objects, Modules and Functors,: is it too much?''
+%\verb|axiom-developer.org/axiom-website/papers/BHR00.pdf|
abstract = "
- Computer algebra systems often have to deal with piecewise continuous
- functions. These are, for example, the absolute value function,
- signum, piecewise defined functions but also functions that are the
- supremum or infimum of two functions. We present a new algebraic
- approach to these types of problems. This paper presents a normal form
- for a function ring containing piecewise polynomial functions of an
- expression. The main result is that this normal form can be used to
- decide extensional equality of two piecewise functions. Also we define
- supremum and infimum for piecewise functions; in fact, we show that
- the function ring forms a lattice. Additionally, a method to solve
- equalities and inequalities in this function ring is
- presented. Finally, we give a ``user interface'' to the algebraic
- representation of the piecewise functions."
+ Abstraction is a powerful tool for developers and it is offered by
+ numerous features such as polymorphism, classes, modules, and
+ functors, $\ldots$ A working programmer may be confused by this
+ abundance. We develop a computer algebra library which is being
+ certificed. Reporting this experience made with a language (Ocaml)
+ offering all these features, we argue that the are all needed
+ together. We compare several ways of using classes to represent
+ algebraic concepts, trying to follow as close as possible mathematical
+ specification. Thenwe show how to combine classes and modules to
+ produce code having very strong typing properties. Currently, this
+ library is made of one hundred units of functional code and behaves
+ faster than analogous ones such as Axiom."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Weber 06]{Webe06} Weber, Andreas
-``Quantifier Elimination on Real Closed Fields and Differential Equations''
-\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber2006a.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Webe06.pdf|
- keywords = "survey",
+\bibitem[Boulme 01]{BHHMR01}
+Boulm\'e, S.; Hardin, T.; Hirschkoff, D.; M\'enissier-Morain, V.; Rioboo, R.
+``On the way to certify Computer Algebra Systems''
+Calculemus-2001
+%\verb|axiom-developer.org/axiom-website/papers/BHHMR01.pdf|
abstract = "
- This paper surveys some recent applications of quantifier elimination
- on real closed fields in the context of differential
- equations. Although polynomial vector fields give rise to solutions
- involving the exponential and other transcendental functions in
- general, many questions can be settled within the real closed field
- without referring to the real exponential field. The technique of
- quantifier elimination on real closed fields is not only of
- theoretical interest, but due to recent advances on the algorithmic
- side including algorithms for the simplification of quantifier-free
- formulae the method has gained practical applications, e.g. in the
- context of computing threshold conditions in epidemic modeling."
+ The FOC project aims at supporting, within a coherent software system,
+ the entire process of mathematical computation, starting with proved
+ theories, ending with certified implementations of algorithms. In this
+ paper, we explain our design requirements for the implementation,
+ using polynomials as a running example. Indeed, proving correctness of
+ implementations depends heavily on the way this design allows
+ mathematical properties to be truly handled at the programming level.
+
+ The FOC project, started at the fall of 1997, is aimed to build a
+ programming environment for the development of certified symbolic
+ computation. The working languages are Coq and Ocaml. In this paper,
+ we present first the motivations of the project. We then explain why
+ and how our concern for proving properties of programs has led us to
+ certain implementation choices in Ocaml. This way, the sources express
+ exactly the mathematical dependencies between different structures.
+ This may ease the achievement of proofs."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Ulmer 03]{Ulm03} Ulmer, Felix
-``Liouvillian solutions of third order differential equations''
-J. Symbolic COmputations 36 pp 855-889 (2003)
-\verb|www.sciencedirect.com/science/article/pii/S0747717103000658|
-%\verb|axiom-developer.org/axiom-website/papers/Ulm03.pdf|
+\bibitem[Daly 10]{Daly10} Daly, Timothy
+``Intel Instruction Semantics Generator''
+\verb|daly.axiom-developer.org/TimothyDaly_files/publications/sei/intel/intel.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Daly10.pdf|
abstract = "
- The Kovacic algorithm and its improvements give explicit formulae for
- the Liouvillian solutions of second order linear differential
- equations. Algorithms for third order differential equations also
- exist, but the tools they use are more sophisticated and the
- computations more involved. In this paper we refine parts of the
- algorithm to find Liouvillian solutions of third order equations. We
- show that,except for four finite groups and a reduction to the second
- order case, it is possible to give a formula in the imprimitve
- case. We also give necessary conditions and several simplifications
- for the computation of the minimal polynomial for the remaining finite
- set of finite groups (or any known finite group) by extracting
- ramification information from the character table. Several examples
- have been constructed, illustrating the possibilities and limitations."
+ Given an Intel x86 binary, extract the semantics of the instruction
+ stream as Conditional Concurrent Assignments (CCAs). These CCAs
+ represent the semantics of each individual instruction. They can be
+ composed to represent higher level semantics."
\end{chunk}
-\subsection{Expression Simplification} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Carette 04]{Car04} Carette, Jacques
-``Understanding Expression Simplification''
-\verb|www.cas.mcmaster.ca/~carette/publications/simplification.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Car04.pdf|
+\bibitem[Danielsson 06]{Dani06} Danielsson, Nils Anders; Hughes, John;
+Jansson, Patrik; Gibbons, Jeremy
+``Fast and Loose Reasoning is Morally Correct''
+ACM POPL'06 January 2005, Charleston, South Carolina, USA
+%\verb|axiom-developer.org/axiom-website/papers/Dani06.pdf|
abstract = "
- We give the first formal definition of the concept of {\sl
- simplification} for general expressions in the context of Computer
- Algebra Systems. The main mathematical tool is an adaptation of the
- theory of Minimum Description Length, which is closely related to
- various theories of complexity, such as Kolmogorov Complexity and
- Algorithmic Information Theory. In particular, we show how this theory
- can justify the use of various ``magic constants'' for deciding
- between some equivalent representations of an expression, as found in
- implementations of simplification routines."
+ Functional programmers often reason about programs as if they were
+ written in a total language, expecting the results to carry over to
+ non-toal (partial) languages. We justify such reasoning.
-\end{chunk}
+ Two languages are defined, one total and one partial, with identical
+ syntax. The semantics of the partial language includes partial and
+ infinite values, and all types are lifted, including the function
+ spaces. A partial equivalence relation (PER) is then defined, the
+ domain of which is the total subset of the partial language. For types
+ not containing function spaces the PER relates equal values, and
+ functions are related if they map related values to related values.
+
+ It is proved that if two closed terms have the same semantics in the
+ total language, then they have related semantics in the partial
+ language. It is also shown that the PER gives rise to a bicartesian
+ closed category which can be used to reason about values in the domain
+ of the relation."
-\subsection{Integration} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Adamchik xx]{Adamxx} Adamchik, Victor
-``Definite Integration''
-\verb|www.cs.cmu.edu/~adamchik/articles/integr/mj.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Adamxx.pdf|
+\bibitem[Davenport 12]{Davenp12} Davenport, James H.; Bradford, Russell;
+England, Matthew; Wilson, David
+``Program Verification in the presence of complex numbers, functions with
+branch cuts etc.''
+\verb|arxiv.org/pdf/1212.5417.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Davenp12.pdf|
+ abstract = "
+ In considering the reliability of numerical programs, it is normal to
+ ``limit our study to the semantics dealing with numerical precision''.
+ On the other hand, there is a great deal of work on the reliability of
+ programs that essentially ignores the numerics. The thesis of this
+ paper is that there is a class of problems that fall between these
+ two, which could be described as ``does the low-level arithmetic
+ implement the high-level mathematics''. Many of these problems arise
+ because mathematics, particularly the mathematics of the complex
+ numbers, is more difficult than expected: for example the complex
+ function log is not continuous, writing down a program to compute an
+ inverse function is more complicated than just solving an equation,
+ and many algebraic simplification rules are not universally valid.
+
+ The good news is that these problems are {\sl theoretically} capable
+ of being solved, and are {\sl practically} close to being solved, but
+ not yet solved, in several real-world examples. However, there is
+ still a long way to go before implementations match the theoretical
+ possibilities."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Adamchik 97]{Adam97} Adamchik, Victor
-``A Class of Logarithmic Integrals''
-\verb|www.cs.cmu.edu/~adamchik/articles/issac/issac97.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Adam97.pdf|
+\bibitem[Dolzmann 97]{Dolz97} Dolzmann, Andreas; Sturm, Thomas
+``Guarded Expressions in Practice''
+\verb|redlog.dolzmann.de/papers/pdf/MIP-9702.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Dolz97.pdf|
abstract = "
- A class of definite integrals involving cyclotomic polynomials and
- nested logarithms is considered. The results are given in terms of
- derivatives of the Hurwitz Zeta function. Some special cases for which
- such derivatives can be expressed in closed form are also considered."
+ Computer algebra systems typically drop some degenerate cases when
+ evaluating expressions, e.g. $x/x$ becomes 1 dropping the case
+ $x=0$. We claim that it is feasible in practice to compute also the
+ degenerate cases yielding {\sl guarded expressions}. We work over real
+ closed fields but our ideas about handling guarded expressions can be
+ easily transferred to other situations. Using formulas as guards
+ provides a powerful tool for heuristically reducing the combinatorial
+ explosion of cases: equivalent, redundant, tautological, and
+ contradictive cases can be detected by simplification and quantifier
+ elimination. Our approach allows to simplify the expressions on the
+ basis of simplification knowledge on the logical side. The method
+ described in this paper is implemented in the REDUCE package GUARDIAN,
+ which is freely available on the WWW."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Avgoustis 77]{Avgo77} Avgoustis, Ioannis Dimitrios
-``Definite Integration using the Generalized Hypergeometric Functions''
-\verb|dspace.mit.edu/handle/1721.1/16269|
-%\verb|axiom-developer.org/axiom-websitep/papers/Avgo77.pdf|
+\bibitem[Dos Reis 11]{DR11} Dos Reis, Gabriel; Matthews, David; Li, Yue
+``Retargeting OpenAxiom to Poly/ML: Towards an Integrated Proof Assistants
+and Computer Algebra System Framework''
+Calculemus (2011) Springer
+\verb|paradise.caltech.edu/~yli/paper/oa-polyml.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/DR11.pdf|
abstract = "
- A design for the definite integration of approximately fifty Special
- Functions is described. The Generalized Hypergeometric Functions are
- utilized as a basis for the representation of the members of the above
- set of Special Functions. Only a relatively small number of formulas
- that generally involve Generalized Hypergeometric Functions are
- utilized for the integration stage. A last and crucial stage is
- required for the integration process: the reduction of the Generalized
- Hypergeometric Function to Elementary and/or Special Functions.
-
- The result of an early implementation which involves Laplace
- transforms are given and some actual examples with their corresponding
- timing are provided."
+ This paper presents an ongoing effort to integrate the Axiom family of
+ computer algebra systems with Poly/ML-based proof assistants in the
+ same framework. A long term goal is to make a large set of efficient
+ implementations of algebraic algorithms available to popular proof
+ assistants, and also to bring the power of mechanized formal
+ verification to a family of strongly typed computer algebra systems at
+ a modest cost. Our approach is based on retargeting the code generator
+ of the OpenAxiom compiler to the Poly/ML abstract machine."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Baddoura 89]{Bad89} Baddoura, Jamil
-``A Dilogarithmic Extension of Liouville's Theorem on Integration in Finite
- Terms''
-\verb|www.dtic.mil/dtic/tr/fulltext/u2/a206681.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Bad89.pdf|
- abstract = "
- The result obtained generalizes Liouville's Theorem by allowing, in
- addition to the elementary functions, dilogarithms to appear in the
- integral of an elementary function. The basic conclusion is that an
- associated function to the dilogarihm, if dilogarithms appear in the
- integral, appears linearly, with logarithms appearing in a non-linear
- way."
-
-\end{chunk}
-
-\begin{chunk}{ignore}
-\bibitem[Baddoura 94]{Bad94} Baddoura, Mohamed Jamil
-``Integration in Finite Terms with Elementary Functions and Dilogarithms''
-\verb|dspace.mit.edu/bitstream/handle/1721.1/26864/30757785.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Bad94.pdf|
+\bibitem[Dunstan 00a]{Dun00a} Dunstan, Martin N.
+``Adding Larch/Aldor Specifications to Aldor''
+%\verb|axiom-developer.org/axiom-website/papers/Dunxx.pdf|
abstract = "
- In this thesis, we report on a new theorem that generalizes
- Liouville's theorem on integration in finite terms. The new theorem
- allows dilogarithms to occur in the integral in addition to elementary
- functions. The proof is base on two identities for the dilogarithm,
- that characterize all the possible algebraic relations among
- dilogarithms of functions that are built up from the rational
- functions by taking transcendental exponentials, dilogarithms, and
- logarithms."
+ We describe a proposal to add Larch-style annotations to the Aldor
+ programming language, based on our PhD research. The annotations
+ are intended to be machine-checkable and may be used for a variety
+ of purposes ranging from compiler optimizations to verification
+ condition (VC) generation. In this report we highlight the options
+ available and describe the changes which would need to be made to
+ the compiler to make use of this technology."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Baddoura 10]{Bad10} Baddoura, Jamil
-``A Note on Symbolic Integration with Polylogarithms''
-J. Math Vol 8 pp229-241 (2011)
-%\verb|axiom-developer.org/axiom-website/papers/Bad10.pdf|
+\bibitem[Dunstan 98]{Dun98} Dunstan, Martin; Kelsey, Tom; Linton, Steve;
+Martin, Ursula
+``Lightweight Formal Methods For Computer Algebra Systems''
+\verb|www.cs.st-andrews.ac.uk/~tom/pub/issac98.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Dun98.pdf|
abstract = "
- We generalize partially Liouville's theorem on integration in finite
- terms to allow polylogarithms of any order to occur in the integral in
- addition to elementary functions. The result is a partial
- generalization of a theorem proved by the author for the
- dilogarithm. It is also a partial proof of a conjecture postulated by
- the author in 1994. The basic conclusion is that an associated
- function to the nth polylogarithm appears linearly with logarithms
- appearing possibly in a polynomial way with non-constant coefficients."
+ Demonstrates the use of formal methods tools to provide a semantics
+ for the type hierarchy of the Axiom computer algebra system, and a
+ methodology for Aldor program analysis and verification. There are
+ examples of abstract specifications of Axiom primitives."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Bajpai 70]{Bajp70} Bajpai, S.D.
-``A contour integral involving legendre polynomial and Meijer's G-function''
-\verb|link.springer.com/article/10.1007/BF03049565|
-%\verb|axiom-developer.org/axiom-website/papers/Bajp70.pdf|
+\bibitem[Dunstan 99a]{Dun99a} Dunstan, MN
+``Larch/Aldor - A Larch BISL for AXIOM and Aldor''
+PhD Thesis, 1999
+\verb|www.cs.st-andrews.uk/files/publications/Dun99.php|
+%\verb|axiom-developer.org/axiom-website/papers/Dun99a.pdf|
abstract = "
- In this paper a countour integral involving Legendre polynomial and
- Meijer's G-function is evaluated. the integral is of general character
- and it is a generalization of results recently given by Meijer,
- MacRobert and others. An integral involving regular radial Coulomb
- wave function is also obtained as a particular case."
+ In this thesis we investigate the use of lightweight formal methods
+ and verification conditions (VCs) to help improve the reliability of
+ components constructed within a computer algebra system. We follow the
+ Larch approach to formal methods and have designed a new behavioural
+ interface specification language (BISL) for use with Aldor: the
+ compiled extension language of Axiom and a fully-featured programming
+ language in its own right. We describe our idea of lightweight formal
+ methods, present a design for a lightweight verification condition
+ generator and review our implementation of a prototype verification
+ condition generator for Larch/Aldor."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Bronstein 89]{Bro89a} Bronstein, M.
-``An Algorithm for the Integration of Elementary Functions''
-Lecture Notes in Computer Science Vol 378 pp491-497 (1989)
-%\verb|axiom-developer.org/axiom-website/papers/Bro89a.pdf|
+\bibitem[Dunstan 00]{Dun00} Dunstan, Martin; Kelsey, Tom; Martin, Ursula;
+Linton, Steve
+``Formal Methods for Extensions to CAS''
+FME'99, Toulouse, France, Sept 20-24, 1999, pp 1758-1777
+\verb|tom.host.cs.st-andrews.ac.uk/pub/fm99.ps|
+%\verb|axiom-developer.org/axiom-website/papers/Dun00.pdf|
abstract = "
- Trager (1984) recently gave a new algorithm for the indefinite
- integration of algebraic functions. His approach was ``rational'' in
- the sense that the only algebraic extension computed in the smallest
- one necessary to express the answer. We outline a generalization of
- this approach that allows us to integrate mixed elementary
- functions. Using only rational techniques, we are able to normalize
- the integrand, and to check a necessary condition for elementary
- integrability."
+ We demonstrate the use of formal methods tools to provide a semantics
+ for the type hierarchy of the AXIOM computer algebra system, and a
+ methodology for Aldor program analysis and verification. We give a
+ case study of abstract specifications of AXIOM primitives, and provide
+ an interface between these abstractions and Aldor code."
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Bronstein 90a]{Bro90a} Bronstein, Manuel
-``Integration of Elementary Functions''
-J. Symbolic Computation (1990) 9, pp117-173 September 1988
-%\verb|axiom-developer.org/axiom-website/papers/Bro90a.pdf|
+\begin{chunk}{axiom.bib}
+@misc{Hard13,
+ author = "Hardin, David S. and McClurg, Jedidiah R. and Davis, Jennifer A.",
+ title = "Creating Formally Verified Components for Layered Assurance with an LLVM to ACL2 Translator",
+ url = "http://www.jrmcclurg.com/papers/law_2013_paper.pdf",
+ paper = "Hard13.pdf",
abstract = "
- We extend a recent algorithm of Trager to a decision procedure for the
- indefinite integration of elementary functions. We can express the
- integral as an elementary function or prove that it is not
- elementary. We show that if the problem of integration in finite terms
- is solvable on a given elementary function field $k$, then it is
- solvable in any algebraic extension of $k(\theta)$, where $\theta$ is
- a logarithm or exponential of an element of $k$. Our proof considers
- an element of such an extension field to be an algebraic function of
- one variable over $k$.
-
- In his algorithm for the integration of algebraic functions, Trager
- describes a Hermite-type reduction to reduce the problem to an
- integrand with only simple finite poles on the associated Riemann
- surface. We generalize that technique to curves over liouvillian
- ground fields, and use it to simplify our integrands. Once the
- multipe finite poles have been removed, we use the Puiseux expansions
- of the integrand at infinity and a generalization of the residues to
- compute the integral. We also generalize a result of Rothstein that
- gives us a necessary condition for elementary integrability, and
- provide examples of its use."
+ This paper describes an effort to create a library of formally
+ verified software component models from code that have been compiled
+ using the Low-Level Virtual Machine (LLVM) intermediate form. The idea
+ is to build a translator from LLVM to the applicative subset of Common
+ Lisp accepted by the ACL2 theorem prover. They perform verification of
+ the component model using ACL2's automated reasoning capabilities."
+}
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Bron90c,
- author = "Bronstein, Manuel",
- title = "On the integration of elementary functions",
- journal = "Journal of Symbolic Computation",
- volume = "9",
- number = "2",
- pages = "117-173",
- year = "1990",
- month = "February"
+@misc{Hard14,
+ author = "Hardin, David S. and Davis, Jennifer A. and Greve, David A. and
+ McClurg, Jedidiah R.",
+ title = "Development of a Translator from LLVM to ACL2",
+ url = "http://arxiv.org/pdf/1406.1566",
+ paper = "Hard14.pdf",
+ abstract = "
+ In our current work a library of formally verified software components
+ is to be created, and assembled, using the Low-Level Virtual Machine
+ (LLVM) intermediate form, into subsystems whose top-level assurance
+ relies on the assurance of the individual components. We have thus
+ undertaken a project to build a translator from LLVM to the
+ applicative subset of Common Lisp accepted by the ACL2 theorem
+ prover. Our translator produces executable ACL2 formal models,
+ allowing us to both prove theorems about the translated models as well
+ as validate those models by testing. The resulting models can be
+ translated and certified without user intervention, even for code with
+ loops, thanks to the use of the def::ung macro which allows us to
+ defer the question of termination. Initial measurements of concrete
+ execution for translated LLVM functions indicate that performance is
+ nearly 2.4 million LLVM instructions per second on a typical laptop
+ computer. In this paper we overview the translation process and
+ illustrate the translator's capabilities by way of a concrete example,
+ including both a functional correctness theorem as well as a
+ validation test for that example."
}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Bronstein 93]{REF-BS93} Bronstein, Manuel; Salvy, Bruno
-``Full partial fraction decomposition of rational functions''
-In Bronstein [Bro93] pp157-160 ISBN 0-89791-604-2 LCCN QA76.95 I59 1993
-\verb|www.acm.org/pubs/citations/proceedings/issac/164081/|
+\bibitem[Lamport 02]{Lamp02} Lamport, Leslie
+``Specifying Systems''
+\verb|research.microsoft.com/en-us/um/people/lamport/tla/book-02-08-08.pdf|
+Addison-Wesley ISBN 0-321-14306-X
+%\verb|axiom-developer.org/axiom-website/papers/Lamp02.pdf|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Bronstein 90]{Bro90b} Bronstein, Manuel
-``A Unification of Liouvillian Extensions''
-%\verb|axiom-developer.org/axiom-website/papers/Bro90b.pdf|
+\bibitem[Martin 97]{Mart97} Martin, U.; Shand, D.
+``Investigating some Embedded Verification Techniques for Computer
+ Algebra Systems''
+\verb|www.risc.jku.at/conferences/Theorema/papers/shand.ps.gz|
+%\verb|axiom-developer.org/axiom-website/papers/Mart97.ps|
abstract = "
- We generalize Liouville's theory of elementary functions to a larger
- class of differential extensions. Elementary, Liouvillian and
- trigonometric extensions are all special cases of our extensions. In
- the transcendental case, we show how the rational techniques of
- integration theory can be applied to our extensions, and we give a
- unified presentation which does not require separate cases for
- different monomials."
+ This paper reports some preliminary ideas on a collaborative project
+ between St. Andrews University in the UK and NAG Ltd. The project aims
+ to use embedded verification techniques to improve the reliability and
+ mathematical soundness of computer algebra systems. We give some
+ history of attempts to integrate computer algebra systems and
+ automated theorem provers and discuss possible advantages and
+ disadvantages of these approaches. We also discuss some possible case
+ studies."
\end{chunk}
\begin{chunk}{axiom.bib}
-@book{Bron97,
- author = "Bronstein, Manuel",
- title = "Symbolic Integration I--Transcendental Functions",
- publisher = "Springer, Heidelberg",
- year = "1997",
- isbn = "3-540-21493-3",
- url = "http://evil-wire.org/arrrXiv/Mathematics/Bronstein,_Symbolic_Integration_I,1997.pdf",
- paper = "Bron97.pdf"
+@book{Maso86,
+ author = "Mason, Ian A.",
+ title = "The Semantics of Destructive Lisp",
+ publisher = "Center for the Study of Language and Information",
+ year = "1986",
+ isbn = "0-937073-06-7",
+ abstract = "
+ Our basic premise is that the ability to construct and modify programs
+ will not improve without a new and comprehensive look at the entire
+ programming process. Past theoretical research, say, in the logic of
+ programs, has tended to focus on methods for reasoning about
+ individual programs; little has been done, it seems to us, to develop
+ a sound understanding of the process of programming -- the process by
+ which programs evolve in concept and in practice. At present, we lack
+ the means to describe the techniques of program construction and
+ improvement in ways that properly link verification, documentation and
+ adaptability."
}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Bronstein 05a]{Bro05a} Bronstein, Manuel
-``The Poor Man's Integrator, a parallel integration heuristic''
-\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/pmint.txt|
-\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/examples|
-%\verb|axiom-developer.org/axiom-website/papers/Bro05a.txt|
-
-\end{chunk}
-
-\begin{chunk}{axiom.bib}
-@article{Bron06,
- author = "Bronstein, M.",
- title = "Parallel integration",
- journal = "Programming and Computer Software",
- year = "2006",
- issn = "0361-7688",
- volume = "32",
- number = "1",
- doi = "10.1134/S0361768806010075",
- url = "http://dx.doi.org/10.1134/S0361768806010075",
- publisher = "Nauka/Interperiodica",
- pages = "59-60",
- paper = "Bron06.pdf",
+\bibitem[Newcombe 13]{Newc13} Newcombe, Chris; Rath, Tim; Zhang, Fan;
+Munteanu, Bogdan; Brooker, Marc; Deardeuff, Michael
+``Use of Formal Methods at Amazon Web Services''
+\verb|research.microsoft.com/en-us/um/people/lamport/tla/|
+\verb|formal-methods-amazon.pdf|
abstract = "
- Parallel integration is an alternative method for symbolic
- integration. While also based on Liouville's theorem, it handles all
- the generators of the differential field containing the integrand ``in
- parallel'', i.e. all at once rather than considering only the topmost
- one in a recursive fasion. Although it still contains heuristic
- aspects, its ease of implementation, speed, high rate of success, and
- ability to integrate functions that cannot be handled by the Risch
- algorithm make it an attractive alternative."
-}
+ In order to find subtle bugs in a system design, it is necessary to
+ have a precise description of that design. There are at least two
+ major benefits to writing a precise design; the author is forced to
+ think more clearly, which helps eliminate ``plausible hand-waving'',
+ and tools can be applied to check for errors in the design, even while
+ it is being written. In contrast, conventional design documents
+ consist of prose, static diagrams, and perhaps pseudo-code in an ad
+ hoc untestable language. Such descriptions are far from precise; they
+ are often ambiguous, or omit critical aspects such as partial failure
+ or the granularity of concurrency (i.e. which constructs are assumed
+ to be atomic). At the other end of the spectrum, the final executable
+ code is unambiguous, but contains an overwhelming amount of detail. We
+ needed to be able to capture the essence of a design in a few hundred
+ lines of precise description. As our designs are unavoidably complex,
+ we need a highly-expressive language, far above the level of code, but
+ with precise semantics. That expressivity must cover real-world
+ concurrency and fault-tolerance. And, as we wish to build services
+ quickly, we wanted a language that is simple to learn and apply,
+ avoiding esoteric concepts. We also very much wanted an existing
+ ecosystem of tools. We found what we were looking for in TLA+, a
+ formal specification language."
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Bron07,
- author = "Bronstein, Manuel",
- title = "Structure theorems for parallel integration",
- journal = "Journal of Symbolic Computation",
- volume = "42",
- number = "7",
- pages = "757-769",
- year = "2007",
- month = "July",
- paper = "Bron07.pdf",
+\begin{chunk}{ignore}
+\bibitem[Poll 99a]{P99a} Poll, Erik
+``The Type System of Axiom''
+\verb|www.cs.ru.nl/E.Poll/talks/axiom.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/P99a.pdf|
abstract = "
- We introduce structure theorems that refine Liouville's Theorem on
- integration in closed form for general derivations on multivariate
- rational function fields. By predicting the arguments of the new
- logarithms that an appear in integrals, as well as the denominator of
- the rational part, those theorems provide theoretical backing for the
- Risch-Norman integration method. They also generalize its applicability
- to non-monomial extensions, for example the Lambert W function."
-}
+ This is a slide deck from a talk on the correspondence between
+ Axiom/Aldor types and Logic."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Charlwood 07]{Charl07} Charlwood, Kevin
-``Integration on Computer Algebra Systems''
-The Electronic J of Math. and Tech. Vol 2, No 3, ISSN 1933-2823
-\verb|12000.org/my_notes/ten_hard_integrals/paper.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Charl07.pdf|
+\bibitem[Poll 99]{PT99} Poll, Erik; Thompson, Simon
+``The Type System of Aldor''
+\verb|www.cs.kent.ac.uk/pubs/1999/874/content.ps|
+%\verb|axiom-developer.org/axiom-website/papers/PT99.pdf|
abstract = "
- In this article, we consider ten indefinite integrals and the ability
- of three computer algebra systems (CAS) to evaluate them in
- closed-form, appealing only to the class of real, elementary
- functions. Although these systems have been widely available for many
- years and have undergone major enhancements in new versions, it is
- interesting to note that there are still indefinite integrals that
- escape the capacity of these systems to provide antiderivatves. When
- this occurs, we consider what a user may do to find a solution with
- the aid of a CAS."
+ This paper gives a formal description of -- at least a part of --
+ the type system of Aldor, the extension language of the Axiom.
+ In the process of doing this a critique of the design of the system
+ emerges."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Charlwood 08]{Charl08} Charlwood, Kevin
-``Symbolic Integration Problems''
-\verb|www.apmaths.uwo.ca/~arich/IndependentTestResults/CharlwoodIntegrationProblems.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Charl08.pdf|
+\bibitem[Poll (a)]{PTxx} Poll, Erik; Thompson, Simon
+``Adding the axioms to Axiom. Toward a system of automated reasoning in
+Aldor''
+\verb|citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.7.1457&rep=rep1&type=ps|
+%\verb|axiom-developer.org/axiom-website/papers/PTxx.pdf|
abstract = "
- A list of the 50 example integration problems from Kevin Charlwood's 2008
- article ``Integration on Computer Algebra Systems''. Each integral along
- with its optimal antiderivative (that is, the best antiderivative found
- so far) is shown."
+ This paper examines the proposal of using the type system of Axiom to
+ represent a logic, and thus to use the constructions of Axiom to
+ handle the logic and represent proofs and propositions, in the same
+ way as is done in theorem provers based on type theory such as Nuprl
+ or Coq.
+
+ The paper shows an interesting way to decorate Axiom with pre- and
+ post-conditions.
+
+ The Curry-Howard correspondence used is
+ \begin{verbatim}
+ PROGRAMMING LOGIC
+ Type Formula
+ Program Proof
+ Product/record type (...,...) Conjunction
+ Sum/union type \/ Disjunction
+ Function type -> Implication
+ Dependent function type (x:A) -> B(x) Universal quantifier
+ Dependent product type (x:A,B(x)) Existential quantifier
+ Empty type Exit Contradictory proposition
+ One element type Triv True proposition
+ \end{verbatim}"
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Cherry 84]{Che84} Cherry, G.W.
-``Integration in Finite Terms with Special Functions: The Error Function''
-J. Symbolic Computation (1985) Vol 1 pp283-302
-%\verb|axiom-developer.org/axiom-website/papers/Che84.pdf|
+\bibitem[Poll 00]{PT00} Poll, Erik; Thompson, Simon
+``Integrating Computer Algebra and Reasoning through the Type System
+of Aldor''
+%\verb|axiom-developer.org/axiom-website/papers/PT00.pdf|
abstract = "
- A decision procedure for integrating a class of transcendental
- elementary functions in terms of elementary functions and error
- functions is described. The procedure consists of three mutually
- exclusive cases. In the first two cases a generalised procedure for
- completing squares is used to limit the error functions which can
- appear in the integral of a finite number. This reduces the problem
- to the solution of a differential equation and we use a result of
- Risch (1969) to solve it. The third case can be reduced to the
- determination of what we have termed $\sum$-decompositions. The resutl
- presented here is the key procuedure to a more general algorithm which
- is described fully in Cherry (1983)."
+ A number of combinations of reasoning and computer algebra systems
+ have been proposed; in this paper we describe another, namely a way to
+ incorporate a logic in the computer algebra system Axiom. We examine
+ the type system of Aldor -- the Axiom Library Compiler -- and show
+ that with some modifications we can use the dependent types of the
+ system to model a logic, under the Curry-Howeard isomorphism. We give
+ a number of example applications of the logi we construct and explain
+ a prototype implementation of a modified type-checking system written
+ in Haskell."
\end{chunk}
+\section{Interval Arithmetic} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Cherry 86]{Che86} Cherry, G.W.
-``Integration in Finite Terms with Special Functions:
-The Logarithmic Integral''
-SIAM J. Comput. Vol 15 pp1-21 February 1986
+\bibitem[Boehm 86]{Boe86} Boehm, Hans-J.; Cartwright, Robert; Riggle, Mark;
+O'Donnell, Michael J.
+``Exact Real Arithmetic: A Case Study in Higher Order Programming''
+\verb|dev.acm.org/pubs/citations/proceedings/lfp/319838/p162-boehm|
+%\verb|axiom-developer.org/axiom-website/papers/Boe86.pdf|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Cherry 89]{Che89} Cherry, G.W.
-``An Analysis of the Rational Exponential Integral''
-SIAM J. Computing Vol 18 pp 893-905 (1989)
-%\verb|axiom-developer.org/axiom-website/papers/Che89.pdf|
- abstract = "
- In this paper an algorithm is presented for integrating expressions of
- the form $\int{ge^f~dx}$, where $f$ and $g$ are rational functions of
- $x$, in terms of a class of special functions called the special
- incomplete $\Gamma$ functions. This class of special functions
- includes the exponential integral, the error functions, the sine and
- cosing integrals, and the Fresnel integrals. The algorithm presented
- here is an improvement over those published previously for integrating
- with special functions in the following ways: (i) This algorithm
- combines all the above special functions into one algorithm, whereas
- previously they were treated separately, (ii) Previous algorithms
- require that the underlying field of constants be algebraically
- closed. This algorithm, however, works over any field of
- characteristic zero in which the basic field operations can be carried
- out. (iii) This algorithm does not rely on Risch's solution of the
- differential equation $y^\prime + fy = g$. Instead, a more direct
- method of undetermined coefficients is used."
+\bibitem[Briggs 04]{Bri04} Briggs, Keith
+``Exact real arithmetic''
+\verb|keithbriggs.info/documents/xr-kent-talk-pp.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Bri04.pdf|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Churchill 06]{Chur06} Churchill, R.C.
-``Liouville's Theorem on Integration Terms of Elementary Functions''
-\verb|www.sci.ccny.cuny.edu/~ksda/PostedPapers/liouv06.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Chur06.pdf|
+\bibitem[Fateman 94]{Fat94} Fateman, Richard J.; Yan, Tak W.
+``Computation with the Extended Rational Numbers and an Application to
+Interval Arithmetic''
+\verb|www.cs.berkeley.edu/~fateman/papers/extrat.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Fat94.pdf|
abstract = "
- This talk should be regarded as an elementary introduction to
- differential algebra. It culminates in a purely algebraic proof, due
- to M. Rosenlicht, of an 1835 theorem of Liouville on the existence of
- ``elementary'' integrals of ``elementary'' functions. The precise
- meaning of elementary will be specified. As an application of that
- theorem we prove that the indefinite integral $\int{e^{x^2}}~dx$
- cannot be expressed in terms of elementary functions.
- \begin{itemize}
- \item Preliminaries on Meromorphic Functions
- \item Basic (Ordinary) Differential Algebra
- \item Differential Ring Extensions with No New Constants
- \item Extending Derivations
- \item Integration in Finite Terms
- \end{itemize}"
+ Programming languages such as Common Lisp, and virtually every
+ computer algebra system (CAS), support exact arbitrary-precision
+ integer arithmetic as well as exect rational number computation.
+ Several CAS include interval arithmetic directly, but not in the
+ extended form indicated here. We explain why changes to the usual
+ rational number system to include infinity and ``not-a-number'' may be
+ useful, especially to support robust interval computation. We describe
+ techniques for implementing these changes."
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Davenport 79b]{Dav79b} Davenport, James Harold
-``On the Integration of Algebraic Functions''
-Springer-Verlag Lecture Notes in Computer Science 102
-ISBN 0-387-10290-6
+\begin{chunk}{axiom.bib}
+@incollection{Lamb06,
+ author = "Lambov, Branimir",
+ title = "Interval Arithmetic Using SSE-2",
+ booktitle = "Lecture Notes in Computer Science",
+ publisher = "Springer-Verlag",
+ year = "2006",
+ isbn = "978-3-540-85520-0",
+ pages = "102-113"
+}
\end{chunk}
+\section{Numerics} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Davenport 79c]{Dav79c} Davenport, J. H.
-``Algorithms for the Integration of Algebraic Functions''
-Lecture Notes in Computer Science V 72 pp415-425 (1979)
-%\verb|axiom-developer.org/axiom-website/papers/Dav79c.pdf|
+\bibitem[Atkinson 09]{Atk09} Atkinson, Kendall; Han, Welmin; Stewear, David
+``Numerical Solution of Ordinary Differential Equations''
+\verb|homepage.math.uiowa.edu/~atkinson/papers/NAODE_Book.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Atk09.pdf|
abstract = "
- The problem of finding elementary integrals of algebraic functions has
- long been recognized as difficult, and has sometimes been thought
- insoluble. Risch stated a theorem characterising the integrands with
- elementary integrals, and we can use the language of algebraic
- geometry and the techniques of Davenport to yield an algorithm that will
- always produce the integral if it exists. We explain the difficulty in
- the way of extending this algorithm, and outline some ways of solving
- it. Using work of Manin we are able to solve the problem in all cases
- where the algebraic expressions depend on a parameter as well as on
- the variable of integration."
+ This book is an expanded version of supplementary notes that we used
+ for a course on ordinary differential equations for upper-division
+ undergraduate students and beginning graduate students in mathematics,
+ engineering, and sciences. The book introduces the numerical analysis
+ of differential equations, describing the mathematical background for
+ understanding numerical methods and giving information on what to
+ expect when using them. As a reason for studying numerical methods as
+ a part of a more general course on differential equations, many of the
+ basic ideas of the numerical analysis of differential equations are
+ tied closely to theoretical behavior associated with the problem being
+ solved. For example, the criteria for the stability of a numerical
+ method is closely connected to the stability of the differential
+ equation problem being solved."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Davenport 82a]{Dav82a} Davenport, J.H.
-``The Parallel Risch Algorithm (I)
-%\verb|axiom-developer.org/axiom-website/papers/Dav82a.pdf|
- abstract = "
- In this paper we review the so-called ``parallel Risch'' algorithm for
- the integration of transcendental functions, and explain what the
- problems with it are. We prove a positive result in the case of
- logarithmic integrands."
+\bibitem[Crank 96]{Cran96} Crank, J.; Nicolson, P.
+``A practical method for numerical evaluations of solutions of partial
+ differential equations of heat-conduction type''
+Advances in Computational Mathematics Vol 6 pp207-226 (1996)
+\verb|www.acms.arizona.edu/FemtoTheory/MK_personal/opti547/literature/|
+\verb|CNMethod-original.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Cran96.pdf|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Davenport 82]{Dav82} Davenport, J.H.
-``On the Parallel Risch Algorithm (III): Use of Tangents''
-SIGSAM V16 no. 3 pp3-6 August 1982
+\bibitem[Lef\'evre 06]{Lef06} Lef\'evre, Vincent; Stehl\'e, Damien;
+Zimmermann, Paul
+``Worst Cases for the Exponential Function
+in the IEEE-754r decimal64 Format''
+in Lecture Notes in Computer Science, Springer ISBN 978-3-540-85520-0
+(2006) pp114-125
+ abstract = "
+ We searched for the worst cases for correct rounding of the
+ exponential function in the IEEE 754r decimal64 format, and computed
+ all the bad cases whose distance from a breakpoint (for all rounding
+ modes) is less than $10^{-15}$ ulp, and we give the worst ones. In
+ particular, the worst case for
+ $\vert{}x\vert{} \ge 3 x 10^{-11}$ is
+ \[
+ exp(9.407822313572878x10^{-2} =
+ 1.09864568206633850000000000000000278\ldots
+ \]
+ This work can be extended to other elementary functions in the decimal64
+ format and allows the design of reasonably fast routines that will
+ evaluate these functions with correct rounding, at least in some
+ situations."
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Davenport 03]{Dav03} Davenport, James H.
-``The Difficulties of Definite Integration''
-\verb|www.researchgate.net/publication/|
-\verb|247837653_The_Diculties_of_Definite_Integration/file/72e7e52a9b1f06e196.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Dav03.pdf|
- abstract = "
- Indefinite integration is the inverse operation to differentiation,
- and, before we can understand what we mean by indefinite integration,
- we need to understand what we mean by differentiation."
+\begin{chunk}{axiom.bib}
+@book{Hamm62,
+ author = "Hamming R W.",
+ title = "Numerical Methods for Scientists and Engineers",
+ publisher = "Dover",
+ year = "1973",
+ isbn = "0-486-65241-6"
+}
\end{chunk}
+\section{Advanced Documentation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Fateman 02]{Fat02} Fateman, Richard
-``Symbolic Integration''
-\verb|inst.eecs.berkeley.edu/~cs282/sp02/lects/14.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Fat02.pdf|
+\bibitem [Bostock 14]{Bos14} Bostock, Mike
+``Visualizing Algorithms''
+\verb|bost.ocks.org/mike/algorithms|
+ abstract = "
+ This website hosts various ways of visualizing algorithms. The hope is
+ that these kind of techniques can be applied to Axiom."
\end{chunk}
-\begin{chunk}{axiom.bib}
-@inproceedings{Gedd89,
- author = "Geddes, K. O. and Stefanus, L. Y.",
- title = "On the Risch-norman Integration Method and Its Implementation
- in MAPLE",
- booktitle = "Proc. of the ACM-SIGSAM 1989 Int. Symp. on Symbolic and
- Algebraic Computation",
- series = "ISSAC '89",
- year = "1989",
- isbn = "0-89791-325-6",
- location = "Portland, Oregon, USA",
- pages = "212--217",
- numpages = "6",
- url = "http://doi.acm.org/10.1145/74540.74567",
- doi = "10.1145/74540.74567",
- acmid = "74567",
- publisher = "ACM",
- address = "New York, NY, USA",
- paper = "Gedd89.pdf",
+\begin{chunk}{ignore}
+\bibitem[Leeuwen]{Leexx} van Leeuwen, Andr\'e M.A.
+``Representation of mathematical object in interactive books''
+%\verb|axiom-developer.org/axiom-website/papers/Leexx.pdf|
abstract = "
- Unlike the Recursive Risch Algorithm for the integration of
- transcendental elementary functions, the Risch-Norman Method processes
- the tower of field extensions directly in one step. In addition to
- logarithmic and exponential field extensions, this method can handle
- extentions in terms of tangents. Consequently, it allows trigonometric
- functions to be treated without converting them to complex exponential
- form. We review this method and describe its implementation in
- MAPLE. A heuristic enhancement to this method is also presented."
-}
+ We present a model for the representation of mathematical objects in
+ structured electronic documents, in a way that allows for interaction
+ with applications such as computer algebra systems and proof checkers.
+ Using a representation that reflects only the intrinsic information of
+ an object, and storing application-dependent information in so-called
+ {\sl application descriptions}, it is shown how the translation from
+ the internal to an external representation and {\sl vice versa} can be
+ achieved. Hereby a formalisation of the concept of {\sl context} is
+ introduced. The proposed scheme allows for a high degree of
+ application integration, e.g., parallel evaluation of subexpressions
+ (by different computer algebra systems), or a proof checker using a
+ computer algebra system to verify an equation involving a symbolic
+ computation."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Geddes 92a]{GCL92a} Geddes, K.O.; Czapor, S.R.; Labahn, G.
-``The Risch Integration Algorithm''
-Algorithms for Computer Algebra, Ch 12 pp511-573 (1992)
-%\verb|axiom-developer.org/axiom-website/papers/GCL92a.pdf|
+\bibitem[Soiffer 91]{Soif91} Soiffer, Neil Morrell
+``The Design of a User Interface for Computer Algebra Systems''
+\verb|www.eecs.berkeley.edu/Pubs/TechRpts/1991/CSD-91-626.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Soif91.pdf|
+ abstract = "
+ This thesis discusses the design and implementation of natural user
+ interfaces for Computer Algebra Systems. Such an interface must not
+ only display expressions generated by the Computer Algebra System in
+ standard mathematical notation, but must also allow easy manipulation
+ and entry of expressions in that notation. The user interface should
+ also assist in understanding of large expressions that are generated
+ by Computer Algebra Systems and should be able to accommodate new
+ notational forms."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Hardy 1916]{Hard16} Hardy, G.H.
-``The Integration of Functions of a Single Variable''
-Cambridge Unversity Press, Cambridge, 1916
-% REF:00002
+\bibitem[Victor 11]{Vict11} Victor, Bret
+``Up and Down the Ladder of Abstraction''
+\verb|worrydream.com/LadderOfAbstraction|
+ abstract = "
+ This interactive essay presents the ladder of abstraction, a technique for
+ thinking explicitly about these levels, so a designer can move among
+ them consciously and confidently. "
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Harrington 78]{Harr87} Harrington, S.J.
-``A new symbolic integration system in reduce''
-\verb|comjnl.oxfordjournals.or/content/22/2/127.full.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Harr87.pdf|
+\bibitem[Victor 12]{Vict12} Victor, Bret
+``Inventing on Principle''
+\verb|www.youtube.com/watch?v=PUv66718DII|
abstract = "
- A new integration system, employing both algorithmic and pattern match
- integration schemes is presented. The organization of the system
- differs from that of earlier programs in its emphasis on the
- algorithmic approach to integration, its modularity and its ease of
- revision. The new Norman-Rish algorithm and its implementation at the
- University of Cambridge are employed, supplemented by a powerful
- collection of simplification and transformation rules. The facility
- for user defined integrals and functions is also included. The program
- is both fast and powerful, and can be easily modified to incorporate
- anticipated developments in symbolic integration."
+ This video raises the level of discussion about human-computer
+ interaction from a technical question to a question of effectively
+ capturing ideas. In particular, this applies well to Axiom's focus on
+ literate programming."
\end{chunk}
+\section{Differential Equations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{axiom.bib}
-@misc{Herm1872,
- author = "Hermite, E.",
- title = "Sur l'int\'{e}gration des fractions rationelles",
- journal = "Nouvelles Annales de Math\'{e}matiques",
- volume = "11",
- pages = "145-148",
- year = "1872"
+@InProceedings{Kalt84,
+ author = "Kaltofen, E.",
+ title = "A Note on the {Risch} Differential Equation",
+ booktitle = "Proc. EUROSAM '84",
+ pages = "359--366",
+ crossref = "EUROSAM84",
+ year = "1984",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/84/Ka84_risch.ps.gz",
+ paper = "Kalt84.ps"
}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Horowitz 71]{Horo71} Horowitz, Ellis
-``Algorithms for Partial Fraction Decomposition and Rational Function
- Integration''
-SYMSAC '71 Proc. ACM Symp. on Symbolic and Algebraic Manipulation (1971)
-pp441-457
-%\verb|axiom-developer.org/axiom-website/papers/Horo71.pdf| REF:00018
- abstract = "
- Algorithms for symbolic partial fraction decomposition and indefinite
- integration of rational functions are described. Two types of
- partial fraction decomposition are investigated, square-free and
- complete square-free. A method is derived, based on the solution of
- a linear system, which produces the square-free decomposition of any
- rational function, say A/B. The computing time is show to be
- $O(n^4(ln nf)^2)$ where ${\rm deg}(A) < {\rm\ deg}(B) = n$ and $f$
- is a number which is closely related to the size of the coefficients
- which occur in A and B. The complete square-free partical fraction
- decomposition can then be directly obtained and it is shown that the
- computing time for this process is also bounded by $O(n^4(ln nf)^2)$."
+\bibitem[Abramov 95]{Abra95} Abramov, Sergei A.; Bronstein, Manuel;
+Petkovsek, Marko
+``On Polynomial Solutions of Linear Operator Equations''
+\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html|
+%\verb|axiom-developer.org/axiom-website/papers/Abra95.pdf|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jeffrey 97]{Jeff97} Jeffrey, D.J.; Rich, A.D.
-``Recursive integration of piecewise-continuous functions''
-\verb|www.cybertester.com/data/recint.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Jeff97.pdf|
+\bibitem[Abramov 01]{Abra01} Abramov, Sergei; Bronstein, Manuel
+``On Solutions of Linear Functional Systems''
+\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html|
+%\verb|axiom-developer.org/axiom-website/papers/Abra01.pdf|
abstract = "
- An algorithm is given for the integration of a class of
- piecewise-continuous functions. The integration is with respect to a
- real variable, because the functions considered do not in general
- allow integration in the complex plane to be defined. The class of
- integrands includes commonly occurring waveforms, such as square
- waves, triangular waves, and the floor function; it also includes the
- signum function. The algorithm can be implemented recursively, and it
- has the property of ensuring that integrals are continuous on domains
- of maximum extent."
+ We describe a new direct algorithm for transforming a linear system of
+ recurrences into an equivalent one with nonsingular leading or
+ trailing matrix. Our algorithm, which is an improvement to the EG
+ elimination method, uses only elementary linear algebra operations
+ (ranks, kernels, and determinants) to produce an equation satisfied by
+ the degress of the solutions with finite support. As a consequence, we
+ can boudn and compute the polynomial and rational solutions of very
+ general linear functional systems such as systems of differential or
+ ($q$)-difference equations."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jeffrey 99]{Jeff99} Jeffrey, D.J.; Labahn, G.; Mohrenschildt, M.v.;
-Rich, A.D.
-``Integration of the signum, piecewise and related functions''
-\verb|cs.uwaterloo.ca/~glabahn/Papers/issac99-2.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Jeff99.pdf|
+\bibitem[Bronstein 96b]{Bro96b} Bronstein, Manuel
+``On the Factorization of Linear Ordinary Differential Operators''
+Mathematics and Computers in Simulation 42 pp 387-389 (1996)
+%\verb|axiom-developer.org/axiom-website/papers/Bro96b.pdf|
abstract = "
- When a computer algebra system has an assumption facility, it is
- possible to distinguish between integration problems with respect to a
- real variable, and those with respect to a complex variable. Here, a
- class of integration problems is defined in which the integrand
- consists of compositions of continuous functions and signum functions,
- and integration is with respect to a real variable. Algorithms are
- given for evaluating such integrals."
+ After reviewing the arithmetic of linear ordinary differential
+ operators, we describe the current status of the factorisation
+ algorithm, specially with respect to factoring over non-algebraically
+ closed constant fields. We also describe recent results from Singer
+ and Ulmer that reduce determining the differential Galois group of an
+ operator to factoring."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Kiymaz 04]{Kiym04} Kiymaz, Onur; Mirasyedioglu, Seref
-``A new symbolic computation for formal integration with exact power series''
-%\verb|axiom-developer.org/axiom-website/Kiym04.pdf|
+\bibitem[Bronstein 96a]{Bro96a} Bronstein, Manuel; Petkovsek, Marko
+``An introduction to pseudo-linear algebra''
+Theoretical Computer Science V157 pp3-33 (1966)
+\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html|
+%\verb|axiom-developer.org/axiom-website/papers/Bro96a.pdf|
abstract = "
- This paper describes a new symbolic algorithm for formal integration
- of a class of functions in the context of exact power series by using
- generalized hypergeometric series and computer algebraic technique."
+ Pseudo-linear algebra is the study of common properties of linear
+ differential and difference operators. We introduce in this paper its
+ basic objects (pseudo-derivations, skew polynomials, and pseudo-linear
+ operators) and describe several recent algorithms on them, which, when
+ applied in the differential and difference cases, yield algorithms for
+ uncoupling and solving systems of linear differential and difference
+ equations in closed form."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Knowles 93]{Know93} Knowles, P.
-``Integration of a class of transcendental liouvillian
-functions with error-functions i''
-Journal of Symbolic Computation Vol 13 pp525-543 (1993)
+\bibitem[Bronstein xb]{Broxb} Bronstein, Manuel
+``Computer Algebra Algorithms for Linear Ordinary Differential and
+Difference equations''
+\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/ecm3.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Broxb.pdf|
+ abstract = "
+ Galois theory has now produced algorithms for solving linear ordinary
+ differential and difference equations in closed form. In addition,
+ recent algorithmic advances have made those algorithms effective and
+ implementable in computer algebra systems. After introducing the
+ relevant parts of the theory, we describe the latest algorithms for
+ solving such equations."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Knowles 95]{Know95} Knowles, P.
-``Integration of a class of transcendental liouvillian
-functions with error-functions ii''
-Journal of Symbolic Computation Vol 16 pp227-241 (1995)
+\bibitem[Bronstein 94]{Bro94} Bronstein, Manuel
+``An improved algorithm for factoring linear ordinary differential
+operators''
+\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html|
+ abstract = "
+ We describe an efficient algorithm for computing the associated
+ equations appearing in the Beke-Schlesinger factorisation method for
+ linear ordinary differential operators. This algorithm, which is based
+ on elementary operations with sets of integers, can be easily
+ implemented for operators of any order, produces several possible
+ associated equations, of which only the simplest can be selected for
+ solving, and often avoids the degenerate case, where the order of the
+ associated equation is less than in the generic case. We conclude with
+ some fast heuristics that can produce some factorizations while using
+ only linear computations."
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Krag09,
- author = "Kragler, R.",
- title = "On Mathematica Program for Poor Man's Integrator Algorithm",
- journal = "Programming and Computer Software",
- volume = "35",
- number = "2",
- pages = "63-78",
- year = "2009",
- issn = "0361-7688",
- paper = "Krag09.pdf",
+\begin{chunk}{ignore}
+\bibitem[Bronstein 90]{Bro90} Bronstein, Manuel
+``On Solutions of Linear Ordinary Differential Equations in their
+Coefficient Field''
+\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html|
+%\verb|axiom-developer.org/axiom-website/papers/Bro90.pdf|
abstract = "
- In this paper by means of computer experiment we study advantages and
- disadvantages of the heuristical method of ``parallel integrator''. For
- this purpose we describe and use implementation of the method in
- Mathematica. In some cases we compare this implementation with the original
- one in Maple."
-}
+ We describe a rational algorithm for finding the denominator of any
+ solution of a linear ordinary differential equation in its coefficient
+ field. As a consequence, there is now a rational algorithm for finding
+ all such solutions when the coefficients can be built up from the
+ rational functions by finitely many algebraic and primitive
+ adjunctions. This also eliminates one of the computational bottlenecks
+ in algorithms that either factor or search for Liouvillian solutions
+ of such equations with Liouvillian coefficients."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lang 93]{Lang93} Lang, S.
-``Algebra''
-Addison-Wesly, New York, 3rd edition 1993
+\bibitem[Bronstein 96]{Bro96} Bronstein, Manuel
+``$\sum^{IT}$ -- A strongly-typed embeddable computer algebra library''
+\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html|
+%\verb|axiom-developer.org/axiom-website/papers/Bro96.pdf|
+ abstract = "
+ We describe the new computer algebra library $\sum^{IT}$ and its
+ underlying design. The development of $\sum^{IT}$ is motivated by the
+ need to provide highly efficient implementations of key algorithms for
+ linear ordinary differential and ($q$)-difference equations to
+ scientific programmers and to computer algebra users, regardless of
+ the programming language or interactive system they use. As such,
+ $\sum^{IT}$ is not a computer algebra system per se, but a library (or
+ substrate) which is designed to be ``plugged'' with minimal efforts
+ into different types of client applications."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Leerawat 02]{Leer02} Leerawat, Utsanee; Laohakosol, Vichian
-``A Generalization of Liouville's Theorem on Integration in Finite Terms''
-\verb|www.mathnet.or.kr/mathnet/kms_tex/113666.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Leer02.pdf|
+\bibitem[Bronstein 99a]{Bro99a} Bronstein, Manuel
+``Solving linear ordinary differential equations over
+$C(x,e^{\int{f(x)dx}})$
+\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html|
+%\verb|axiom-developer.org/axiom-website/papers/Bro99a.pdf|
abstract = "
- A generalization of Liouville's theorem on integration in finite
- terms, by enlarging the class of fields to an extension called
- Ei-Gamma extension is established. This extension includes the
- $\mathcal{E}\mathcal{L}$-elementary extensions of Singer, Saunders and
- Caviness and contains the Gamma function."
+ We describe a new algorithm for computing the solutions in
+ \[F=C(x,e^{\int{f(x)dx}})\] of linear ordinary differential equations
+ with coefficients in $F$. Compared to the general algorithm, our
+ algorithm avoids the computation of exponential solutions of equations
+ with coefficients in $C(x)$, as well as the solving of linear
+ differential systems over $C(x)$. Our method is effective and has been
+ implemented."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Leslie 09]{Lesl09} Leslie, Martin
-``Why you can't integrate exp($x^2$)''
-\verb|math.arizona.edu/~mleslie/files/integrationtalk.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Lesl09.pdf|
+\bibitem[Bronstein 00]{Bro00} Bronstein, Manuel
+``On Solutions of Linear Ordinary Differential Equations in their
+ Coefficient Field''
+\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html|
+%\verb|axiom-developer.org/axiom-website/papers/Bro00.pdf|
+ abstract = "
+ We extend the notion of monomial extensions of differential fields,
+ i.e. simple transcendental extensions in which the polynomials are
+ closed under differentiation, to difference fields. The structure of
+ such extensions provides an algebraic framework for solving
+ generalized linear difference equations with coefficients in such
+ fields. We then describe algorithms for finding the denominator of any
+ solution of those equations in an important subclass of monomial
+ extensions that includes transcendental indefinite sums and
+ products. This reduces the general problem of finding the solutions of
+ such equations in their coefficient fields to bounding their
+ degrees. In the base case, this yields in particular a new algorithm
+ for computing the rational solutions of $q$-difference equations with
+ polynomial coefficients."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lichtblau 11]{Lich11} Lichtblau, Daniel
-``Symbolic definite (and indefinite) integration: methods and open issues''
-ACM Comm. in Computer Algebra Issue 175, Vol 45, No.1 (2011)
-\verb|www.sigsam.org/bulletin/articles/175/issue175.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Lich11.pdf|
+\bibitem[Bronstein 02]{Bro02} Bronstein, Manuel; Lafaille, S\'ebastien
+``Solutions of linear ordinary differential equations in terms of
+special functions''
+\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac2002.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Bro02.pdf|
abstract = "
- The computation of definite integrals presents one with a variety of
- choices. There are various methods such as Newton-Leibniz or Slater's
- convolution method. There are questions such as whether to split or
- merge sums, how to search for singularities on the path of
- integration, when to issue conditional results, how to assess
- (possibly conditional) convergence, and more. These various
- considerations moreover interact with one another in a multitude of
- ways. Herein we discuss these various issues and illustrate with examples."
+ We describe a new algorithm for computing special function solutions
+ of the form $y(x) = m(x)F(\eta(x))$ of second order linear ordinary
+ differential equations, where $m(x)$ is an arbitrary Liouvillian
+ function, $\eta(x)$ is an arbitrary rational function, and $F$
+ satisfies a given second order linear ordinary differential
+ equations. Our algorithm, which is base on finding an appropriate
+ point transformation between the equation defining $F$ and the one to
+ solve, is able to find all rational transformations for a large class
+ of functions $F$, in particular (but not only) the $_0F_1$ and $_1F_1$
+ special functions of mathematical physics, such as Airy, Bessel,
+ Kummer and Whittaker functions. It is also able to identify the values
+ of the parameters entering those special functions, and can be
+ generalized to equations of higher order."
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Liou1833a,
- author = "Liouville, Joseph",
- title = "Premier m\'{e}moire sur la d\'{e}termination des int\'{e}grales
- dont la valeur est alg\'{e}brique",
- journal = "Journal de l'Ecole Polytechnique",
- volume = "14",
- pages = "124-128",
- year = "1833"
-}
+\begin{chunk}{ignore}
+\bibitem[Bronstein 03]{Bro03} Bronstein, Manuel; Trager, Barry M.
+``A Reduction for Regular Differential Systems''
+\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mega2003.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Bro03.pdf|
+ abstract = "
+ We propose a definition of regularity of a linear differential system
+ with coefficients in a monomial extension of a differential field, as
+ well as a global and truly rational (i.e. factorisation-free)
+ iteration that transforms a system with regular finite singularites
+ into an equivalent one with simple finite poles. We then apply our
+ iteration to systems satisfied by bases of algebraic function fields,
+ obtaining algorithms for computing the number of irreducible
+ components and the genus of algebraic curves."
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Liou1833b,
- author = "Liouville, Joseph",
- title = "Second m\'{e}moire sur la d\'{e}termination des int\'{e}grales
- dont la valeur est alg\'{e}brique",
- journal = "Journal de l'Ecole Polytechnique",
- volume = "14",
- pages = "149-193",
- year = "1833"
-}
+\begin{chunk}{ignore}
+\bibitem[Bronstein 03a]{Bro03a} Bronstein, Manuel; Sol\'e, Patrick
+``Linear recurrences with polynomial coefficients''
+\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html|
+%\verb|axiom-developer.org/axiom-website/papers/Bro03a.pdf|
+ abstract = "
+ We relate sequences generated by recurrences with polynomial
+ coefficients to interleaving and multiplexing of sequences generated
+ by recurrences with constant coefficients. In the special case of
+ finite fields, we show that such sequences are periodic and provide
+ linear complexity estimates for all three constructions."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Liouville 1833c]{Lio1833c} Liouville, Joseph
-``Note sur la determination des int\'egrales dont la
-valeur est alg\'ebrique''
-Journal f\"ur die Reine und Angewandte Mathematik,
-Vol 10 pp 247-259, (1833)
+\bibitem[Bronstein 05]{Bro05} Bronstein, Manuel; Li, Ziming; Wu, Min
+``Picard-Vessiot Extensions for Linear Functional Systems''
+\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac2005.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Bro05.pdf|
+ abstract = "
+ Picard-Vessiot extensions for ordinary differential and difference
+ equations are well known and are at the core of the associated Galois
+ theories. In this paper, we construct fundamental matrices and
+ Picard-Vessiot extensions for systems of linear partial functional
+ equations having finite linear dimension. We then use those extensions
+ to show that all the solutions of a factor of such a system can be
+ completed to solutions of the original system."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Liouville 1833d]{Lio1833d} Liouville, Joseph
-``Sur la determination des int\'egrales dont la valeur est
-alg\'ebrique''
-{\sl Journal de l'Ecole Polytechnique}, 14:124-193, 1833
+\bibitem[Davenport 86]{Dav86} Davenport, J.H.
+``The Risch Differential Equation Problem''
+SIAM J. COMPUT. Vol 15, No. 4 1986
+%\verb|axiom-developer.org/axiom-website/papers/Dav86.pdf|
+ abstract = "
+ We propose a new algorithm, similar to Hermite's method for the
+ integration of rational functions, for the resolution of Risch
+ differential equations in closed form, or proving that they have no
+ resolution. By requiring more of the presentation of our differential
+ fields (in particular that the exponentials be weakly normalized), we
+ can avoid the introduction of arbitrary constants which have to be
+ solved for later.
+
+ We also define a class of fields known as exponentially reduced, and
+ show that solutions of Risch differential equations which arise from
+ integrating in these fields satisfy the ``natural'' degree constraints
+ in their main variables, and we conjecture (after Risch and Norman)
+ that this is true in all variables."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Liouville 1835]{Lio1835} Liouville, Joseph
-``M\'emoire sur l'int\'gration d'une classe de fonctions
-transcendentes''
-Journal f\"ur die Reine und Angewandte Mathematik,
-Vol 13(2) pp 93-118, (1835)
+\bibitem[Singer 9]{Sing91.pdf} singer, Michael F.
+``Liouvillian Solutions of Linear Differential Equations with Liouvillian
+ Coefficients''
+J. Symbolic Computation V11 No 3 pp251-273 (1991)
+\verb|www.sciencedirect.com/science/article/pii/S074771710880048X|
+%\verb|axiom-developer.org/axiom-website/papers/Sing91.pdf|
+ abstract = "
+ Let $L(y)=b$ be a linear differential equation with coefficients in a
+ differential field $K$. We discuss the problem of deciding if such an
+ equation has a non-zero solution in $K$ and give a decision procedure
+ in case $K$ is an elementary extension of the field of rational
+ functions or is an algebraic extension of a transcendental liouvillian
+ extension of the field of rational functions We show how one can use
+ this result to give a procedure to find a basis for the space of
+ solutions, liouvillian over $K$, of $L(y)=0$ where $K$ is such a field
+ and $L(y)$ has coefficients in $K$."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Marc 94]{Marc94} Marchisotto, Elena Anne; Zakeri, Gholem-All
-``An Invitation to Integration in Finite Terms''
-College Mathematics Journal Vol 25 No 4 (1994) pp295-308
-\verb|www.rangevoting.org/MarchisottoZint.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Marc94.pdf|
+\bibitem[Von Mohrenschildt 94]{Mohr94} Von Mohrenschildt, Martin
+``Symbolic Solutions of Discontinuous Differential Equations''
+\verb|e-collection.library.ethz.ch/eserv/eth:39463/eth-39463-01.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Mohr94.pdf|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Marik 91]{Mari91} Marik, Jan
-``A note on integration of rational functions''
-\verb|dml.cz/bitstream/handle/10338.dmlcz/126024/MathBohem_116-1991-4_9.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Mari91.pdf|
+\bibitem[Von Mohrenschildt 98]{Mohr98} Von Mohrenschildt, Martin
+``A Normal Form for Function Rings of Piecewise Functions''
+J. Symbolic Computation (1998) Vol 26 pp607-619
+\verb|www.cas.mcmaster.ca/~mohrens/JSC.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Mohr98.pdf|
abstract = "
- Let $P$ and $Q$ be polynomials in one variable with complex coefficients
- and let $n$ be a natural number. Suppose that $Q$ is not constant and
- has only simple roots. Then there is a rational function $\varphi$
- with $\varphi^\prime=P/Q^{n+1}$ if and only if the Wronskian of the
- functions $Q^\prime$, $(Q^2)^\prime,\ldots\,(Q^n)^\prime$,$P$ is
- divisible by $Q$."
+ Computer algebra systems often have to deal with piecewise continuous
+ functions. These are, for example, the absolute value function,
+ signum, piecewise defined functions but also functions that are the
+ supremum or infimum of two functions. We present a new algebraic
+ approach to these types of problems. This paper presents a normal form
+ for a function ring containing piecewise polynomial functions of an
+ expression. The main result is that this normal form can be used to
+ decide extensional equality of two piecewise functions. Also we define
+ supremum and infimum for piecewise functions; in fact, we show that
+ the function ring forms a lattice. Additionally, a method to solve
+ equalities and inequalities in this function ring is
+ presented. Finally, we give a ``user interface'' to the algebraic
+ representation of the piecewise functions."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Moses 76]{Mos76} Moses, Joel
-``An introduction to the Risch Integration Algorithm''
-ACM Proc. 1976 annual conference pp425-428
-%\verb|axiom-developer.org/axiom-website/papers/Mos76.pdf| REF:00048
+\bibitem[Weber 06]{Webe06} Weber, Andreas
+``Quantifier Elimination on Real Closed Fields and Differential Equations''
+\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber2006a.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Webe06.pdf|
+ keywords = "survey",
abstract = "
- Risch's decision procedure for determining the integrability in closed
- form of the elementary functions of the calculus is presented via
- examples. The exponential and logarithmic cases of the algorithsm had
- been implemented for the MACSYMA system several years ago. The
- implementation of the algebraic case of the algorithm is the subject
- of current research."
+ This paper surveys some recent applications of quantifier elimination
+ on real closed fields in the context of differential
+ equations. Although polynomial vector fields give rise to solutions
+ involving the exponential and other transcendental functions in
+ general, many questions can be settled within the real closed field
+ without referring to the real exponential field. The technique of
+ quantifier elimination on real closed fields is not only of
+ theoretical interest, but due to recent advances on the algorithmic
+ side including algorithms for the simplification of quantifier-free
+ formulae the method has gained practical applications, e.g. in the
+ context of computing threshold conditions in epidemic modeling."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Moses 71a]{Mos71a} Moses, Joel
-``Symbolic Integration: The Stormy Decade''
-CACM Aug 1971 Vol 14 No 8 pp548-560
-\verb|www-inst.eecs.berkeley.edu/~cs282/sp02/readings/moses-int.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Mos71a.pdf| REF:00017
+\bibitem[Ulmer 03]{Ulm03} Ulmer, Felix
+``Liouvillian solutions of third order differential equations''
+J. Symbolic COmputations 36 pp 855-889 (2003)
+\verb|www.sciencedirect.com/science/article/pii/S0747717103000658|
+%\verb|axiom-developer.org/axiom-website/papers/Ulm03.pdf|
abstract = "
- Three approaches to symbolic integration in the 1960's are
- described. The first, from artificial intelligence, led to Slagle's
- SAINT and to a large degree to Moses' SIN. The second, from algebraic
- manipulation, led to Monove's implementation and to Horowitz' and
- Tobey's reexamination of the Hermite algorithm for integrating
- rational functions. The third, from mathematics, led to Richardson's
- proof of the unsolvability of the problem for a class of functions and
- for Risch's decision procedure for the elementary functions.
- Generalizations of Risch's algorithm to a class of special
- functions and programs for solving differential equations and for
- finding the definite integral are also described."
+ The Kovacic algorithm and its improvements give explicit formulae for
+ the Liouvillian solutions of second order linear differential
+ equations. Algorithms for third order differential equations also
+ exist, but the tools they use are more sophisticated and the
+ computations more involved. In this paper we refine parts of the
+ algorithm to find Liouvillian solutions of third order equations. We
+ show that,except for four finite groups and a reduction to the second
+ order case, it is possible to give a formula in the imprimitve
+ case. We also give necessary conditions and several simplifications
+ for the computation of the minimal polynomial for the remaining finite
+ set of finite groups (or any known finite group) by extracting
+ ramification information from the character table. Several examples
+ have been constructed, illustrating the possibilities and limitations."
\end{chunk}
+\section{Expression Simplification} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Norman 79]{Nor79} Norman, A.C.; Davenport, J.H.
-``Symbolic Integration -- The Dust Settles?''
-%\verb|axiom-developer.org/axiom-website/papers/Nor79.pdf|
+\bibitem[Carette 04]{Car04} Carette, Jacques
+``Understanding Expression Simplification''
+\verb|www.cas.mcmaster.ca/~carette/publications/simplification.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Car04.pdf|
abstract = "
- By the end of the 1960s it had been shown that a computer could find
- indefinite integrals with a competence exceeding that of typical
- undergraduates. This practical advance was backed up by algorithmic
- interpretations of a number of clasical results on integration, and by
- some significant mathematical extensions to these same results. At
- that time it would have been possible to claim that all the major
- barriers in the way of a complete system for automated analysis had
- been breached. In this paper we survey the work that has grown out of
- the above-mentioned early results, showing where the development has
- been smooth and where it has spurred work in seemingly unrelated fields."
+ We give the first formal definition of the concept of {\sl
+ simplification} for general expressions in the context of Computer
+ Algebra Systems. The main mathematical tool is an adaptation of the
+ theory of Minimum Description Length, which is closely related to
+ various theories of complexity, such as Kolmogorov Complexity and
+ Algorithmic Information Theory. In particular, we show how this theory
+ can justify the use of various ``magic constants'' for deciding
+ between some equivalent representations of an expression, as found in
+ implementations of simplification routines."
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Ostrowski 46]{Ost46} Ostrowski, A.
-``Sur l'int\'egrabilit\'e \'el\'ementaire de quelques classes
-d'expressions''
-Comm. Math. Helv., Vol 18 pp 283-308, (1946)
-% REF:00008
+\section{Integration} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@TechReport{Kalt84b,
+ author = "Kaltofen, E.",
+ title = "The Algebraic Theory of Integration",
+ institution = "RPI",
+ address = "Dept. Comput. Sci., Troy, New York",
+ year = "1984",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/84/Ka84_integration.pdf",
+ paper = "Kalt84b.pdf"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Raab 12]{Raab12} Raab, Clemens G.
-``Definite Integration in Differential Fields''
-\verb|www.risc.jku.at/publications/download/risc_4583/PhD_CGR.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Raab12.pdf|
- abstract = "
- The general goal of this thesis is to investigate and develop computer
- algebra tools for the simplification resp. evaluation of definite
- integrals. One way of finding the value of a def- inite integral is
- via the evaluation of an antiderivative of the integrand. In the
- nineteenth century Joseph Liouville was among the first who analyzed
- the structure of elementary antiderivatives of elementary functions
- systematically. In the early twentieth century the algebraic structure
- of differential fields was introduced for modeling the differential
- properties of functions. Using this framework Robert H. Risch
- published a complete algorithm for transcendental elementary
- integrands in 1969. Since then this result has been extended to
- certain other classes of integrands as well by Michael F. Singer,
- Manuel Bronstein, and several others. On the other hand, if no
- antiderivative of suitable form is available, then linear relations
- that are satisfied by the parameter integral of interest may be found
- based on the principle of parametric integration (often called
- differentiating under the integral sign or creative telescoping).
-
- The main result of this thesis extends the results mentioned above to
- a complete algo- rithm for parametric elementary integration for a
- certain class of integrands covering a majority of the special
- functions appearing in practice such as orthogonal polynomials,
- polylogarithms, Bessel functions, etc. A general framework is provided
- to model those functions in terms of suitable differential fields. If
- the integrand is Liouvillian, then the present algorithm considerably
- improves the efficiency of the corresponding algorithm given by Singer
- et al. in 1985. Additionally, a generalization of Czichowski’s
- algorithm for computing the logarithmic part of the integral is
- presented. Moreover, also partial generalizations to include other
- types of integrands are treated.
+\bibitem[Adamchik xx]{Adamxx} Adamchik, Victor
+``Definite Integration''
+\verb|www.cs.cmu.edu/~adamchik/articles/integr/mj.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Adamxx.pdf|
- As subproblems of the integration algorithm one also has to find
- solutions of linear or- dinary differential equations of a certain
- type. Some contributions are also made to solve those problems in our
- setting, where the results directly dealing with systems of
- differential equations have been joint work with Moulay A. Barkatou.
+\end{chunk}
- For the case of Liouvillian integrands we implemented the algorithm in
- form of our Mathematica package Integrator. Parts of the
- implementation also deal with more general functions. Our procedures
- can be applied to a significant amount of the entries in integral
- tables, both indefinite and definite integrals. In addition, our
- procedures have been successfully applied to interesting examples of
- integrals that do not appear in these tables or for which current
- standard computer algebra systems like Mathematica or Maple do not
- succeed. We also give examples of how parameter integrals coming from
- the work of other researchers can be solved with the software, e.g.,
- an integral arising in analyzing the entropy of certain processes."
+\begin{chunk}{ignore}
+\bibitem[Adamchik 97]{Adam97} Adamchik, Victor
+``A Class of Logarithmic Integrals''
+\verb|www.cs.cmu.edu/~adamchik/articles/issac/issac97.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Adam97.pdf|
+ abstract = "
+ A class of definite integrals involving cyclotomic polynomials and
+ nested logarithms is considered. The results are given in terms of
+ derivatives of the Hurwitz Zeta function. Some special cases for which
+ such derivatives can be expressed in closed form are also considered."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Raab 13]{Raab13} Raab, Clemens G.
-``Generalization of Risch's Algorithm to Special Functions''
-\verb|arxiv.org/pdf/1305.1481|
-%\verb|axiom-developer.org/axiom-website/papers/Raab13.pdf|
+\bibitem[Avgoustis 77]{Avgo77} Avgoustis, Ioannis Dimitrios
+``Definite Integration using the Generalized Hypergeometric Functions''
+\verb|dspace.mit.edu/handle/1721.1/16269|
+%\verb|axiom-developer.org/axiom-websitep/papers/Avgo77.pdf|
abstract = "
- Symbolic integration deals with the evaluation of integrals in closed
- form. We present an overview of Risch's algorithm including recent
- developments. The algorithms discussed are suited for both indefinite
- and definite integration. They can also be used to compute linear
- relations among integrals and to find identities for special functions
- given by parameter integrals. The aim of this presentation is twofold:
- to introduce the reader to some basic idea of differential algebra in
- the context of integration and to raise awareness in the physics
- community of computer algebra algorithms for indefinite and definite
- integration."
+ A design for the definite integration of approximately fifty Special
+ Functions is described. The Generalized Hypergeometric Functions are
+ utilized as a basis for the representation of the members of the above
+ set of Special Functions. Only a relatively small number of formulas
+ that generally involve Generalized Hypergeometric Functions are
+ utilized for the integration stage. A last and crucial stage is
+ required for the integration process: the reduction of the Generalized
+ Hypergeometric Function to Elementary and/or Special Functions.
+
+ The result of an early implementation which involves Laplace
+ transforms are given and some actual examples with their corresponding
+ timing are provided."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Raab xx]{Raabxx} Raab, Clemens G.
-``Integration in finite terms for Liouvillian functions''
-\verb|www.mmrc.iss.ac.cn/~dart4/posters/Raab.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Raabxx.pdf|
+\bibitem[Baddoura 89]{Bad89} Baddoura, Jamil
+``A Dilogarithmic Extension of Liouville's Theorem on Integration in Finite
+ Terms''
+\verb|www.dtic.mil/dtic/tr/fulltext/u2/a206681.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Bad89.pdf|
abstract = "
- Computing integrals is a common task in many areas of science,
- antiderivatives are one way to accomplish this. The problem of
- integration in finite terms can be states as follows. Given a
- differential field $(F,D)$ and $f \in F$, compute $g$ in some
- elementary extension of $(F,D)$ such that $Dg = f$ if such a $g$
- exists.
-
- This problem has been solved for various classes of fields $F$. For
- rational functions $(C(x), \frac{d}{dx})$ such a $g$ always exists and
- algorithms to compute it are known already for a long time. In 1969
- Risch published an algorithm that solves this problem when $(F,D)$ is
- a transcendental elementary extension of $(C(x),\frac{d}{dx})$. Later
- this has been extended towards integrands being Liouvillian functions
- by Singer et. al. via the use of regular log-explicit extensions of
- $(C(x),\frac{d}{dx})$. Our algorithm extends this to handling
- transcendental Liouvillian extensions $(F,D)$ of $(C,0)$ directly
- without the need to embed them into log-explicit extensions. For
- example, this means that
- \[\int{(z-x)x^{z-1}e^{-x}dx} = x^ze^{-x}\]
- can be computed without including log(x) in the differential field."
+ The result obtained generalizes Liouville's Theorem by allowing, in
+ addition to the elementary functions, dilogarithms to appear in the
+ integral of an elementary function. The basic conclusion is that an
+ associated function to the dilogarihm, if dilogarithms appear in the
+ integral, appears linearly, with logarithms appearing in a non-linear
+ way."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Rich 09]{Rich09} Rich, A.D.; Jeffrey, D.J.
-``A Knowledge Repository for Indefinite Integration Based on Transformation Rules''
-\verb|www.apmaths.uwo.ca/~arich/A%2520Rule-based%2520Knowedge%2520Repository.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Rich09.pdf|
+\bibitem[Baddoura 94]{Bad94} Baddoura, Mohamed Jamil
+``Integration in Finite Terms with Elementary Functions and Dilogarithms''
+\verb|dspace.mit.edu/bitstream/handle/1721.1/26864/30757785.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Bad94.pdf|
abstract = "
- Taking the specific problem domain of indefinite integration, we
- describe the on-going development of a repository of mathematical
- knowledge based on transformation rules. It is important that the
- repository be not confused with a look-up table. The database of
- transformation rules is at present encoded in Mathematica, but this is
- only one convenient form of the repository, and it could be readily
- translated into other formats. The principles upon which the set of
- rules is compiled is described. One important principle is
- minimality. The benefits of the approach are illustrated with
- examples, and with the results of comparisons with other approaches."
+ In this thesis, we report on a new theorem that generalizes
+ Liouville's theorem on integration in finite terms. The new theorem
+ allows dilogarithms to occur in the integral in addition to elementary
+ functions. The proof is base on two identities for the dilogarithm,
+ that characterize all the possible algebraic relations among
+ dilogarithms of functions that are built up from the rational
+ functions by taking transcendental exponentials, dilogarithms, and
+ logarithms."
\end{chunk}
-\begin{chunk}{axiom.bib}
-@techreport{Risc68,
- author = "Risch, Robert",
- title = "On the integration of elementary functions which are built up
- using algebraic operations",
- type = "Research Report",
- number = "SP-2801/002/00",
- institution = "System Development Corporation, Santa Monica, CA, USA",
- year = "1968"
-}
+\begin{chunk}{ignore}
+\bibitem[Baddoura 10]{Bad10} Baddoura, Jamil
+``A Note on Symbolic Integration with Polylogarithms''
+J. Math Vol 8 pp229-241 (2011)
+%\verb|axiom-developer.org/axiom-website/papers/Bad10.pdf|
+ abstract = "
+ We generalize partially Liouville's theorem on integration in finite
+ terms to allow polylogarithms of any order to occur in the integral in
+ addition to elementary functions. The result is a partial
+ generalization of a theorem proved by the author for the
+ dilogarithm. It is also a partial proof of a conjecture postulated by
+ the author in 1994. The basic conclusion is that an associated
+ function to the nth polylogarithm appears linearly with logarithms
+ appearing possibly in a polynomial way with non-constant coefficients."
\end{chunk}
-\begin{chunk}{axiom.bib}
-@techreport{Risc69a,
- author = "Risch, Robert",
- title = "Further results on elementary functions",
- type = "Research Report",
- number = "RC-2042",
- institution = "IBM Research, Yorktown Heights, NY, USA",
- year = "1969"
-
-}
+\begin{chunk}{ignore}
+\bibitem[Bajpai 70]{Bajp70} Bajpai, S.D.
+``A contour integral involving legendre polynomial and Meijer's G-function''
+\verb|link.springer.com/article/10.1007/BF03049565|
+%\verb|axiom-developer.org/axiom-website/papers/Bajp70.pdf|
+ abstract = "
+ In this paper a countour integral involving Legendre polynomial and
+ Meijer's G-function is evaluated. the integral is of general character
+ and it is a generalization of results recently given by Meijer,
+ MacRobert and others. An integral involving regular radial Coulomb
+ wave function is also obtained as a particular case."
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Risc69b,
- author = "Risch, Robert",
- title = "The problem of integration in finite terms",
- journal = "Transactions of the American Mathematical Society",
- volume = "139",
- year = "1969",
- pages = "167-189",
- paper = "Ris69b.pdf",
- abstract = "This paper deals with the problem of telling whether a
- given elementary function, in the sense of analysis, has an elementary
- indefinite integral."
-}
+\begin{chunk}{ignore}
+\bibitem[Bronstein 89]{Bro89a} Bronstein, M.
+``An Algorithm for the Integration of Elementary Functions''
+Lecture Notes in Computer Science Vol 378 pp491-497 (1989)
+%\verb|axiom-developer.org/axiom-website/papers/Bro89a.pdf|
+ abstract = "
+ Trager (1984) recently gave a new algorithm for the indefinite
+ integration of algebraic functions. His approach was ``rational'' in
+ the sense that the only algebraic extension computed in the smallest
+ one necessary to express the answer. We outline a generalization of
+ this approach that allows us to integrate mixed elementary
+ functions. Using only rational techniques, we are able to normalize
+ the integrand, and to check a necessary condition for elementary
+ integrability."
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Risc70,
- author = "Risch, Robert",
- title = "The Solution of the Problem of Integration in Finite Terms",
- journal = "Bull. AMS",
- year = "1970",
- issn = "0002-9904",
- volume = "76",
- number = "3",
- pages = "605-609",
- paper = "Risc70.pdf",
+\begin{chunk}{ignore}
+\bibitem[Bronstein 90a]{Bro90a} Bronstein, Manuel
+``Integration of Elementary Functions''
+J. Symbolic Computation (1990) 9, pp117-173 September 1988
+%\verb|axiom-developer.org/axiom-website/papers/Bro90a.pdf|
abstract = "
- The problem of integration in finite terms asks for an algorithm for
- deciding whether an elementary function has an elementary indefinite
- integral and for finding the integral if it does. ``Elementary'' is
- used here to denote those functions build up from the rational
- functions using only exponentiation, logarithms, trigonometric,
- inverse trigonometric and algebraic operations. This vaguely worded
- question has several precise, but inequivalent formulations. The
- writer has devised an algorithm which solves the classical problem of
- Liouville. A complete account is planned for a future publication. The
- present note is intended to indiciate some of the ideas and techniques
- involved."
-}
+ We extend a recent algorithm of Trager to a decision procedure for the
+ indefinite integration of elementary functions. We can express the
+ integral as an elementary function or prove that it is not
+ elementary. We show that if the problem of integration in finite terms
+ is solvable on a given elementary function field $k$, then it is
+ solvable in any algebraic extension of $k(\theta)$, where $\theta$ is
+ a logarithm or exponential of an element of $k$. Our proof considers
+ an element of such an extension field to be an algebraic function of
+ one variable over $k$.
+
+ In his algorithm for the integration of algebraic functions, Trager
+ describes a Hermite-type reduction to reduce the problem to an
+ integrand with only simple finite poles on the associated Riemann
+ surface. We generalize that technique to curves over liouvillian
+ ground fields, and use it to simplify our integrands. Once the
+ multipe finite poles have been removed, we use the Puiseux expansions
+ of the integrand at infinity and a generalization of the residues to
+ compute the integral. We also generalize a result of Rothstein that
+ gives us a necessary condition for elementary integrability, and
+ provide examples of its use."
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Risc79,
- author = "Risch, Robert",
- title = "Algebraic properties of the elementary functions of analysis",
- journal = "American Journal of Mathematics",
- volume = "101",
- pages = "743-759",
- year = "1979"
+@article{Bron90c,
+ author = "Bronstein, Manuel",
+ title = "On the integration of elementary functions",
+ journal = "Journal of Symbolic Computation",
+ volume = "9",
+ number = "2",
+ pages = "117-173",
+ year = "1990",
+ month = "February"
}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Ritt 48]{Ritt48} Ritt, J.F.
-``Integration in Finite Terms''
-Columbia University Press, New York 1948
-% REF:00046
+\bibitem[Bronstein 93]{REF-BS93} Bronstein, Manuel; Salvy, Bruno
+``Full partial fraction decomposition of rational functions''
+In Bronstein [Bro93] pp157-160 ISBN 0-89791-604-2 LCCN QA76.95 I59 1993
+\verb|www.acm.org/pubs/citations/proceedings/issac/164081/|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Rosenlicht 68]{Ro68} Rosenlicht, Maxwell
-``Liouville's Theorem on Functions with Elementary Integrals''
-Pacific Journal of Mathematics Vol 24 No 1 (1968)
-\verb|msp.org/pjm/1968/24-1/pjm-v24-n1-p16-p.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Ro68.pdf| REF:00047
+\bibitem[Bronstein 90]{Bro90b} Bronstein, Manuel
+``A Unification of Liouvillian Extensions''
+%\verb|axiom-developer.org/axiom-website/papers/Bro90b.pdf|
abstract = "
- Defining a function with one variable to be elemetary if it has an
- explicit representation in terms of a finite number of algebraic
- operations, logarithms, and exponentials. Liouville's theorem in its
- simplest case says that if an algebraic function has an elementary
- integral then the latter is itself an algebraic function plus a sum of
- constant multiples of logarithms of algebraic functions. Ostrowski has
- generalized Liouville's results to wider classes of meromorphic
- functions on regions of the complex plane and J.F. Ritt has given the
- classical account of the entire subject in his Integraion in Finite
- Terms, Columbia University Press, 1948. In spite of the essentially
- algebraic nature of the problem, all proofs so far have been analytic.
- This paper gives a self contained purely algebraic exposition of the
- probelm, making a few new points in addition to the resulting
- simplicity and generalization."
+ We generalize Liouville's theory of elementary functions to a larger
+ class of differential extensions. Elementary, Liouvillian and
+ trigonometric extensions are all special cases of our extensions. In
+ the transcendental case, we show how the rational techniques of
+ integration theory can be applied to our extensions, and we give a
+ unified presentation which does not require separate cases for
+ different monomials."
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Rose72,
- author = "Rosenlicht, Maxwell",
- title = "Integration in finite terms",
- journal = "American Mathematical Monthly",
- year = "1972",
- volume = "79",
- pages = "963-972",
- paper = "Rose72.pdf"
+@book{Bron97,
+ author = "Bronstein, Manuel",
+ title = "Symbolic Integration I--Transcendental Functions",
+ publisher = "Springer, Heidelberg",
+ year = "1997",
+ isbn = "3-540-21493-3",
+ url = "http://evil-wire.org/arrrXiv/Mathematics/Bronstein,_Symbolic_Integration_I,1997.pdf",
+ paper = "Bron97.pdf"
}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Rothstein 76]{Ro76} Rothstein, Michael
-``Aspects of symbolic integration and simplifcation of exponential
-and primitive functions''
-PhD thesis, University of Wisconsin-Madison (1976)
-\verb|www.cs.kent.edu/~rothstei/dis.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Ro76.pdf| REF:00051
- abstract = "
- In this thesis we cover some aspects of the theory necessary to obtain
- a canonical form for functions obtained by integration and
- exponentiation from the set of rational functions.
-
- These aspects include a new algorithm for symbolic integration of
- functions involving logarithms and exponentials which avoids
- factorization of polynomials in those cases where algebraic extension
- of the constant field is not required, avoids partial fraction
- decompositions, and only solves linear systems with a small number of
- unknowns.
-
- We have also found a theorem which states, roughly speaking, that if
- integrals which can be represented as logarithms are represented as
- such, the only algebraic dependence that a new exponential or
- logarithm can satify is given by the law of exponents or the law of
- logarithms."
-
-\end{chunk}
-
-\begin{chunk}{ignore}
-\bibitem[Rothstein 76a]{Ro76a} Rothstein, Michael; Caviness, B.F.
-``A structure theorem for exponential and primitive functions: a preliminary
- report''
-ACM Sigsam Bulletin Vol 10 Issue 4 (1976)
-%\verb|axiom-developer.org/axiom-website/papers/Ro76a.pdf|
- abstract = "
- In this paper a generalization of the Risch Structure Theorem is reported.
- The generalization applies to fields $F(t_1,\ldots,t_n)$ where $F$
- is a differential field (in our applications $F$ will be a finitely
- generated extension of $Q$, the field of rational numbers) and each $t_i$
- is either algebraic over $F_{i-1}=F(t_1,\ldots,t_{i-1})$, is an
- exponential of an element in $F_{i-1}$, or is an integral of an element
- in $F_{i-1}$. If $t_i$ is an integral and can be expressed using
- logarithms, it must be so expressed for the generalized structure
- theorem to apply."
+\bibitem[Bronstein 05a]{Bro05a} Bronstein, Manuel
+``The Poor Man's Integrator, a parallel integration heuristic''
+\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/pmint.txt|
+\verb|www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/examples|
+%\verb|axiom-developer.org/axiom-website/papers/Bro05a.txt|
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Rothstein 76b]{Ro76b} Rothstein, Michael; Caviness, B.F.
-``A structure theorem for exponential and primitive functions''
-SIAM J. Computing Vol 8 No 3 (1979)
-%\verb|axiom-developer.org/axiom-website/papers/Ro76b.pdf| REF:00104
+\begin{chunk}{axiom.bib}
+@article{Bron06,
+ author = "Bronstein, M.",
+ title = "Parallel integration",
+ journal = "Programming and Computer Software",
+ year = "2006",
+ issn = "0361-7688",
+ volume = "32",
+ number = "1",
+ doi = "10.1134/S0361768806010075",
+ url = "http://dx.doi.org/10.1134/S0361768806010075",
+ publisher = "Nauka/Interperiodica",
+ pages = "59-60",
+ paper = "Bron06.pdf",
abstract = "
- In this paper a new theorem is proved that generalizes a result of
- Risch. The new theorem gives all the possible algebraic relationships
- among functions that can be built up from the rational functions by
- algebraic operations, by taking exponentials, and by integration. The
- functions so generated are called exponential and primitive functions.
- From the theorem an algorithm for determining algebraic dependence
- among a given set of exponential and primitive functions is derived.
- The algorithm is then applied to a problem in computer algebra."
+ Parallel integration is an alternative method for symbolic
+ integration. While also based on Liouville's theorem, it handles all
+ the generators of the differential field containing the integrand ``in
+ parallel'', i.e. all at once rather than considering only the topmost
+ one in a recursive fasion. Although it still contains heuristic
+ aspects, its ease of implementation, speed, high rate of success, and
+ ability to integrate functions that cannot be handled by the Risch
+ algorithm make it an attractive alternative."
+}
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Roth77,
- author = "Rothstein, Michael",
- title = "A new algorithm for the integration of exponential and
- logarithmic functions",
- journal = "Proceedings of the 1977 MACSYMA Users Conference",
- year = "1977",
- pages = "263-274",
- publisher = "NASA Pub CP-2012"
+@article{Bron07,
+ author = "Bronstein, Manuel",
+ title = "Structure theorems for parallel integration",
+ journal = "Journal of Symbolic Computation",
+ volume = "42",
+ number = "7",
+ pages = "757-769",
+ year = "2007",
+ month = "July",
+ paper = "Bron07.pdf",
+ abstract = "
+ We introduce structure theorems that refine Liouville's Theorem on
+ integration in closed form for general derivations on multivariate
+ rational function fields. By predicting the arguments of the new
+ logarithms that an appear in integrals, as well as the denominator of
+ the rational part, those theorems provide theoretical backing for the
+ Risch-Norman integration method. They also generalize its applicability
+ to non-monomial extensions, for example the Lambert W function."
}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Seidenberg 58]{Sei58} Seidenberg, Abraham
-``Abstract differential algebra and the analytic case''
-Proc. Amer. Math. Soc. Vol 9 pp159-164 (1958)
+\bibitem[Charlwood 07]{Charl07} Charlwood, Kevin
+``Integration on Computer Algebra Systems''
+The Electronic J of Math. and Tech. Vol 2, No 3, ISSN 1933-2823
+\verb|12000.org/my_notes/ten_hard_integrals/paper.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Charl07.pdf|
+ abstract = "
+ In this article, we consider ten indefinite integrals and the ability
+ of three computer algebra systems (CAS) to evaluate them in
+ closed-form, appealing only to the class of real, elementary
+ functions. Although these systems have been widely available for many
+ years and have undergone major enhancements in new versions, it is
+ interesting to note that there are still indefinite integrals that
+ escape the capacity of these systems to provide antiderivatves. When
+ this occurs, we consider what a user may do to find a solution with
+ the aid of a CAS."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Seidenberg 69]{Sei69} Seidenberg, Abraham
-``Abstract differential algebra and the analytic case. II''
-Proc. Amer. Math. Soc. Vol 23 pp689-691 (1969)
+\bibitem[Charlwood 08]{Charl08} Charlwood, Kevin
+``Symbolic Integration Problems''
+\verb|www.apmaths.uwo.ca/~arich/IndependentTestResults/CharlwoodIntegrationProblems.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Charl08.pdf|
+ abstract = "
+ A list of the 50 example integration problems from Kevin Charlwood's 2008
+ article ``Integration on Computer Algebra Systems''. Each integral along
+ with its optimal antiderivative (that is, the best antiderivative found
+ so far) is shown."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Singer 85]{Sing85} Singer, M.F.; Saunders, B.D.; Caviness, B.F.
-``An extension of Liouville's theorem on integration in finite terms''
-SIAM J. of Comp. Vol 14 pp965-990 (1985)
-\verb|www4.ncsu.edu/~singer/papers/singer_saunders_caviness.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Sing85.pdf|
+\bibitem[Cherry 84]{Che84} Cherry, G.W.
+``Integration in Finite Terms with Special Functions: The Error Function''
+J. Symbolic Computation (1985) Vol 1 pp283-302
+%\verb|axiom-developer.org/axiom-website/papers/Che84.pdf|
abstract = "
- In Part 1 of this paper, we give an extension of Liouville's Theorem
- and give a number of examples which show that integration with special
- functions involves some phenomena that do not occur in integration
- with the elementary functions alone. Our main result generalizes
- Liouville's Theorem by allowing, in addition to the elementary
- functions, special functions such as the error function, Fresnel
- integrals and the logarithmic integral (but not the dilogarithm or
- exponential integral) to appear in the integral of an elementary
- function. The basic conclusion is that these functions, if they
- appear, appear linearly. We give an algorithm which decides if an
- elementary function, built up using only exponential functions and
- rational operations has an integral which can be expressed in terms of
- elementary functions and error functions."
+ A decision procedure for integrating a class of transcendental
+ elementary functions in terms of elementary functions and error
+ functions is described. The procedure consists of three mutually
+ exclusive cases. In the first two cases a generalised procedure for
+ completing squares is used to limit the error functions which can
+ appear in the integral of a finite number. This reduces the problem
+ to the solution of a differential equation and we use a result of
+ Risch (1969) to solve it. The third case can be reduced to the
+ determination of what we have termed $\sum$-decompositions. The resutl
+ presented here is the key procuedure to a more general algorithm which
+ is described fully in Cherry (1983)."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Slagle 61]{Slag61} Slagle, J.
-``A heuristic program that solves symbolic integration problems in
- freshman calculus''
-Ph.D Diss. MIT, May 1961; also Computers and Thought, Feigenbaum and Feldman.
-% REF:00014
+\bibitem[Cherry 86]{Che86} Cherry, G.W.
+``Integration in Finite Terms with Special Functions:
+The Logarithmic Integral''
+SIAM J. Comput. Vol 15 pp1-21 February 1986
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Terelius 09]{Tere09} Terelius, Bjorn
-``Symbolic Integration''
-%\verb|axiom-developer.org/axiom-website/papers/Tere09.pdf|
+\bibitem[Cherry 89]{Che89} Cherry, G.W.
+``An Analysis of the Rational Exponential Integral''
+SIAM J. Computing Vol 18 pp 893-905 (1989)
+%\verb|axiom-developer.org/axiom-website/papers/Che89.pdf|
abstract = "
- Symbolic integration is the problem of expressing an indefinite integral
- $\int{f}$ of a given function $f$ as a finite combination $g$ of elementary
- functions, or more generally, to determine whether a certain class of
- functions contains an element $g$ such that $g^\prime = f$.
-
- In the first part of this thesis, we compare different algorithms for
- symbolic integration. Specifically, we review the integration rules
- taught in calculus courses and how they can be used systematically to
- create a reasonable, but somewhat limited, integration method. Then we
- present the differential algebra required to prove the transcendental
- cases of Risch's algorithm. Risch's algorithm decides if the integral
- of an elementary function is elementary and if so computes it. The
- presentation is mostly self-contained and, we hope, simpler than
- previous descriptions of the algorithm. Finally, we describe
- Risch-Norman's algorithm which, although it is not a decision
- procedure, works well in practice and is considerably simpler than the
- full Risch algorithm.
-
- In the second part of this thesis, we briefly discuss an
- implementation of a computer algebra system and some of the
- experiences it has given us. We also demonstrate an implementation of
- the rule-based approach and how it can be used, not only to compute
- integrals, but also to generate readable derivations of the results."
+ In this paper an algorithm is presented for integrating expressions of
+ the form $\int{ge^f~dx}$, where $f$ and $g$ are rational functions of
+ $x$, in terms of a class of special functions called the special
+ incomplete $\Gamma$ functions. This class of special functions
+ includes the exponential integral, the error functions, the sine and
+ cosing integrals, and the Fresnel integrals. The algorithm presented
+ here is an improvement over those published previously for integrating
+ with special functions in the following ways: (i) This algorithm
+ combines all the above special functions into one algorithm, whereas
+ previously they were treated separately, (ii) Previous algorithms
+ require that the underlying field of constants be algebraically
+ closed. This algorithm, however, works over any field of
+ characteristic zero in which the basic field operations can be carried
+ out. (iii) This algorithm does not rely on Risch's solution of the
+ differential equation $y^\prime + fy = g$. Instead, a more direct
+ method of undetermined coefficients is used."
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Trag76,
- author = "Trager, Barry",
- title = "Algebraic factoring and rational function integration",
- journal = "Proceedings of SYMSAC'76",
- year = "1976",
- pages = "219-226",
- paper = "Trag76.pdf",
+\begin{chunk}{ignore}
+\bibitem[Churchill 06]{Chur06} Churchill, R.C.
+``Liouville's Theorem on Integration Terms of Elementary Functions''
+\verb|www.sci.ccny.cuny.edu/~ksda/PostedPapers/liouv06.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Chur06.pdf|
abstract = "
- This paper presents a new, simple, and efficient algorithm for
- factoring polynomials in several variables over an algebraic number
- field. The algorithm is then used interatively to construct the
- splitting field of a polynomial over the integers. Finally the
- factorization and splitting field algorithms are applied to the
- problem of determining the transcendental part of the integral of a
- rational function. In particular, a constructive procedure is given
- for finding a least degree extension field in which the integral can
- be expressed."
-}
+ This talk should be regarded as an elementary introduction to
+ differential algebra. It culminates in a purely algebraic proof, due
+ to M. Rosenlicht, of an 1835 theorem of Liouville on the existence of
+ ``elementary'' integrals of ``elementary'' functions. The precise
+ meaning of elementary will be specified. As an application of that
+ theorem we prove that the indefinite integral $\int{e^{x^2}}~dx$
+ cannot be expressed in terms of elementary functions.
+ \begin{itemize}
+ \item Preliminaries on Meromorphic Functions
+ \item Basic (Ordinary) Differential Algebra
+ \item Differential Ring Extensions with No New Constants
+ \item Extending Derivations
+ \item Integration in Finite Terms
+ \end{itemize}"
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Trager 76a]{Tr76a} Trager, Barry Marshall
-``Algorithms for Manipulating Algebraic Functions''
-MIT Master's Thesis.
-\verb|www.dm.unipi.it/pages/gianni/public_html/Alg-Comp/fattorizzazione-EA.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Tr76a.pdf| REF:00050
- abstract = "
- Given a base field $k$, of characteristic zero, with effective
- procedures for performing arithmetic and factoring polynomials, this
- thesis presents algorithms for extending those capabilities to
- elements of a finite algebraic symbolic manipulation system. An
- algebraic factorization algorithm along with a constructive version of
- the primitive element theorem is used to construct splitting fields of
- polynomials. These fields provide a context in which we can operate
- symbolically with all the roots of a set of polynomials. One
- application for this capability is rational function integrations.
- Previously presented symbolic algorithms concentrated on finding the
- rational part and were only able to compute the complete
- integral in special cases. This thesis presents an algorithm for
- finding an algebraic extension field of least degreee in which the
- integral can be expressed, and then constructs the integral in that
- field. The problem of algebraic function integration is also
- examined, and a highly efficient procedure is presented for generating
- the algebraic part of integrals whose function fields are defined by a
- single radical extension of the rational functions."
+\bibitem[Davenport 79b]{Dav79b} Davenport, James Harold
+``On the Integration of Algebraic Functions''
+Springer-Verlag Lecture Notes in Computer Science 102
+ISBN 0-387-10290-6
\end{chunk}
-\begin{chunk}{axiom.bib}
-@phdthesis{Trag84,
- author = "Trager, Barry",
- title = "On the integration of algebraic functions",
- school = "MIT",
- year = "1984",
- url = "http://www.dm.unipi.it/pages/gianni/public_html/Alg-Comp/thesis.pdf",
- paper = "Trag76.pdf",
+\begin{chunk}{ignore}
+\bibitem[Davenport 79c]{Dav79c} Davenport, J. H.
+``Algorithms for the Integration of Algebraic Functions''
+Lecture Notes in Computer Science V 72 pp415-425 (1979)
+%\verb|axiom-developer.org/axiom-website/papers/Dav79c.pdf|
abstract = "
- We show how the ``rational'' approach for integrating algebraic
- functions can be extended to handle elementary functions. The
- resulting algorithm is a practical decision procedure for determining
- whether a given elementary function has an elementary antiderivative,
- and for computing it if it exists."
-}
+ The problem of finding elementary integrals of algebraic functions has
+ long been recognized as difficult, and has sometimes been thought
+ insoluble. Risch stated a theorem characterising the integrands with
+ elementary integrals, and we can use the language of algebraic
+ geometry and the techniques of Davenport to yield an algorithm that will
+ always produce the integral if it exists. We explain the difficulty in
+ the way of extending this algorithm, and outline some ways of solving
+ it. Using work of Manin we are able to solve the problem in all cases
+ where the algebraic expressions depend on a parameter as well as on
+ the variable of integration."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[W\"urfl 07]{Wurf07} W\"urfl, Andreas
-``Basic Concepts of Differential Algebra''
-\verb|www14.in.tum.de/konferenzen/Jass07/courses/1/Wuerfl/wuerfl_paper.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Wurf07.pdf|
+\bibitem[Davenport 82a]{Dav82a} Davenport, J.H.
+``The Parallel Risch Algorithm (I)
+%\verb|axiom-developer.org/axiom-website/papers/Dav82a.pdf|
abstract = "
- Modern computer algebra systems symbolically integrate a vast variety
- of functions. To reveal the underlying structure it is necessary to
- understand infinite integration not only as an analytical problem but
- as an algebraic one. Introducing the differential field of elementary
- functions we sketch the mathematical tools like Liouville's Principle
- used in modern algorithms. We present Hermite's method for integration
- of rational functions as well as the Rothstein/Trager method for
- rational and for elementary functions. Further applications of the
- mentioned algorithms in the field of ODE's conclude this paper."
+ In this paper we review the so-called ``parallel Risch'' algorithm for
+ the integration of transcendental functions, and explain what the
+ problems with it are. We prove a positive result in the case of
+ logarithmic integrands."
\end{chunk}
-\subsection{Partial Fraction Decomposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Angell]{Angell} Angell, Tom
-``Guidelines for Partial Fraction Decomposition''
-\verb|www.math.udel.edu/~angell/partfrac_I.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Angell.pdf|
+\bibitem[Davenport 82]{Dav82} Davenport, J.H.
+``On the Parallel Risch Algorithm (III): Use of Tangents''
+SIGSAM V16 no. 3 pp3-6 August 1982
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Laval 08]{Lava08} Laval, Philippe B.
-``Partial Fractions Decomposition''
-\verb|www.math.wisc.edu/~park/Fall2011/integration/Partial%20Fraction.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Lava08.pdf|
+\bibitem[Davenport 03]{Dav03} Davenport, James H.
+``The Difficulties of Definite Integration''
+\verb|www.researchgate.net/publication/|
+\verb|247837653_The_Diculties_of_Definite_Integration/file/72e7e52a9b1f06e196.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Dav03.pdf|
+ abstract = "
+ Indefinite integration is the inverse operation to differentiation,
+ and, before we can understand what we mean by indefinite integration,
+ we need to understand what we mean by differentiation."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Mudd 14]{Mudd14} Harvey Mudd College
-``Partial Fractions''
-\verb|www.math.hmc.edu/calculus/tutorials/partial_fractions/partial_fractions.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Mudd14.pdf|
+\bibitem[Fateman 02]{Fat02} Fateman, Richard
+``Symbolic Integration''
+\verb|inst.eecs.berkeley.edu/~cs282/sp02/lects/14.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Fat02.pdf|
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Rajasekaran 14]{Raja14} Rajasekaran, Raja
-``Partial Fraction Expansion''
-\verb|www.utdallas.edu/~raja1/EE4361%20Spring%2014/Lecture%20Notes/|
-\verb|Partial%20Fractions.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Raja14.pdf|
+\begin{chunk}{axiom.bib}
+@inproceedings{Gedd89,
+ author = "Geddes, K. O. and Stefanus, L. Y.",
+ title = "On the Risch-norman Integration Method and Its Implementation
+ in MAPLE",
+ booktitle = "Proc. of the ACM-SIGSAM 1989 Int. Symp. on Symbolic and
+ Algebraic Computation",
+ series = "ISSAC '89",
+ year = "1989",
+ isbn = "0-89791-325-6",
+ location = "Portland, Oregon, USA",
+ pages = "212--217",
+ numpages = "6",
+ url = "http://doi.acm.org/10.1145/74540.74567",
+ doi = "10.1145/74540.74567",
+ acmid = "74567",
+ publisher = "ACM",
+ address = "New York, NY, USA",
+ paper = "Gedd89.pdf",
+ abstract = "
+ Unlike the Recursive Risch Algorithm for the integration of
+ transcendental elementary functions, the Risch-Norman Method processes
+ the tower of field extensions directly in one step. In addition to
+ logarithmic and exponential field extensions, this method can handle
+ extentions in terms of tangents. Consequently, it allows trigonometric
+ functions to be treated without converting them to complex exponential
+ form. We review this method and describe its implementation in
+ MAPLE. A heuristic enhancement to this method is also presented."
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Wootton 14]{Woot14} Wootton, Aaron
-``Integration of Rational Functions by Partial Fractions''
-\verb|faculty.up.edu/wootton/calc2/section7.4.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Woot14.pdf|
+\bibitem[Geddes 92a]{GCL92a} Geddes, K.O.; Czapor, S.R.; Labahn, G.
+``The Risch Integration Algorithm''
+Algorithms for Computer Algebra, Ch 12 pp511-573 (1992)
+%\verb|axiom-developer.org/axiom-website/papers/GCL92a.pdf|
\end{chunk}
-\subsection{Ore Rings} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-This is used as a reference for the LeftOreRing category, in particular,
-the least left common multiple (lcmCoef) function.
\begin{chunk}{ignore}
-\bibitem[Abramov 97]{Abra97} Abramov, Sergei A.; van Hoeij, Mark
-``A method for the Integration of Solutions of Ore Equations''
-Proc ISSAC 97 pp172-175 (1997)
-%\verb|axiom-developer.org/axiom-website/papers/Abra97.pdf|
- abstract = "
- We introduce the notion of the adjoint Ore ring and give a definition
- of adjoint polynomial, operator and equation. We apply this for
- integrating solutions of Ore equations."
+\bibitem[Hardy 1916]{Hard16} Hardy, G.H.
+``The Integration of Functions of a Single Variable''
+Cambridge Unversity Press, Cambridge, 1916
+% REF:00002
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Delenclos 06]{DL06} Delenclos, Jonathon; Leroy, Andr\'e
-``Noncommutative Symmetric functions and $W$-polynomials''
-\verb|arxiv.org/pdf/math/0606614.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/DL06.pdf|
+\bibitem[Harrington 78]{Harr87} Harrington, S.J.
+``A new symbolic integration system in reduce''
+\verb|comjnl.oxfordjournals.or/content/22/2/127.full.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Harr87.pdf|
abstract = "
- Let $K$, $S$, $D$ be a division ring an endomorphism and a
- $S$-derivation of $K$, respectively. In this setting we introduce
- generalized noncommutative symmetric functions and obtain Vi\'ete
- formula and decompositions of different operators. $W$-polynomials
- show up naturally, their connetions with $P$-independency. Vandermonde
- and Wronskian matrices are briefly studied. The different linear
- factorizations of $W$-polynomials are analysed. Connections between
- the existence of LLCM (least left common multiples) of monic linear
- polynomials with coefficients in a ring and the left duo property are
- established at the end of the paper."
+ A new integration system, employing both algorithmic and pattern match
+ integration schemes is presented. The organization of the system
+ differs from that of earlier programs in its emphasis on the
+ algorithmic approach to integration, its modularity and its ease of
+ revision. The new Norman-Rish algorithm and its implementation at the
+ University of Cambridge are employed, supplemented by a powerful
+ collection of simplification and transformation rules. The facility
+ for user defined integrals and functions is also included. The program
+ is both fast and powerful, and can be easily modified to incorporate
+ anticipated developments in symbolic integration."
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Abramov 05]{Abra05} Abramov, S.A.; Le, H.Q.; Li, Z.
-``Univariate Ore Polynomial Rings in Computer Algebra''
-\verb|www.mmrc.iss.ac.cn/~zmli/papers/oretools.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Abra05.pdf|
- abstract = "
- We present some algorithms related to rings of Ore polynomials (or,
- briefly, Ore rings) and describe a computer algebra library for basic
- operations in an arbitrary Ore ring. The library can be used as a
- basis for various algorithms in Ore rings, in particular, in
- differential, shift, and $q$-shift rings."
+\begin{chunk}{axiom.bib}
+@misc{Herm1872,
+ author = "Hermite, E.",
+ title = "Sur l'int\'{e}gration des fractions rationelles",
+ journal = "Nouvelles Annales de Math\'{e}matiques",
+ volume = "11",
+ pages = "145-148",
+ year = "1872"
+}
\end{chunk}
-\subsection{Number Theory} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Shoup 08]{Sho08} Shoup, Victor
-``A Computational Introduction to Number Theory''
-\verb|shoup.net/ntb/ntb-v2.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Sho08.pdf|
+\bibitem[Horowitz 71]{Horo71} Horowitz, Ellis
+``Algorithms for Partial Fraction Decomposition and Rational Function
+ Integration''
+SYMSAC '71 Proc. ACM Symp. on Symbolic and Algebraic Manipulation (1971)
+pp441-457
+%\verb|axiom-developer.org/axiom-website/papers/Horo71.pdf| REF:00018
+ abstract = "
+ Algorithms for symbolic partial fraction decomposition and indefinite
+ integration of rational functions are described. Two types of
+ partial fraction decomposition are investigated, square-free and
+ complete square-free. A method is derived, based on the solution of
+ a linear system, which produces the square-free decomposition of any
+ rational function, say A/B. The computing time is show to be
+ $O(n^4(ln nf)^2)$ where ${\rm deg}(A) < {\rm\ deg}(B) = n$ and $f$
+ is a number which is closely related to the size of the coefficients
+ which occur in A and B. The complete square-free partical fraction
+ decomposition can then be directly obtained and it is shown that the
+ computing time for this process is also bounded by $O(n^4(ln nf)^2)$."
\end{chunk}
-\subsection{Polynomial Factorization} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{chunk}{ignore}
+\bibitem[Jeffrey 97]{Jeff97} Jeffrey, D.J.; Rich, A.D.
+``Recursive integration of piecewise-continuous functions''
+\verb|www.cybertester.com/data/recint.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Jeff97.pdf|
+ abstract = "
+ An algorithm is given for the integration of a class of
+ piecewise-continuous functions. The integration is with respect to a
+ real variable, because the functions considered do not in general
+ allow integration in the complex plane to be defined. The class of
+ integrands includes commonly occurring waveforms, such as square
+ waves, triangular waves, and the floor function; it also includes the
+ signum function. The algorithm can be implemented recursively, and it
+ has the property of ensuring that integrals are continuous on domains
+ of maximum extent."
-\subsection{Branch Cuts} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Beau03,
- author = "Beaumont, James and Bradford, Russell and Davenport, James H.",
- title = "Better simplification of elementary functions through power series",
- journal = "2003 International Symposium on Symbolic and Algebraic Computation",
- series = "ISSAC'03",
- year = "2003",
- month = "August",
- paper = "Beau03.pdf",
+\begin{chunk}{ignore}
+\bibitem[Jeffrey 99]{Jeff99} Jeffrey, D.J.; Labahn, G.; Mohrenschildt, M.v.;
+Rich, A.D.
+``Integration of the signum, piecewise and related functions''
+\verb|cs.uwaterloo.ca/~glabahn/Papers/issac99-2.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Jeff99.pdf|
+ abstract = "
+ When a computer algebra system has an assumption facility, it is
+ possible to distinguish between integration problems with respect to a
+ real variable, and those with respect to a complex variable. Here, a
+ class of integration problems is defined in which the integrand
+ consists of compositions of continuous functions and signum functions,
+ and integration is with respect to a real variable. Algorithms are
+ given for evaluating such integrals."
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Kiymaz 04]{Kiym04} Kiymaz, Onur; Mirasyedioglu, Seref
+``A new symbolic computation for formal integration with exact power series''
+%\verb|axiom-developer.org/axiom-website/Kiym04.pdf|
+ abstract = "
+ This paper describes a new symbolic algorithm for formal integration
+ of a class of functions in the context of exact power series by using
+ generalized hypergeometric series and computer algebraic technique."
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Knowles 93]{Know93} Knowles, P.
+``Integration of a class of transcendental liouvillian
+functions with error-functions i''
+Journal of Symbolic Computation Vol 13 pp525-543 (1993)
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Knowles 95]{Know95} Knowles, P.
+``Integration of a class of transcendental liouvillian
+functions with error-functions ii''
+Journal of Symbolic Computation Vol 16 pp227-241 (1995)
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@article{Krag09,
+ author = "Kragler, R.",
+ title = "On Mathematica Program for Poor Man's Integrator Algorithm",
+ journal = "Programming and Computer Software",
+ volume = "35",
+ number = "2",
+ pages = "63-78",
+ year = "2009",
+ issn = "0361-7688",
+ paper = "Krag09.pdf",
+ abstract = "
+ In this paper by means of computer experiment we study advantages and
+ disadvantages of the heuristical method of ``parallel integrator''. For
+ this purpose we describe and use implementation of the method in
+ Mathematica. In some cases we compare this implementation with the original
+ one in Maple."
+}
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Lang 93]{Lang93} Lang, S.
+``Algebra''
+Addison-Wesly, New York, 3rd edition 1993
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Leerawat 02]{Leer02} Leerawat, Utsanee; Laohakosol, Vichian
+``A Generalization of Liouville's Theorem on Integration in Finite Terms''
+\verb|www.mathnet.or.kr/mathnet/kms_tex/113666.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Leer02.pdf|
+ abstract = "
+ A generalization of Liouville's theorem on integration in finite
+ terms, by enlarging the class of fields to an extension called
+ Ei-Gamma extension is established. This extension includes the
+ $\mathcal{E}\mathcal{L}$-elementary extensions of Singer, Saunders and
+ Caviness and contains the Gamma function."
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Leslie 09]{Lesl09} Leslie, Martin
+``Why you can't integrate exp($x^2$)''
+\verb|math.arizona.edu/~mleslie/files/integrationtalk.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Lesl09.pdf|
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Lichtblau 11]{Lich11} Lichtblau, Daniel
+``Symbolic definite (and indefinite) integration: methods and open issues''
+ACM Comm. in Computer Algebra Issue 175, Vol 45, No.1 (2011)
+\verb|www.sigsam.org/bulletin/articles/175/issue175.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Lich11.pdf|
+ abstract = "
+ The computation of definite integrals presents one with a variety of
+ choices. There are various methods such as Newton-Leibniz or Slater's
+ convolution method. There are questions such as whether to split or
+ merge sums, how to search for singularities on the path of
+ integration, when to issue conditional results, how to assess
+ (possibly conditional) convergence, and more. These various
+ considerations moreover interact with one another in a multitude of
+ ways. Herein we discuss these various issues and illustrate with examples."
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@article{Liou1833a,
+ author = "Liouville, Joseph",
+ title = "Premier m\'{e}moire sur la d\'{e}termination des int\'{e}grales
+ dont la valeur est alg\'{e}brique",
+ journal = "Journal de l'Ecole Polytechnique",
+ volume = "14",
+ pages = "124-128",
+ year = "1833"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@article{Liou1833b,
+ author = "Liouville, Joseph",
+ title = "Second m\'{e}moire sur la d\'{e}termination des int\'{e}grales
+ dont la valeur est alg\'{e}brique",
+ journal = "Journal de l'Ecole Polytechnique",
+ volume = "14",
+ pages = "149-193",
+ year = "1833"
+}
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Liouville 1833c]{Lio1833c} Liouville, Joseph
+``Note sur la determination des int\'egrales dont la
+valeur est alg\'ebrique''
+Journal f\"ur die Reine und Angewandte Mathematik,
+Vol 10 pp 247-259, (1833)
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Liouville 1833d]{Lio1833d} Liouville, Joseph
+``Sur la determination des int\'egrales dont la valeur est
+alg\'ebrique''
+{\sl Journal de l'Ecole Polytechnique}, 14:124-193, 1833
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Liouville 1835]{Lio1835} Liouville, Joseph
+``M\'emoire sur l'int\'gration d'une classe de fonctions
+transcendentes''
+Journal f\"ur die Reine und Angewandte Mathematik,
+Vol 13(2) pp 93-118, (1835)
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Marc 94]{Marc94} Marchisotto, Elena Anne; Zakeri, Gholem-All
+``An Invitation to Integration in Finite Terms''
+College Mathematics Journal Vol 25 No 4 (1994) pp295-308
+\verb|www.rangevoting.org/MarchisottoZint.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Marc94.pdf|
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Marik 91]{Mari91} Marik, Jan
+``A note on integration of rational functions''
+\verb|dml.cz/bitstream/handle/10338.dmlcz/126024/MathBohem_116-1991-4_9.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Mari91.pdf|
+ abstract = "
+ Let $P$ and $Q$ be polynomials in one variable with complex coefficients
+ and let $n$ be a natural number. Suppose that $Q$ is not constant and
+ has only simple roots. Then there is a rational function $\varphi$
+ with $\varphi^\prime=P/Q^{n+1}$ if and only if the Wronskian of the
+ functions $Q^\prime$, $(Q^2)^\prime,\ldots\,(Q^n)^\prime$,$P$ is
+ divisible by $Q$."
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Moses 76]{Mos76} Moses, Joel
+``An introduction to the Risch Integration Algorithm''
+ACM Proc. 1976 annual conference pp425-428
+%\verb|axiom-developer.org/axiom-website/papers/Mos76.pdf| REF:00048
+ abstract = "
+ Risch's decision procedure for determining the integrability in closed
+ form of the elementary functions of the calculus is presented via
+ examples. The exponential and logarithmic cases of the algorithsm had
+ been implemented for the MACSYMA system several years ago. The
+ implementation of the algebraic case of the algorithm is the subject
+ of current research."
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Moses 71a]{Mos71a} Moses, Joel
+``Symbolic Integration: The Stormy Decade''
+CACM Aug 1971 Vol 14 No 8 pp548-560
+\verb|www-inst.eecs.berkeley.edu/~cs282/sp02/readings/moses-int.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Mos71a.pdf| REF:00017
+ abstract = "
+ Three approaches to symbolic integration in the 1960's are
+ described. The first, from artificial intelligence, led to Slagle's
+ SAINT and to a large degree to Moses' SIN. The second, from algebraic
+ manipulation, led to Monove's implementation and to Horowitz' and
+ Tobey's reexamination of the Hermite algorithm for integrating
+ rational functions. The third, from mathematics, led to Richardson's
+ proof of the unsolvability of the problem for a class of functions and
+ for Risch's decision procedure for the elementary functions.
+ Generalizations of Risch's algorithm to a class of special
+ functions and programs for solving differential equations and for
+ finding the definite integral are also described."
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Norman 79]{Nor79} Norman, A.C.; Davenport, J.H.
+``Symbolic Integration -- The Dust Settles?''
+%\verb|axiom-developer.org/axiom-website/papers/Nor79.pdf|
+ abstract = "
+ By the end of the 1960s it had been shown that a computer could find
+ indefinite integrals with a competence exceeding that of typical
+ undergraduates. This practical advance was backed up by algorithmic
+ interpretations of a number of clasical results on integration, and by
+ some significant mathematical extensions to these same results. At
+ that time it would have been possible to claim that all the major
+ barriers in the way of a complete system for automated analysis had
+ been breached. In this paper we survey the work that has grown out of
+ the above-mentioned early results, showing where the development has
+ been smooth and where it has spurred work in seemingly unrelated fields."
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Ostrowski 46]{Ost46} Ostrowski, A.
+``Sur l'int\'egrabilit\'e \'el\'ementaire de quelques classes
+d'expressions''
+Comm. Math. Helv., Vol 18 pp 283-308, (1946)
+% REF:00008
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Raab 12]{Raab12} Raab, Clemens G.
+``Definite Integration in Differential Fields''
+\verb|www.risc.jku.at/publications/download/risc_4583/PhD_CGR.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Raab12.pdf|
+ abstract = "
+ The general goal of this thesis is to investigate and develop computer
+ algebra tools for the simplification resp. evaluation of definite
+ integrals. One way of finding the value of a def- inite integral is
+ via the evaluation of an antiderivative of the integrand. In the
+ nineteenth century Joseph Liouville was among the first who analyzed
+ the structure of elementary antiderivatives of elementary functions
+ systematically. In the early twentieth century the algebraic structure
+ of differential fields was introduced for modeling the differential
+ properties of functions. Using this framework Robert H. Risch
+ published a complete algorithm for transcendental elementary
+ integrands in 1969. Since then this result has been extended to
+ certain other classes of integrands as well by Michael F. Singer,
+ Manuel Bronstein, and several others. On the other hand, if no
+ antiderivative of suitable form is available, then linear relations
+ that are satisfied by the parameter integral of interest may be found
+ based on the principle of parametric integration (often called
+ differentiating under the integral sign or creative telescoping).
+
+ The main result of this thesis extends the results mentioned above to
+ a complete algo- rithm for parametric elementary integration for a
+ certain class of integrands covering a majority of the special
+ functions appearing in practice such as orthogonal polynomials,
+ polylogarithms, Bessel functions, etc. A general framework is provided
+ to model those functions in terms of suitable differential fields. If
+ the integrand is Liouvillian, then the present algorithm considerably
+ improves the efficiency of the corresponding algorithm given by Singer
+ et al. in 1985. Additionally, a generalization of Czichowski’s
+ algorithm for computing the logarithmic part of the integral is
+ presented. Moreover, also partial generalizations to include other
+ types of integrands are treated.
+
+ As subproblems of the integration algorithm one also has to find
+ solutions of linear or- dinary differential equations of a certain
+ type. Some contributions are also made to solve those problems in our
+ setting, where the results directly dealing with systems of
+ differential equations have been joint work with Moulay A. Barkatou.
+
+ For the case of Liouvillian integrands we implemented the algorithm in
+ form of our Mathematica package Integrator. Parts of the
+ implementation also deal with more general functions. Our procedures
+ can be applied to a significant amount of the entries in integral
+ tables, both indefinite and definite integrals. In addition, our
+ procedures have been successfully applied to interesting examples of
+ integrals that do not appear in these tables or for which current
+ standard computer algebra systems like Mathematica or Maple do not
+ succeed. We also give examples of how parameter integrals coming from
+ the work of other researchers can be solved with the software, e.g.,
+ an integral arising in analyzing the entropy of certain processes."
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Raab 13]{Raab13} Raab, Clemens G.
+``Generalization of Risch's Algorithm to Special Functions''
+\verb|arxiv.org/pdf/1305.1481|
+%\verb|axiom-developer.org/axiom-website/papers/Raab13.pdf|
+ abstract = "
+ Symbolic integration deals with the evaluation of integrals in closed
+ form. We present an overview of Risch's algorithm including recent
+ developments. The algorithms discussed are suited for both indefinite
+ and definite integration. They can also be used to compute linear
+ relations among integrals and to find identities for special functions
+ given by parameter integrals. The aim of this presentation is twofold:
+ to introduce the reader to some basic idea of differential algebra in
+ the context of integration and to raise awareness in the physics
+ community of computer algebra algorithms for indefinite and definite
+ integration."
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Raab xx]{Raabxx} Raab, Clemens G.
+``Integration in finite terms for Liouvillian functions''
+\verb|www.mmrc.iss.ac.cn/~dart4/posters/Raab.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Raabxx.pdf|
+ abstract = "
+ Computing integrals is a common task in many areas of science,
+ antiderivatives are one way to accomplish this. The problem of
+ integration in finite terms can be states as follows. Given a
+ differential field $(F,D)$ and $f \in F$, compute $g$ in some
+ elementary extension of $(F,D)$ such that $Dg = f$ if such a $g$
+ exists.
+
+ This problem has been solved for various classes of fields $F$. For
+ rational functions $(C(x), \frac{d}{dx})$ such a $g$ always exists and
+ algorithms to compute it are known already for a long time. In 1969
+ Risch published an algorithm that solves this problem when $(F,D)$ is
+ a transcendental elementary extension of $(C(x),\frac{d}{dx})$. Later
+ this has been extended towards integrands being Liouvillian functions
+ by Singer et. al. via the use of regular log-explicit extensions of
+ $(C(x),\frac{d}{dx})$. Our algorithm extends this to handling
+ transcendental Liouvillian extensions $(F,D)$ of $(C,0)$ directly
+ without the need to embed them into log-explicit extensions. For
+ example, this means that
+ \[\int{(z-x)x^{z-1}e^{-x}dx} = x^ze^{-x}\]
+ can be computed without including log(x) in the differential field."
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Rich 09]{Rich09} Rich, A.D.; Jeffrey, D.J.
+``A Knowledge Repository for Indefinite Integration Based on Transformation Rules''
+\verb|www.apmaths.uwo.ca/~arich/A%2520Rule-based%2520Knowedge%2520Repository.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Rich09.pdf|
+ abstract = "
+ Taking the specific problem domain of indefinite integration, we
+ describe the on-going development of a repository of mathematical
+ knowledge based on transformation rules. It is important that the
+ repository be not confused with a look-up table. The database of
+ transformation rules is at present encoded in Mathematica, but this is
+ only one convenient form of the repository, and it could be readily
+ translated into other formats. The principles upon which the set of
+ rules is compiled is described. One important principle is
+ minimality. The benefits of the approach are illustrated with
+ examples, and with the results of comparisons with other approaches."
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@techreport{Risc68,
+ author = "Risch, Robert",
+ title = "On the integration of elementary functions which are built up
+ using algebraic operations",
+ type = "Research Report",
+ number = "SP-2801/002/00",
+ institution = "System Development Corporation, Santa Monica, CA, USA",
+ year = "1968"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@techreport{Risc69a,
+ author = "Risch, Robert",
+ title = "Further results on elementary functions",
+ type = "Research Report",
+ number = "RC-2042",
+ institution = "IBM Research, Yorktown Heights, NY, USA",
+ year = "1969"
+
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@article{Risc69b,
+ author = "Risch, Robert",
+ title = "The problem of integration in finite terms",
+ journal = "Transactions of the American Mathematical Society",
+ volume = "139",
+ year = "1969",
+ pages = "167-189",
+ paper = "Ris69b.pdf",
+ abstract = "This paper deals with the problem of telling whether a
+ given elementary function, in the sense of analysis, has an elementary
+ indefinite integral."
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@article{Risc70,
+ author = "Risch, Robert",
+ title = "The Solution of the Problem of Integration in Finite Terms",
+ journal = "Bull. AMS",
+ year = "1970",
+ issn = "0002-9904",
+ volume = "76",
+ number = "3",
+ pages = "605-609",
+ paper = "Risc70.pdf",
+ abstract = "
+ The problem of integration in finite terms asks for an algorithm for
+ deciding whether an elementary function has an elementary indefinite
+ integral and for finding the integral if it does. ``Elementary'' is
+ used here to denote those functions build up from the rational
+ functions using only exponentiation, logarithms, trigonometric,
+ inverse trigonometric and algebraic operations. This vaguely worded
+ question has several precise, but inequivalent formulations. The
+ writer has devised an algorithm which solves the classical problem of
+ Liouville. A complete account is planned for a future publication. The
+ present note is intended to indiciate some of the ideas and techniques
+ involved."
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@article{Risc79,
+ author = "Risch, Robert",
+ title = "Algebraic properties of the elementary functions of analysis",
+ journal = "American Journal of Mathematics",
+ volume = "101",
+ pages = "743-759",
+ year = "1979"
+}
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Ritt 48]{Ritt48} Ritt, J.F.
+``Integration in Finite Terms''
+Columbia University Press, New York 1948
+% REF:00046
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Rosenlicht 68]{Ro68} Rosenlicht, Maxwell
+``Liouville's Theorem on Functions with Elementary Integrals''
+Pacific Journal of Mathematics Vol 24 No 1 (1968)
+\verb|msp.org/pjm/1968/24-1/pjm-v24-n1-p16-p.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Ro68.pdf| REF:00047
+ abstract = "
+ Defining a function with one variable to be elemetary if it has an
+ explicit representation in terms of a finite number of algebraic
+ operations, logarithms, and exponentials. Liouville's theorem in its
+ simplest case says that if an algebraic function has an elementary
+ integral then the latter is itself an algebraic function plus a sum of
+ constant multiples of logarithms of algebraic functions. Ostrowski has
+ generalized Liouville's results to wider classes of meromorphic
+ functions on regions of the complex plane and J.F. Ritt has given the
+ classical account of the entire subject in his Integraion in Finite
+ Terms, Columbia University Press, 1948. In spite of the essentially
+ algebraic nature of the problem, all proofs so far have been analytic.
+ This paper gives a self contained purely algebraic exposition of the
+ probelm, making a few new points in addition to the resulting
+ simplicity and generalization."
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@article{Rose72,
+ author = "Rosenlicht, Maxwell",
+ title = "Integration in finite terms",
+ journal = "American Mathematical Monthly",
+ year = "1972",
+ volume = "79",
+ pages = "963-972",
+ paper = "Rose72.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Rothstein 76]{Ro76} Rothstein, Michael
+``Aspects of symbolic integration and simplifcation of exponential
+and primitive functions''
+PhD thesis, University of Wisconsin-Madison (1976)
+\verb|www.cs.kent.edu/~rothstei/dis.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Ro76.pdf| REF:00051
+ abstract = "
+ In this thesis we cover some aspects of the theory necessary to obtain
+ a canonical form for functions obtained by integration and
+ exponentiation from the set of rational functions.
+
+ These aspects include a new algorithm for symbolic integration of
+ functions involving logarithms and exponentials which avoids
+ factorization of polynomials in those cases where algebraic extension
+ of the constant field is not required, avoids partial fraction
+ decompositions, and only solves linear systems with a small number of
+ unknowns.
+
+ We have also found a theorem which states, roughly speaking, that if
+ integrals which can be represented as logarithms are represented as
+ such, the only algebraic dependence that a new exponential or
+ logarithm can satify is given by the law of exponents or the law of
+ logarithms."
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Rothstein 76a]{Ro76a} Rothstein, Michael; Caviness, B.F.
+``A structure theorem for exponential and primitive functions: a preliminary
+ report''
+ACM Sigsam Bulletin Vol 10 Issue 4 (1976)
+%\verb|axiom-developer.org/axiom-website/papers/Ro76a.pdf|
+ abstract = "
+ In this paper a generalization of the Risch Structure Theorem is reported.
+ The generalization applies to fields $F(t_1,\ldots,t_n)$ where $F$
+ is a differential field (in our applications $F$ will be a finitely
+ generated extension of $Q$, the field of rational numbers) and each $t_i$
+ is either algebraic over $F_{i-1}=F(t_1,\ldots,t_{i-1})$, is an
+ exponential of an element in $F_{i-1}$, or is an integral of an element
+ in $F_{i-1}$. If $t_i$ is an integral and can be expressed using
+ logarithms, it must be so expressed for the generalized structure
+ theorem to apply."
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Rothstein 76b]{Ro76b} Rothstein, Michael; Caviness, B.F.
+``A structure theorem for exponential and primitive functions''
+SIAM J. Computing Vol 8 No 3 (1979)
+%\verb|axiom-developer.org/axiom-website/papers/Ro76b.pdf| REF:00104
+ abstract = "
+ In this paper a new theorem is proved that generalizes a result of
+ Risch. The new theorem gives all the possible algebraic relationships
+ among functions that can be built up from the rational functions by
+ algebraic operations, by taking exponentials, and by integration. The
+ functions so generated are called exponential and primitive functions.
+ From the theorem an algorithm for determining algebraic dependence
+ among a given set of exponential and primitive functions is derived.
+ The algorithm is then applied to a problem in computer algebra."
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@article{Roth77,
+ author = "Rothstein, Michael",
+ title = "A new algorithm for the integration of exponential and
+ logarithmic functions",
+ journal = "Proceedings of the 1977 MACSYMA Users Conference",
+ year = "1977",
+ pages = "263-274",
+ publisher = "NASA Pub CP-2012"
+}
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Seidenberg 58]{Sei58} Seidenberg, Abraham
+``Abstract differential algebra and the analytic case''
+Proc. Amer. Math. Soc. Vol 9 pp159-164 (1958)
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Seidenberg 69]{Sei69} Seidenberg, Abraham
+``Abstract differential algebra and the analytic case. II''
+Proc. Amer. Math. Soc. Vol 23 pp689-691 (1969)
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Singer 85]{Sing85} Singer, M.F.; Saunders, B.D.; Caviness, B.F.
+``An extension of Liouville's theorem on integration in finite terms''
+SIAM J. of Comp. Vol 14 pp965-990 (1985)
+\verb|www4.ncsu.edu/~singer/papers/singer_saunders_caviness.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Sing85.pdf|
+ abstract = "
+ In Part 1 of this paper, we give an extension of Liouville's Theorem
+ and give a number of examples which show that integration with special
+ functions involves some phenomena that do not occur in integration
+ with the elementary functions alone. Our main result generalizes
+ Liouville's Theorem by allowing, in addition to the elementary
+ functions, special functions such as the error function, Fresnel
+ integrals and the logarithmic integral (but not the dilogarithm or
+ exponential integral) to appear in the integral of an elementary
+ function. The basic conclusion is that these functions, if they
+ appear, appear linearly. We give an algorithm which decides if an
+ elementary function, built up using only exponential functions and
+ rational operations has an integral which can be expressed in terms of
+ elementary functions and error functions."
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Slagle 61]{Slag61} Slagle, J.
+``A heuristic program that solves symbolic integration problems in
+ freshman calculus''
+Ph.D Diss. MIT, May 1961; also Computers and Thought, Feigenbaum and Feldman.
+% REF:00014
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Terelius 09]{Tere09} Terelius, Bjorn
+``Symbolic Integration''
+%\verb|axiom-developer.org/axiom-website/papers/Tere09.pdf|
+ abstract = "
+ Symbolic integration is the problem of expressing an indefinite integral
+ $\int{f}$ of a given function $f$ as a finite combination $g$ of elementary
+ functions, or more generally, to determine whether a certain class of
+ functions contains an element $g$ such that $g^\prime = f$.
+
+ In the first part of this thesis, we compare different algorithms for
+ symbolic integration. Specifically, we review the integration rules
+ taught in calculus courses and how they can be used systematically to
+ create a reasonable, but somewhat limited, integration method. Then we
+ present the differential algebra required to prove the transcendental
+ cases of Risch's algorithm. Risch's algorithm decides if the integral
+ of an elementary function is elementary and if so computes it. The
+ presentation is mostly self-contained and, we hope, simpler than
+ previous descriptions of the algorithm. Finally, we describe
+ Risch-Norman's algorithm which, although it is not a decision
+ procedure, works well in practice and is considerably simpler than the
+ full Risch algorithm.
+
+ In the second part of this thesis, we briefly discuss an
+ implementation of a computer algebra system and some of the
+ experiences it has given us. We also demonstrate an implementation of
+ the rule-based approach and how it can be used, not only to compute
+ integrals, but also to generate readable derivations of the results."
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@article{Trag76,
+ author = "Trager, Barry",
+ title = "Algebraic factoring and rational function integration",
+ journal = "Proceedings of SYMSAC'76",
+ year = "1976",
+ pages = "219-226",
+ paper = "Trag76.pdf",
+ abstract = "
+ This paper presents a new, simple, and efficient algorithm for
+ factoring polynomials in several variables over an algebraic number
+ field. The algorithm is then used interatively to construct the
+ splitting field of a polynomial over the integers. Finally the
+ factorization and splitting field algorithms are applied to the
+ problem of determining the transcendental part of the integral of a
+ rational function. In particular, a constructive procedure is given
+ for finding a least degree extension field in which the integral can
+ be expressed."
+}
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Trager 76a]{Tr76a} Trager, Barry Marshall
+``Algorithms for Manipulating Algebraic Functions''
+MIT Master's Thesis.
+\verb|www.dm.unipi.it/pages/gianni/public_html/Alg-Comp/fattorizzazione-EA.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Tr76a.pdf| REF:00050
+ abstract = "
+ Given a base field $k$, of characteristic zero, with effective
+ procedures for performing arithmetic and factoring polynomials, this
+ thesis presents algorithms for extending those capabilities to
+ elements of a finite algebraic symbolic manipulation system. An
+ algebraic factorization algorithm along with a constructive version of
+ the primitive element theorem is used to construct splitting fields of
+ polynomials. These fields provide a context in which we can operate
+ symbolically with all the roots of a set of polynomials. One
+ application for this capability is rational function integrations.
+ Previously presented symbolic algorithms concentrated on finding the
+ rational part and were only able to compute the complete
+ integral in special cases. This thesis presents an algorithm for
+ finding an algebraic extension field of least degreee in which the
+ integral can be expressed, and then constructs the integral in that
+ field. The problem of algebraic function integration is also
+ examined, and a highly efficient procedure is presented for generating
+ the algebraic part of integrals whose function fields are defined by a
+ single radical extension of the rational functions."
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@phdthesis{Trag84,
+ author = "Trager, Barry",
+ title = "On the integration of algebraic functions",
+ school = "MIT",
+ year = "1984",
+ url = "http://www.dm.unipi.it/pages/gianni/public_html/Alg-Comp/thesis.pdf",
+ paper = "Trag76.pdf",
+ abstract = "
+ We show how the ``rational'' approach for integrating algebraic
+ functions can be extended to handle elementary functions. The
+ resulting algorithm is a practical decision procedure for determining
+ whether a given elementary function has an elementary antiderivative,
+ and for computing it if it exists."
+}
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[W\"urfl 07]{Wurf07} W\"urfl, Andreas
+``Basic Concepts of Differential Algebra''
+\verb|www14.in.tum.de/konferenzen/Jass07/courses/1/Wuerfl/wuerfl_paper.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Wurf07.pdf|
+ abstract = "
+ Modern computer algebra systems symbolically integrate a vast variety
+ of functions. To reveal the underlying structure it is necessary to
+ understand infinite integration not only as an analytical problem but
+ as an algebraic one. Introducing the differential field of elementary
+ functions we sketch the mathematical tools like Liouville's Principle
+ used in modern algorithms. We present Hermite's method for integration
+ of rational functions as well as the Rothstein/Trager method for
+ rational and for elementary functions. Further applications of the
+ mentioned algorithms in the field of ODE's conclude this paper."
+
+\end{chunk}
+
+\section{Partial Fraction Decomposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{ignore}
+\bibitem[Angell]{Angell} Angell, Tom
+``Guidelines for Partial Fraction Decomposition''
+\verb|www.math.udel.edu/~angell/partfrac_I.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Angell.pdf|
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Laval 08]{Lava08} Laval, Philippe B.
+``Partial Fractions Decomposition''
+\verb|www.math.wisc.edu/~park/Fall2011/integration/Partial%20Fraction.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Lava08.pdf|
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Mudd 14]{Mudd14} Harvey Mudd College
+``Partial Fractions''
+\verb|www.math.hmc.edu/calculus/tutorials/partial_fractions/partial_fractions.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Mudd14.pdf|
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Rajasekaran 14]{Raja14} Rajasekaran, Raja
+``Partial Fraction Expansion''
+\verb|www.utdallas.edu/~raja1/EE4361%20Spring%2014/Lecture%20Notes/|
+\verb|Partial%20Fractions.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Raja14.pdf|
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Wootton 14]{Woot14} Wootton, Aaron
+``Integration of Rational Functions by Partial Fractions''
+\verb|faculty.up.edu/wootton/calc2/section7.4.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Woot14.pdf|
+
+\end{chunk}
+\section{Ore Rings} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+This is used as a reference for the LeftOreRing category, in particular,
+the least left common multiple (lcmCoef) function.
+
+\begin{chunk}{ignore}
+\bibitem[Abramov 97]{Abra97} Abramov, Sergei A.; van Hoeij, Mark
+``A method for the Integration of Solutions of Ore Equations''
+Proc ISSAC 97 pp172-175 (1997)
+%\verb|axiom-developer.org/axiom-website/papers/Abra97.pdf|
abstract = "
- In [5], we introduced an algorithm for deciding whether a proposed
- simplification of elementary functions was correct in the presence of
- branch cuts. This algorithm used multivalued function simplification
- followed by verification that the branches were consistent.
+ We introduce the notion of the adjoint Ore ring and give a definition
+ of adjoint polynomial, operator and equation. We apply this for
+ integrating solutions of Ore equations."
- In [14] an algorithm was presented for zero-testing functions defined
- by ordinary differential equations, in terms of their power series.
+\end{chunk}
- The purpose of the current paper is to investigate merging the two
- techniques. In particular, we will show an explicit reduction to the
- constant problem [16]."
+\begin{chunk}{ignore}
+\bibitem[Delenclos 06]{DL06} Delenclos, Jonathon; Leroy, Andr\'e
+``Noncommutative Symmetric functions and $W$-polynomials''
+\verb|arxiv.org/pdf/math/0606614.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/DL06.pdf|
+ abstract = "
+ Let $K$, $S$, $D$ be a division ring an endomorphism and a
+ $S$-derivation of $K$, respectively. In this setting we introduce
+ generalized noncommutative symmetric functions and obtain Vi\'ete
+ formula and decompositions of different operators. $W$-polynomials
+ show up naturally, their connetions with $P$-independency. Vandermonde
+ and Wronskian matrices are briefly studied. The different linear
+ factorizations of $W$-polynomials are analysed. Connections between
+ the existence of LLCM (least left common multiples) of monic linear
+ polynomials with coefficients in a ring and the left duo property are
+ established at the end of the paper."
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Abramov 05]{Abra05} Abramov, S.A.; Le, H.Q.; Li, Z.
+``Univariate Ore Polynomial Rings in Computer Algebra''
+\verb|www.mmrc.iss.ac.cn/~zmli/papers/oretools.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Abra05.pdf|
+ abstract = "
+ We present some algorithms related to rings of Ore polynomials (or,
+ briefly, Ore rings) and describe a computer algebra library for basic
+ operations in an arbitrary Ore ring. The library can be used as a
+ basis for various algorithms in Ore rings, in particular, in
+ differential, shift, and $q$-shift rings."
+
+\end{chunk}
+
+\section{Number Theory} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt89d,
+ author = "Kaltofen, E. and Valente, T. and Yui, N.",
+ title = "An improved {Las Vegas} primality test",
+ booktitle = "Proc. 1989 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC89",
+ pages = "26--33",
+ year = "1989",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/KVY89.pdf",
+ paper = "Kalt89d.pdf"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Beau07,
- author = "Beaumont, James C. and Bradford, Russell J. and
- Davenport, James H. and Phisanbut, Nalina",
- title = "Testing elementary function identities using CAD",
- journal = "Applicable Algebra in Engineering, Communication and Computing",
- year = "2007",
- volume = "18",
- number = "6",
- issn = "0938-1279",
+@InCollection{Kalt91b,
+ author = "Kaltofen, E. and Yui, N.",
+ editor = "D. V. Chudnovsky and G. V. Chudnovsky and H. Cohn and
+ M. B. Nathanson",
+ title = "Explicit construction of {Hilbert} class fields of imaginary
+ quadratic fields by integer lattice reduction",
+ booktitle = "Number Theory New York Seminar 1989--1990",
+ pages = "150--202",
publisher = "Springer-Verlag",
- pages = "513-543",
- paper = "Beau07.pdf",
- abstract = "
- One of the problems with manipulating function identities in computer
- algebra systems is that they often involve functions which are
- multivalued, whilst most users tend to work with single-valued
- functions. The problem is that many well-known identities may no
- longer be true everywhere in the complex plane when working with their
- single-valued counterparts. Conversely, we cannot ignore them, since
- in particular contexts they may be valid. We investigate the
- practicality of a method to verify such identities by means of an
- experiment; this is based on a set of test examples which one might
- realistically meet in practice. Essentially, the method works as
- follows. We decompose the complex plane via means of cylindrical
- algebraic decomposition into regions with respect to the branch cuts
- of the functions. We then test the identity numerically at a sample
- point in the region. The latter step is facilitated by the notion of
- the {\sl adherence} of a branch cut, which was previously introduced
- by the authors. In addition to presenting the results of the
- experiment, we explain how adherence relates to the proposal of
- {\sl signed zeros} by W. Kahan, and develop this idea further in order to
- allow us to cover previously untreatable cases. Finally, we discuss
- other ways to improve upon our general methodology as well as topics
- for future research."
+ year = "1991",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaYui91.pdf",
+ paper = "Kalt91b.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt84a,
+ author = "Kaltofen, E. and Yui, N.",
+ title = "Explicit construction of the {Hilbert} class field of imaginary
+ quadratic fields with class number 7 and 11",
+ booktitle = "Proc. EUROSAM '84",
+ pages = "310--320",
+ crossref = "EUROSAM84",
+ year = "1984",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/84/KaYui84_eurosam.ps.gz",
+ paper = "Kalt84a.ps"
+}
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Shoup 08]{Sho08} Shoup, Victor
+``A Computational Introduction to Number Theory''
+\verb|shoup.net/ntb/ntb-v2.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Sho08.pdf|
+
+\end{chunk}
+
+\section{Sparse Polynomial Interpolation} %%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt07a,
+ author = "Kaltofen, Erich and Yang, Zhengfeng and Zhi, Lihong",
+ title = "On probabilistic analysis of randomization in hybrid
+ symbolic-numeric algorithms",
+ year = "2007",
+ booktitle = "Proc. 2007 Internat. Workshop on Symbolic-Numeric Comput.",
+ crossref = "SNC07",
+ pages = "11--17",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/07/KYZ07.pdf",
+ paper = "Kalt07a.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt07b,
+ author = "Kaltofen, Erich and Yang, Zhengfeng",
+ title = "On Exact and Approximate Interpolation of Sparse
+ Rational Functions",
+ year = "2007",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'07",
+ crossref = "ISSAC07",
+ pages = "203--210",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/07/KaYa07.pdf",
+ paper = "Kalt07b.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@Article{Gies03,
+ author = "Giesbrecht, Mark and Kaltofen, Erich and Lee, Wen-shin",
+ title = "Algorithms for Computing Sparsest Shifts of Polynomials in
+ Power, {Chebychev}, and {Pochhammer} Bases",
+ year = "2003",
+ journal = "Journal of Symbolic Computation",
+ volume = "36",
+ number = "3--4",
+ pages = "401--424",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/03/GKL03.pdf",
+ paper = "Gies03.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Gies02,
+ author = "Giesbrecht, Mark and Kaltofen, Erich and Lee, Wen-shin",
+ title = "Algorithms for Computing the Sparsest Shifts for Polynomials via the
+ {Berlekamp}/{Massey} Algorithm",
+ booktitle = "Proc. 2002 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC02",
+ pages = "101--108",
+ year = "2002",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/02/GKL02.pdf",
+ paper = "Gies02.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@Article{Kalt03b,
+ author = "Kaltofen, Erich and Lee, Wen-shin",
+ title = "Early Termination in Sparse Interpolation Algorithms",
+ year = "2003",
+ journal = "Journal of Symbolic Computation",
+ volume = "36",
+ number = "3--4",
+ pages = "365--400",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/03/KL03.pdf",
+ paper = "Kalt03b.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt00a,
+ author = "Kaltofen, E. and Lee, W.-s. and Lobo, A.A.",
+ title = "Early termination in {Ben-Or/Tiwari} sparse interpolation
+ and a hybrid of {Zippel}'s algorithm",
+ booktitle = "Proc. 2000 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC2K",
+ pages = "192--201",
+ year = "2000",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/2K/KLL2K.pdf",
+ paper = "Kalt00a.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt10b,
+ author = "Kaltofen, Erich L.",
+ title = "Fifteen years after {DSC} and {WLSS2} {What} parallel
+ computations {I} do today [{Invited} Lecture at {PASCO} 2010]",
+ year = "2010",
+ booktitle = "Proc. 2010 Internat. Workshop on Parallel Symbolic Comput.",
+ crossref = "PASCO10",
+ pages = "10--17",
+ month = "July",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/10/Ka10_pasco.pdf",
+ paper = "Kalt10b.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt90,
+ author = "Kaltofen, E. and Lakshman, Y.N. and Wiley, J.M.",
+ editor = "S. Watanabe and M. Nagata",
+ title = "Modular rational sparse multivariate polynomial interpolation",
+ booktitle = "Proc. 1990 Internat. Symp. Symbolic Algebraic Comput.",
+ pages = "135--139",
+ publisher = "ACM Press",
+ year = "1990",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/KLW90.pdf",
+ paper = "Kalt90.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt88a,
+ author = "Kaltofen, E. and Yagati, Lakshman",
+ title = "Improved sparse multivariate polynomial interpolation algorithms",
+ booktitle = "Symbolic Algebraic Comput. Internat. Symp. ISSAC '88 Proc.",
+ crossref = "ISSAC88",
+ pages = "467--474",
+ year = "1988",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/KaLa88.pdf",
+ paper = "Kalt88a.pdf"
}
-
+
\end{chunk}
+\section{Divisions and Algebraic Complexity} %%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{axiom.bib}
-@article{Brad02,
- author="Bradford, Russell and Corless, RobertM. and Davenport, JamesH. and
- Jeffrey, DavidJ. and Watt, StephenM.",
- title="Reasoning about the Elementary Functions of Complex Analysis",
- journal="Annals of Mathematics and Artificial Intelligence",
- year="2002",
- issn="1012-2443",
- volume="36",
- number="3",
- doi="10.1023/A:1016007415899",
- url="http://dx.doi.org/10.1023/A%3A1016007415899",
- publisher="Kluwer Academic Publishers",
- keywords="elementary functions; branch cuts; complex identities",
- pages="303-318",
- paper = "Brad02.pdf",
- abstract = "
- There are many problems with the simplification of elementary
- functions, particularly over the complex plane, though not
- exclusively. Systems tend to make ``howlers'' or not to simplify
- enough. In this paper we outline the ``unwinding number'' approach to
- such problems, and show how it can be used to prevent errors and to
- systematise such simplification, even though we have not yet reduced
- the simplification process to a complete algorithm. The unsolved
- problems are probably more amenable to the techniques of artificial
- intelligence and theorem proving than the original problem of complex
- variable analysis."
+@InCollection{Gren11,
+ author = "Grenet, Bruno and Kaltofen, Erich L. and Koiran, Pascal
+ and Portier, Natacha",
+ title = "Symmetric Determinantal Representation of Formulas and Weakly
+ Skew Circuits",
+ booktitle = "Randomization, Relaxation, and Complexity in Polynomial
+ Equation Solving",
+ year = "2011",
+ editor = "Leonid Gurvits and Philippe P\'{e}bay and J. Maurice Rojas
+ and David Thompson",
+ pages = "61--96",
+ publisher = "American Mathematical Society",
+ address = "Providence, Rhode Island, USA",
+ isbn = "978-0-8218-5228-6",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/10/GKKP10.pdf",
+ paper = "Gren11.pdf"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@inproceedings{Chyz11,
- author = "Chyzak, Fr\'ed\'eric and Davenport, James H. and Koutschan, Christoph and Salvy, Bruno",
- title = "On Kahan's Rules for Determining Branch Cuts",
- booktitle = "Proc. 13th Int. Symp. on Symbolic and Numeric Algorithms for Scientific Computing",
- year = "2011",
- isbn = "978-1-4673-0207-4",
- location = "Timisoara",
- pages = "47-51",
- doi = "10.1109/SYNASC.2011.51",
- acmid = "258794",
- publisher = "IEEE",
- paper = "Chyz11.pdf",
- abstract = "
- In computer algebra there are different ways of approaching the
- mathematical concept of functions, one of which is by defining them as
- solutions of differential equations. We compare different such
- appraoches and discuss the occurring problems. The main focus is on
- the question of determining possible branch cuts. We explore the
- extent to which the treatment of branch cuts can be rendered (more)
- algorithmic, by adapting Kahan's rules to the differential equation
- setting."
+@InProceedings{Kalt08a,
+ author = "Kaltofen, Erich and Koiran, Pascal",
+ title = "Expressing a Fraction of Two Determinants as a Determinant",
+ year = "2008",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'08",
+ crossref = "ISSAC08",
+ pages = "141--146",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/08/KaKoi08.pdf",
+ paper = "Kalt08a.pdf"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Dave10,
- author = "Davenport, James",
- title = {The Challenges of Multivalued "Functions"},
- journal = "Lecture Notes in Computer Science",
- volume = "6167",
- year = "2010",
- pages = "1-12",
- paper = "Dave10.pdf",
- abstract = "
- Although, formally, mathematics is clear that a function is a
- single-valued object, mathematical practice is looser, particularly
- with n-th roots and various inverse functions. In this paper, we point
- out some of the looseness, and ask what the implications are, both for
- Artificial Intelligence and Symbolic Computation, of these practices.
- In doing so, we look at the steps necessary to convert existing tests
- into
- \begin{itemize}
- \item (a) rigorous statements
- \item (b) rigorously proved statements
- \end{itemize}
- In particular we ask whether there might be a constant ``de Bruij factor''
- [18] as we make these texts more formal, and conclude that the answer
- depends greatly on the interpretation being placed on the symbols."
+@Article{Hitz95,
+ author = "Kitz, M.A. and Kaltofen, E.",
+ title = "Integer division in residue number systems",
+ journal = "IEEE Trans. Computers",
+ year = "1995",
+ volume = "44",
+ number = "8",
+ pages = "983--989",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/HiKa95.pdf",
+ paper = "Hitz95.pdf"
}
-
+
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Dave12,
- author = "Davenport, James H. and Bradford, Russell and England, Matthew
- and Wilson, David",
- title = "Program Verification in the presence of complex numbers, functions
- with branch cuts etc",
- journal = "14th Int. Symp. on Symbolic and Numeric Algorithms for
- Scientific Computing",
- year = "2012",
- series = "SYNASC'12",
- pages = "83-88",
- publisher = "IEEE",
- paper = "Dave12.pdf",
- abstract = "
- In considering the reliability of numerical programs, it is normal to
- ``limit our study to the semantics dealing with numerical precision''.
- On the other hand, there is a great deal of work on the reliability of
- programs that essentially ignores the numerics. The thesis of this
- paper is that there is a class of problems that fall between the two,
- which could be described as ``does the low-level arithmetic implement
- the high-level mathematics''. Many of these problems arise because
- mathematics, particularly the mathematics of the complex numbers, is
- more difficult than expected; for example the complex function log is
- not continuous, writing down a program to compute an inverse function
- is more complicated than just solving an equation, and many algebraic
- simplification rules are not universally valid.
+@InProceedings{Kalt92a,
+ author = "Kaltofen, E.",
+ title = "On computing determinants of matrices without divisions",
+ booktitle = "Proc. 1992 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC92",
+ pages = "342--349",
+ year = "1992",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/92/Ka92_issac.pdf",
+ paper = "Kalt92a.pdf"
+}
- The good news is that these problems are theoretically capable of
- being solved, and are practically close to being solved, but not yet
- solved, in several real-world examples. However, there is still a long
- way to go before implementations match the theoretical possibilities."
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@Article{Cant91,
+ author = "Cantor, D.G. and Kaltofen, E.",
+ title = "On fast multiplication of polynomials over arbitrary algebras",
+ journal = "Acta Inform.",
+ year = "1991",
+ volume = "28",
+ number = "7",
+ pages = "693--701",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/CaKa91.pdf",
+ paper = "Cant91.pdf"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Jeff04,
- author = "Jeffrey, D. J. and Norman, A. C.",
- title = "Not Seeing the Roots for the Branches: Multivalued Functions in
- Computer Algebra",
- journal = "SIGSAM Bull.",
- issue_date = "September 2004",
- volume = "38",
- number = "3",
- month = "September",
- year = "2004",
- issn = "0163-5824",
- pages = "57--66",
- numpages = "10",
- url = "http://doi.acm.org/10.1145/1040034.1040036",
- doi = "10.1145/1040034.1040036",
- acmid = "1040036",
- publisher = "ACM",
- address = "New York, NY, USA",
- paper = "Jeff04.pdf",
- abstract = "
- We discuss the multiple definitions of multivalued functions and their
- suitability for computer algebra systems. We focus the discussion by
- taking one specific problem and considering how it is solved using
- different definitions. Our example problem is the classical one of
- calculating the roots of a cubic polynomial from the Cardano formulae,
- which contains fractional powers. We show that some definitions of
- these functions result in formulae that are correct only in the sense
- that they give candidates for solutions; these candidates must then be
- tested. Formulae that are based on single-valued functions, in
- contract, are efficient and direct."
+@Article{Kalt88b,
+ author = "Kaltofen, E.",
+ title = "Greatest common divisors of polynomials given by
+ straight-line programs",
+ journal = "J. ACM",
+ year = "1988",
+ volume = "35",
+ number = "1",
+ pages = "231--264",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/Ka88_jacm.pdf",
+ paper = "Kalt88b.pdf"
}
\end{chunk}
+\section{Polynomial Factorization} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{axiom.bib}
-@inproceedings{Kaha86,
- author = "Kahan, W.",
- title = "Branch cuts for complex elementary functions",
- booktitle = "The State of the Art in Numerical Analysis",
- year = "1986",
- month = "April",
- editor = "Powell, M.J.D and Iserles, A.",
- publisher = "Oxford University Press"
+@PhdThesis{Kalt82,
+ author = "Kaltofen, E.",
+ title = "On the complexity of factoring polynomials with integer
+ coefficients",
+ school = "RPI",
+ address = "Troy, N. Y.",
+ year = "1982",
+ month = "December",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/82/Ka82_thesis.pdf",
+ paper = "Kalt82.pdf"
}
-\end{chunk}
+\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Rich96,
- author = "Rich, Albert D. and Jeffrey, David J.",
- title = "Function Evaluation on Branch Cuts",
- journal = "SIGSAM Bull.",
- issue_date = "June 1996",
- volume = "30",
- number = "2",
- month = "June",
- year = "1996",
- issn = "0163-5824",
- pages = "25--27",
- numpages = "3",
- url = "http://doi.acm.org/10.1145/235699.235704",
- doi = "10.1145/235699.235704",
- acmid = "235704",
- publisher = "ACM",
- address = "New York, NY, USA",
- abstract = "
- Once it is decided that a CAS will evaluate multivalued functions on
- their principal branches, questions arise concerning the branch
- definitions. The first questions concern the standardization of the
- positions of the branch cuts. These questions have largely been
- resolved between the various algebra systems and the numerical
- libraries, although not completely. In contrast to the computer
- systems, many mathematical textbooks are much further behind: for
- example, many popular textbooks still specify that the argument of a
- complex number lies between 0 and $2\pi$. We do not intend to discuss
- these first questions here, however. Once the positions of the branch
- cuts have been fixed, a second set of questions arises concerning the
- evaluation of functions on their branch cuts."
+@Article{Gath85,
+ author = "Gathen, Joachim von zur; Kaltofen, E.",
+ title = "Factoring sparse multivariate polynomials",
+ journal = "J. Comput. System Sci.",
+ year = "1985",
+ volume = "31",
+ pages = "265--287",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/85/GaKa85_mathcomp.ps.gz",
+ paper = "Gath85.ps"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@InCollection{Kalt11c,
+ author = "Kaltofen, Erich and Lecerf, Gr{\'e}goire",
+ title = "Section 11.5. {Factorization} of multivariate polynomials",
+ booktitle = "Handbook of Finite Fields",
+ crossref = "HFF11",
+ pages = "382--392",
+ year = "2011",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/11/KL11.pdf",
+ paper = "Kalt11c.pdf"
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt05b,
+ author = "Kaltofen, Erich and Koiran, Pascal",
+ title = "On the complexity of factoring bivariate supersparse
+ (lacunary) polynomials",
+ year = "2005",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'05",
+ crossref = "ISSAC05",
+ pages = "208--215",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/05/KaKoi05.pdf",
+ paper = "Kalt05b.pdf"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Patt96,
- author = "Patton, Charles M.",
- title = "A Representation of Branch-cut Information",
- journal = "SIGSAM Bull.",
- issue_date = "June 1996",
- volume = "30",
- number = "2",
- month = "June",
- year = "1996",
- issn = "0163-5824",
- pages = "21--24",
- numpages = "4",
- url = "http://doi.acm.org/10.1145/235699.235703",
- doi = "10.1145/235699.235703",
- acmid = "235703",
- publisher = "ACM",
- address = "New York, NY, USA",
- paper = "Patt96.pdf",
- abstract = "
- Handling (possibly) multi-valued functions is a problem in all current
- computer algebra systems. The problem is not an issue of technology.
- Its solution, however, is tied to a uniform handling of the issues by
- the mathematics community."
+@InProceedings{Kalt06a,
+ author = "Kaltofen, Erich and Koiran, Pascal",
+ title = "Finding Small Degree Factors of Multivariate Supersparse
+ (Lacunary) Polynomials Over Algebraic Number Fields",
+ year = "2006",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'06",
+ crossref = "ISSAC06",
+ pages = "162--168",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/06/KaKoi06.pdf",
+ paper = "Kalt06a.pdf"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Squi91,
- author = "Squire, Jon S.",
- title = "Rationale for the Proposed Standard for a Generic Package of
- Complex Elementary Functions",
- journal = "Ada Lett.",
- issue_date = "Fall 1991",
- volume = "XI",
- number = "7",
- month = "September",
- year = "1991",
- issn = "1094-3641",
- pages = "166--179",
- numpages = "14",
- url = "http://doi.acm.org/10.1145/123533.123545",
- doi = "10.1145/123533.123545",
- acmid = "123545",
- publisher = "ACM",
- address = "New York, NY, USA",
- paper = "Squi91.pdf",
- abstract = "
- This document provides the background on decisions that were made
- during the development of the specification for Generic Complex
- Elementary fuctions. It also rovides some information that was used to
- develop error bounds, range, domain and definitions of complex
- elementary functions."
+@InProceedings{Kalt97a,
+ author = "Kaltofen, E. and Shoup, V.",
+ title = "Fast polynomial factorization over high algebraic extensions of
+ finite fields",
+ booktitle = "Proc. 1997 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC97",
+ year = "1997",
+ pages = "184--188",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/KaSh97.pdf",
+ paper = "Kalt97a.pdf"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Squi91a,
- editor = "Squire, Jon S.",
- title = "Proposed Standard for a Generic Package of Complex
- Elementary Functions",
- journal = "Ada Lett.",
- issue_date = "Fall 1991",
- volume = "XI",
- number = "7",
- month = "September",
- year = "1991",
- issn = "1094-3641",
- pages = "140--165",
- numpages = "26",
- url = "http://doi.acm.org/10.1145/123533.123544",
- doi = "10.1145/123533.123544",
- acmid = "123544",
- publisher = "ACM",
- address = "New York, NY, USA",
- abstract = "
- This document defines the specification of a generic package of
- complex elementary functions called Generic Complex Elementary
- Functions. It does not provide the body of the package."
+@Article{Kalt98,
+ author = "Kaltofen, E. and Shoup, V.",
+ title = "Subquadratic-time factoring of polynomials over finite fields",
+ journal = "Math. Comput.",
+ month = "July",
+ year = "1998",
+ volume = "67",
+ number = "223",
+ pages = "1179--1197",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/98/KaSh98.pdf",
+ paper = "Kalt98.pdf"
}
\end{chunk}
-\subsection{Square-free Decomposition } %%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{axiom.bib}
-@article{Bern97,
- author = "Bernardin, Laurent",
- title = "On square-free factorization of multivariate polynomials over a
- finite field",
- journal = "Theoretical Computer Science",
- volume = "187",
- number = "1-2",
- year = "1997",
- month = "November",
- pages = "105-116",
- keywords = "axiomref",
- paper = "Bern97.pdf",
- abstract = "
- In this paper we present a new deterministic algorithm for computing
- the square-free decomposition of multivariate polynomials with
- coefficients from a finite field.
-
- Our algorithm is based on Yun's square-free factorization algorithm
- for characteristic 0. The new algorithm is more efficient than
- existing, deterministic algorithms based on Musser's squarefree
- algorithm
+@InProceedings{Kalt95a,
+ author = "Kaltofen, E. and Shoup, V.",
+ title = "Subquadratic-time factoring of polynomials over finite fields",
+ booktitle = "Proc. 27th Annual ACM Symp. Theory Comput.",
+ year = "1995",
+ publisher = "ACM Press",
+ address = "New York, N.Y.",
+ pages = "398--406",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/KaSh95.ps.gz",
+ paper = "Kalt95a.ps"
+}
- We will show that the modular approach presented by Yun has no
- significant performance advantage over our algorithm. The new
- algorithm is also simpler to implement and it can rely on any existing
- GCD algorithm without having to worry about choosing ``good'' evaluation
- points.
+\end{chunk}
- To demonstrate this, we present some timings using implementations in
- Maple (Char et al. 1991), where the new algorithm is used for Release
- 4 onwards, and Axiom (Jenks and Sutor, 1992) which is the only system
- known to the author to use and implementation of Yun's modular
- algorithm mentioned above."
+\begin{chunk}{axiom.bib}
+@InProceedings{Diaz95,
+ author = "Diaz, A. and Kaltofen, E.",
+ title = "On computing greatest common divisors with polynomials given by
+ black boxes for their evaluation",
+ booktitle = "Proc. 1995 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC95",
+ pages = "232--239",
+ year = "1995",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/DiKa95.ps.gz",
+ paper = "Diaz95.ps"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Chez07,
- author = "Ch\'eze, Guillaume and Lecerf, Gr\'egoire",
- title = "Lifting and recombination techniques for absolute factorization",
- journal = "Journal of Complexity",
- volume = "23",
- number = "3",
- year = "2007",
- month = "June",
- pages = "380-420",
- paper = "Chez07.pdf",
- abstract = "
- In the vein of recent algorithmic advances in polynomial factorization
- based on lifting and recombination techniques, we present new faster
- algorithms for computing the absolute factorization of a bivariate
- polynomial. The running time of our probabilistic algorithm is less
- than quadratic in the dense size of the polynomial to be factored."
+@InProceedings{Kalt88,
+ author = "Kaltofen, E. and Trager, B.",
+ title = "Computing with polynomials given by black boxes for their
+ evaluations: Greatest common divisors, factorization, separation of
+ numerators and denominators",
+ booktitle = "Proc. 29th Annual Symp. Foundations of Comp. Sci.",
+ pages = "296--305",
+ year = "1988",
+ organization = "IEEE",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/focs88.ps.gz",
+ paper = "Kalt88.ps"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Lece07,
- author = "Lecerf, Gr\'egoire",
- title = "Improved dense multivariate polynomial factorization algorithms",
- journal = "Journal of Symbolic Computation",
- volume = "42",
- number = "4",
- year = "2007",
- month = "April",
- pages = "477-494",
- paper = "Lece07.pdf",
- abstract = "
- We present new deterministic and probabilistic algorithms that reduce
- the factorization of dense polynomials from several variables to one
- variable. The deterministic algorithm runs in sub-quadratic time in
- the dense size of the input polynomial, and the probabilistic
- algorithm is softly optimal when the number of variables is at least
- three. We also investigate the reduction from several to two variables
- and improve the quantitative versions of Bertini's irreducibility theorem."
+@InProceedings{Kalt85b,
+ author = "Kaltofen, E.",
+ title = "Computing with polynomials given by straight-line programs {II};
+ sparse factorization",
+ booktitle = "Proc. 26th Annual Symp. Foundations of Comp. Sci.",
+ year = "1985",
+ pages = "451--458",
+ organization = "IEEE",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_focs.ps.gz",
+ paper = "Kalt85b.ps"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Wang77,
- author = "Wang, Paul S.",
- title = "An efficient squarefree decomposition algorithm",
- journal = "ACM SIGSAM Bulletin",
- volume = "11",
- number = "2",
- year = "1977",
- month = "May",
- pages = "4-6",
- paper = "Wang77.pdf",
- abstract = "
- The concept of polynomial squarefree decomposition is an important one
- in algebraic computation. The squarefree decomposition process has
- many uses in computer symbolic computation. A recent survey by D. Yun
- [3] describes many useful algorithms for this purpose. All of these
- methods depend on computing the greated common divisor (gcd) of the
- polynomial to be decomposed and its first derivative (with repect to
- some variable). In the multivariate case, this gcd computation is
- non-trivial and dominates the cost for the squarefree decompostion."
+@InProceedings{Kalt86,
+ author = "Kaltofen, E.",
+ title = "Uniform closure properties of p-computable functions",
+ booktitle = "Proc. 18th Annual ACM Symp. Theory Comput.",
+ year = "1986",
+ pages = "330--337",
+ organization = "ACM",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/86/Ka86_stoc.pdf",
+ paper = "Kalt86.pdf"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Wang79,
- author = "Wang, Paul S. and Trager, Barry M.",
- title = "New Algorithms for Polynomial Square-Free Decomposition
- over the Integers",
- journal = "SIAM Journal on Computing",
- volume = "8",
- number = "3",
- year = "1979",
- publisher = "Society for Industrial and Applied Mathematics",
- issn = "00975397",
- paper = "Wang79.pdf",
- abstract = "
- Previously known algorithms for polynomial square-free decomposition
- rely on greatest common divisor (gcd) computations over the same
- coefficient domain where the decomposition is to be performed. In
- particular, gcd of the given polynomial and its first derivative (with
- respect to some variable) is obtained to begin with. Application of
- modular homomorphism and $p$-adic construction (multivariate case) or
- the Chinese remainder algorithm (univariate case) results in new
- square-free decomposition algorithms which, generally speaking, take
- less time than a single gcd between the given polynomial and its first
- derivative. The key idea is to obtain one or several ``correct''
- homomorphic images of the desired square-free decomposition
- first. This provides information as to how many different square-free
- factors there are, their multiplicities and their homomorphic
- images. Since the multiplicities are known, only the square-free
- factors need to be constructed. Thus, these new algorithms are
- relatively insensitive to the multiplicities of the square-free factors."
+@InProceedings{Kalt87b,
+ author = "Kaltofen, E.",
+ title = "Single-factor Hensel lifting and its application to the
+ straight-line complexity of certain polynomials",
+ booktitle = "Proc. 19th Annual ACM Symp. Theory Comput.",
+ year = "1987",
+ pages = "443--452",
+ organization = "ACM",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_stoc.pdf",
+ paper = "Kalt87b.pdf"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@inproceedings{Yun76,
- author = "Yun, D.Y.Y",
- title = "On square-free decomposition algorithms",
- booktitle = "Proceedings of SYMSAC'76",
- year = "1976",
- keywords = "survey",
- pages = "26-35"
+@InCollection{Kalt89,
+ author = "Kaltofen, E.",
+ editor = "S. Micali",
+ title = "Factorization of polynomials given by straight-line programs",
+ booktitle = "Randomness and Computation",
+ pages = "375--412",
+ publisher = "JAI Press Inc.",
+ year = "1989",
+ volume = "5",
+ series = "Advances in Computing Research",
+ address = "Greenwhich, Connecticut",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_slpfac.pdf",
+ paper = "Kalt89.pdf"
}
\end{chunk}
-\section{Axiom Citations in the Literature}
-
-\subsection{A} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{chunk}{axiom.bib}
+@Article{Gao04,
+ author = "Gao, Shuhong and Kaltofen, E. and Lauder, A.",
+ title = "Deterministic distinct degree factorization for polynomials
+ over finite fields",
+ year = "2004",
+ journal = "Journal of Symbolic Computation",
+ volume = "38",
+ number = "6",
+ pages = "1461--1470",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/01/GKL01.pdf",
+ paper = "Gao04.pdf"
+}
-\begin{chunk}{ignore}
-\bibitem[ACM 89]{ACM89} ACM, editor
-Proceedings of the ACM-SIGSAM 1989 International
-Symposium on Symbolic and Algebraic Computation, ISSAC '89 ACM Press,
-New York, NY 10036, USA, 1989, , LCCN QA76.95.I59
- year = "1989",
- isbn = "0-89791-325-6",
- keywords = "axiomref",
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[ACM 94]{ACM94} ACM, editor
-ISSAC '94. Proceedings of the International
-Symposium on Symbolic and Algebraic Computation. ACM Press, New York, NY,
-10036, USA, 1994, . LCCN QA76.95.I59
- year = "1994",
- isbn = "0-89791-638-7",
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@Article{Kalt87c,
+ author = "Kaltofen, E.",
+ title = "Deterministic irreducibility testing of polynomials over
+ large finite fields",
+ journal = "Journal of Symbolic Computation",
+ year = "1987",
+ volume = "4",
+ pages = "77--82",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_jsc.ps.gz",
+ paper = "Kalt87c.ps"
+}
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Augo91,
- author = "Augot, D. and Charpin, P. and Sendrier, N.",
- title = "The miniumum distance of some binary codes via the
- Newton's identities",
- journal = "Cohen and Charping [CC91]",
- year = "1991",
- pages = "65-73",
- isbn = "0-387-54303-1",
- misc = "3-540-54303-1 (Berlin). LCCN QA268.E95 1990",
- keywords = "axiomref",
- paper = "Augo91.pdf"
+@Article{Kalt95b,
+ author = "Kaltofen, E.",
+ title = "Effective {Noether} irreducibility forms and applications",
+ journal = "J. Comput. System Sci.",
+ year = "1995",
+ volume = "50",
+ number = "2",
+ pages = "274--295",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/Ka95_jcss.pdf",
+ paper = "Kalt95b.pdf"
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Adams 94]{AL94}
- author = "Adams, William W. and Loustaunau, Philippe",
- title = "An Introduction to Gr\"obner Bases",
- year = "1994",
-American Mathematical Society (1994)
- isbn = "0-8218-3804-0",
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@Article{Kalt85a,
+ author = "Kaltofen, E.",
+ title = "Fast parallel absolute irreducibility testing",
+ journal = "Journal of Symbolic Computation",
+ year = "1985",
+ volume = "1",
+ number = "1",
+ pages = "57--67",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_jsc.pdf",
+ paper = "Kalt85a.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Andrews 84]{And84}
- author = "Andrews, George E.",
- title = "Ramanujan and SCRATCHPAD",
- year = "1984",
- pages = "383-??",
- keywords = "axiomref",
-In Golden and Hussain [GH84]
-
-\end{chunk}
+\begin{chunk}{axiom.bib}
+@Article{Gath85a,
+ author = "{von zur Gathen}, Joachim and Kaltofen, E.",
+ title = "Factoring multivariate polynomials over finite fields",
+ journal = "Math. Comput.",
+ year = "1985",
+ volume = "45",
+ pages = "251--261",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/85/GaKa85_mathcomp.ps.gz",
+ paper = "Gath85a.ps"
+}
-\begin{chunk}{ignore}
-\bibitem[Andrews 88]{And88}
- author = "Andrews, G. E.",
- title = "Application of Scratchpad to problems in special functions and
- combinatorics",
- year = "1988"
- pages = "158-??",
- isbn = "3-540-18928-9",
- keywords = "axiomref",
-In Janssen [Jan88], pages 158-?? ISBN
-0-387-18928-9 LCCN QA155.7.E4T74
+\begin{chunk}{axiom.bib}
+@Article{Kalt85e,
+ author = "Kaltofen, E.",
+ title = "Polynomial-time reductions from multivariate to bi- and univariate
+ integral polynomial factorization",
+ journal = "{SIAM} J. Comput.",
+ year = "1985",
+ volume = "14",
+ number = "2",
+ pages = "469--489",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_sicomp.pdf",
+ paper = "Kalt85e.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Anon 91]{Ano91}
- author = "Anonymous",
- year = "1991,
- keywords = "axiomref",
-Proceedings 1991 Annual Conference, American Society for
-Engineering Education. Challenges of a Changing World. ASEE, Washington, DC
- 2 vol.
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt82a,
+ author = "Kaltofen, E.",
+ title = "A polynomial-time reduction from bivariate to univariate
+ integral polynomial factorization",
+ booktitle = "Proc. 23rd Annual Symp. Foundations of Comp. Sci.",
+ year = "1982",
+ pages = "57--64",
+ organization = "IEEE",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/82/Ka82_focs.pdf",
+ paper = "Kalt82a.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Anon 92]{Ano92}
- author = "Anonymous",
- year = "1992",
- keywords = "axiomref",
-Programming environments for high-level scientific problem solving.
-IFIP TC2/WG 2.5 working conference. IFIP Transactions. A Computer Science
-and Technology, A-2:??, CODEN ITATEC. ISSN 0926-5473
-
-\end{chunk}
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt03,
+ author = "Kaltofen, Erich",
+ title = "Polynomial Factorization: a Success Story",
+ year = "2003",
+ booktitle = "Symbolic Algebraic Comput. Internat. Symp. ISSAC '88 Proc.",
+ crossref = "ISSAC03",
+ pages = "3--4",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/03/Ka03.pdf",
+ keywords = "survey",
+ paper = "Kalt03.pdf"
+}
-\begin{chunk}{ignore}
-\bibitem[Anono 95]{Ano95}
- author =Anonymous
- keywords = "axiomref",
- year = "1995",
-GAMM 94 annual meeting. Zeitschrift fur Angewandte Mathematik und
-Physik, 75 (suppl. 2), CODEN ZAMMAX, ISSN 0044-2267
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt92b,
+ author = "Kaltofen, E.",
+ title = "Polynomial factorization 1987-1991",
+ booktitle = "Proc. LATIN '92",
+ editor = "I. Simon",
+ series = "Lect. Notes Comput. Sci.",
+ volume = "583",
+ pages = "294--313",
+ publisher = "Springer-Verlag",
+ year = "1992",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/92/Ka92_latin.pdf",
+ keywords = "survey",
+ paper = "Kalt92b.pdf"
+}
\end{chunk}
-\subsection{B} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{axiom.bib}
-@article{Bacl14,
- author = "Baclawski, Krystian",
- title = "SPAD language type checker",
- journal = "unknown",
- year = "2014",
- url = "http://github.com/cahirwpz/phd",
- keywords = "axiomref",
- abstract = "
- The project aims to deliver a new type checker for SPAD language.
- Several improvements over current type checker are planned.
- \begin{itemize}
- \item introduce better type inference
- \item introduce modern language constructs
- \item produce understandable diagnostic messages
- \item eliminate well known bugs in the type system
- \item find new type errors
- \end{itemize}"
+@InCollection{Kalt90c,
+ author = "Kaltofen, E.",
+ editor = "D. V. Chudnovsky and R. D. Jenks",
+ title = "Polynomial Factorization 1982-1986",
+ booktitle = "Computers in Mathematics",
+ pages = "285--309",
+ publisher = "Marcel Dekker, Inc.",
+ year = "1990",
+ volume = "125",
+ series = "Lecture Notes in Pure and Applied Mathematics",
+ address = "New York, N. Y.",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/Ka90_survey.ps.gz",
+ keywords = "survey",
+ paper = "Kalt90c.ps"
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Blair 70]{BGJ70}
- author = "Blair, Fred W and Griesmer, James H. and Jenks, Richard D.",
- title = "An interactive facility for symbolic mathematics",
- year = "1970",
- pages = "394-419",
- keywords = "axiomref",
-Proc. International Computing Symposium, Bonn, Germany,
+\begin{chunk}{axiom.bib}
+@InCollection{Kalt82b,
+ author = "Kaltofen, E.",
+ title = "Polynomial factorization",
+ editor = "B. Buchberger and G. Collins and R. Loos",
+ booktitle = "Computer Algebra",
+ edition = "2",
+ pages = "95--113",
+ publisher = "Springer-Verlag",
+ year = "1982",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/82/Ka82_survey.ps.gz",
+ keywords = "survey",
+ paper = "Kalt82b.ps"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Blair 70a]{BJ70}
- author = "Blair, Fred W. and Jenks, Richard D.",
- title = "LPL: LISP programming language",
- year = "1970",
- keywords = "axiomref",
-IBM Research Report, RC3062 Sept
-
-\end{chunk}
+\section{Branch Cuts} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{axiom.bib}
-\bibitem[Broadbery 95]{BGDW95}
- author = "Broadbery, P. A. and G{\'o}mez-D{\'\i}az, T. and Watt, S. M.",
- title = "On the Implementation of Dynamic Evaluation",
- year = "1995",
- pages = "77-84",
- keywords = "axiomref",
- isbn = "0-89791-699-9",
- url = "http://pdf.aminer.org/000/449/014/on_the_implementation_of_dynamic_evaluation.pdf",
- paper = "BGDW95.pdf",
+@article{Beau03,
+ author = "Beaumont, James and Bradford, Russell and Davenport, James H.",
+ title = "Better simplification of elementary functions through power series",
+ journal = "2003 International Symposium on Symbolic and Algebraic Computation",
+ series = "ISSAC'03",
+ year = "2003",
+ month = "August",
+ paper = "Beau03.pdf",
abstract = "
- Dynamic evaluation is a technique for producing multiple results
- according to a decision tree which evolves with program execution.
- Sometimes it is desired to produce results for all possible branches
- in the decision tree, while on other occasions, it may be sufficient
- to compute a single result which satisfies certain properties. This
- techinique finds use in computer algebra where computing the correct
- result depends on recognizing and properly handling special cases of
- parameters. In previous work, programs using dynamic evaluation have
- explored all branches of decision trees by repeating the computations
- prior to decision points.
+ In [5], we introduced an algorithm for deciding whether a proposed
+ simplification of elementary functions was correct in the presence of
+ branch cuts. This algorithm used multivalued function simplification
+ followed by verification that the branches were consistent.
- This paper presents two new implementations of dynamic evaluation
- which avoid recomputing intermediate results. The first approach uses
- Scheme ``continuations'' to record state for resuming program
- execution. The second implementation uses the Unix ``fork'' operation
- to form new processes to explore alternative branches in parallel."
+ In [14] an algorithm was presented for zero-testing functions defined
+ by ordinary differential equations, in terms of their power series.
+
+ The purpose of the current paper is to investigate merging the two
+ techniques. In particular, we will show an explicit reduction to the
+ constant problem [16]."
}
\end{chunk}
\begin{chunk}{axiom.bib}
-\bibitem[Boehm 89]{Boe89}
-@inproceedings{Boe89,
- author = "Boehm, Hans-J.",
- title = "Type Inference in the Presence of Type Abstraction",
- year = "1989",
- pages = "192-206",
- keywords = "axiomref",
- url = "http://www.acm.org/pubs/citations/proceedings/pldi/73141/p192-boehm",
- paper = "Boe89.pdf",
- booktitle = "ACM SIGPLAN Notices",
- volume = "24",
- number = "7",
- month = "July",
+@article{Beau07,
+ author = "Beaumont, James C. and Bradford, Russell J. and
+ Davenport, James H. and Phisanbut, Nalina",
+ title = "Testing elementary function identities using CAD",
+ journal = "Applicable Algebra in Engineering, Communication and Computing",
+ year = "2007",
+ volume = "18",
+ number = "6",
+ issn = "0938-1279",
+ publisher = "Springer-Verlag",
+ pages = "513-543",
+ paper = "Beau07.pdf",
abstract = "
- A number of recent programming language designs incorporate a type
- checking system based on the Girard-Reynolds polymorphic
- $\lambda$-calculus. This allows the construction of general purpose,
- reusable software without sacrificing compile-time type checking. A
- major factor constraining the implementation of these languages is the
- difficulty of automatically inferring the lengthy type information
- that is otherwise required if full use is made of these
- languages. There is no known algorithm to solve any natural and fully
- general formulation of the ``type inference'' problem. One very
- reasonable formulation of the problem is known to be undecidable.
-
- Here we define a restricted version of the type inference problem and
- present an efficient algorithm for its solution. We argue that the
- restriction is sufficiently weak to be unobtrusive in practice."
+ One of the problems with manipulating function identities in computer
+ algebra systems is that they often involve functions which are
+ multivalued, whilst most users tend to work with single-valued
+ functions. The problem is that many well-known identities may no
+ longer be true everywhere in the complex plane when working with their
+ single-valued counterparts. Conversely, we cannot ignore them, since
+ in particular contexts they may be valid. We investigate the
+ practicality of a method to verify such identities by means of an
+ experiment; this is based on a set of test examples which one might
+ realistically meet in practice. Essentially, the method works as
+ follows. We decompose the complex plane via means of cylindrical
+ algebraic decomposition into regions with respect to the branch cuts
+ of the functions. We then test the identity numerically at a sample
+ point in the region. The latter step is facilitated by the notion of
+ the {\sl adherence} of a branch cut, which was previously introduced
+ by the authors. In addition to presenting the results of the
+ experiment, we explain how adherence relates to the proposal of
+ {\sl signed zeros} by W. Kahan, and develop this idea further in order to
+ allow us to cover previously untreatable cases. Finally, we discuss
+ other ways to improve upon our general methodology as well as topics
+ for future research."
}
-
+
\end{chunk}
\begin{chunk}{axiom.bib}
-@inproceedings{BHGM04,
- author = "Boulton, Richard and Hardy, Ruth and Gottliebsen, Hanne
- and Martin, Ursula",
- title = "Design verification for control engineering",
- year = "2004",
- month = "April",
- booktitle = "Proc 4th Int. Conf. on Integrated Formal Methods",
- keywords = "axiomref",
+@article{Brad02,
+ author="Bradford, Russell and Corless, RobertM. and Davenport, JamesH. and
+ Jeffrey, DavidJ. and Watt, StephenM.",
+ title="Reasoning about the Elementary Functions of Complex Analysis",
+ journal="Annals of Mathematics and Artificial Intelligence",
+ year="2002",
+ issn="1012-2443",
+ volume="36",
+ number="3",
+ doi="10.1023/A:1016007415899",
+ url="http://dx.doi.org/10.1023/A%3A1016007415899",
+ publisher="Kluwer Academic Publishers",
+ keywords="elementary functions; branch cuts; complex identities",
+ pages="303-318",
+ paper = "Brad02.pdf",
abstract = "
- We introduce control engineering as a new domain of application for
- formal methods. We discuss design verification, drawing attention to
- the role played by diagrammatic evaluation criteria involving numeric
- plots of a design, such as Nichols and Bode plots. We show that
- symbolic computation and computational logic can be used to discharge
- these criteria and provide symbolic, automated, and very general
- alternatives to these standard numeric tests. We illustrate our work
- with reference to a standard reference model drawn from military
- avionics."
+ There are many problems with the simplification of elementary
+ functions, particularly over the complex plane, though not
+ exclusively. Systems tend to make ``howlers'' or not to simplify
+ enough. In this paper we outline the ``unwinding number'' approach to
+ such problems, and show how it can be used to prevent errors and to
+ systematise such simplification, even though we have not yet reduced
+ the simplification process to a complete algorithm. The unsolved
+ problems are probably more amenable to the techniques of artificial
+ intelligence and theorem proving than the original problem of complex
+ variable analysis."
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Boulanger 91]{Bou91}
- author = "Boulanger, Jean-Louis",
- title = "Etude de la compilation de scratchpad 2",
- year = "1991",
- month = "September",
-Rapport de DEA Universite dl lille 1
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@inproceedings{Chyz11,
+ author = "Chyzak, Fr\'ed\'eric and Davenport, James H. and Koutschan, Christoph and Salvy, Bruno",
+ title = "On Kahan's Rules for Determining Branch Cuts",
+ booktitle = "Proc. 13th Int. Symp. on Symbolic and Numeric Algorithms for Scientific Computing",
+ year = "2011",
+ isbn = "978-1-4673-0207-4",
+ location = "Timisoara",
+ pages = "47-51",
+ doi = "10.1109/SYNASC.2011.51",
+ acmid = "258794",
+ publisher = "IEEE",
+ paper = "Chyz11.pdf",
+ abstract = "
+ In computer algebra there are different ways of approaching the
+ mathematical concept of functions, one of which is by defining them as
+ solutions of differential equations. We compare different such
+ appraoches and discuss the occurring problems. The main focus is on
+ the question of determining possible branch cuts. We explore the
+ extent to which the treatment of branch cuts can be rendered (more)
+ algorithmic, by adapting Kahan's rules to the differential equation
+ setting."
+}
\end{chunk}
\begin{chunk}{axiom.bib}
-@misc{Bou93a,
- author = "Boulanger, Jean-Louis",
- title = "Axiom, language fonctionnel \`a d\'evelopement objet",
- year = "1993",
- month = "October",
- paper = "Bou93a.pdf",
- keywords = "axiomref"
+@article{Dave10,
+ author = "Davenport, James",
+ title = {The Challenges of Multivalued "Functions"},
+ journal = "Lecture Notes in Computer Science",
+ volume = "6167",
+ year = "2010",
+ pages = "1-12",
+ paper = "Dave10.pdf",
+ abstract = "
+ Although, formally, mathematics is clear that a function is a
+ single-valued object, mathematical practice is looser, particularly
+ with n-th roots and various inverse functions. In this paper, we point
+ out some of the looseness, and ask what the implications are, both for
+ Artificial Intelligence and Symbolic Computation, of these practices.
+ In doing so, we look at the steps necessary to convert existing tests
+ into
+ \begin{itemize}
+ \item (a) rigorous statements
+ \item (b) rigorously proved statements
+ \end{itemize}
+ In particular we ask whether there might be a constant ``de Bruij factor''
+ [18] as we make these texts more formal, and conclude that the answer
+ depends greatly on the interpretation being placed on the symbols."
}
-
+
\end{chunk}
\begin{chunk}{axiom.bib}
-@misc{Bou93b,
- author = "Boulanger, Jean-Louis",
- title = "AXIOM, A Functional Language with Object Oriented Development",
- year = "1993",
- paper = "Bou93b.pdf",
- keywords = "axiomref",
+@article{Dave12,
+ author = "Davenport, James H. and Bradford, Russell and England, Matthew
+ and Wilson, David",
+ title = "Program Verification in the presence of complex numbers, functions
+ with branch cuts etc",
+ journal = "14th Int. Symp. on Symbolic and Numeric Algorithms for
+ Scientific Computing",
+ year = "2012",
+ series = "SYNASC'12",
+ pages = "83-88",
+ publisher = "IEEE",
+ paper = "Dave12.pdf",
abstract = "
- We present in this paper, a study about the computer algebra system
- Axiom, which gives us many very interesting Software engineering
- concepts. This language is a functional language with an Object
- Oriented Development. This feature is very important for modeling the
- mathematical world (Hierarchy) and provides a running with
- mathematical sense. (All objects are functions). We present many
- problems of running and development in Axiom. We can note that Aiom is
- the only system of this category."
+ In considering the reliability of numerical programs, it is normal to
+ ``limit our study to the semantics dealing with numerical precision''.
+ On the other hand, there is a great deal of work on the reliability of
+ programs that essentially ignores the numerics. The thesis of this
+ paper is that there is a class of problems that fall between the two,
+ which could be described as ``does the low-level arithmetic implement
+ the high-level mathematics''. Many of these problems arise because
+ mathematics, particularly the mathematics of the complex numbers, is
+ more difficult than expected; for example the complex function log is
+ not continuous, writing down a program to compute an inverse function
+ is more complicated than just solving an equation, and many algebraic
+ simplification rules are not universally valid.
+
+ The good news is that these problems are theoretically capable of
+ being solved, and are practically close to being solved, but not yet
+ solved, in several real-world examples. However, there is still a long
+ way to go before implementations match the theoretical possibilities."
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Boulanger 94]{Bou94}
- author = "Boulanger, J.L.",
- title = "Object Oriented Method for Axiom",
- year = "1995",
- month = "February",
- pages = "33-41",
- paper = "Bou94.pdf",
-ACM SIGPLAN Notices, 30(2) CODEN SINODQ ISSN 0362-1340
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Jeff04,
+ author = "Jeffrey, D. J. and Norman, A. C.",
+ title = "Not Seeing the Roots for the Branches: Multivalued Functions in
+ Computer Algebra",
+ journal = "SIGSAM Bull.",
+ issue_date = "September 2004",
+ volume = "38",
+ number = "3",
+ month = "September",
+ year = "2004",
+ issn = "0163-5824",
+ pages = "57--66",
+ numpages = "10",
+ url = "http://doi.acm.org/10.1145/1040034.1040036",
+ doi = "10.1145/1040034.1040036",
+ acmid = "1040036",
+ publisher = "ACM",
+ address = "New York, NY, USA",
+ paper = "Jeff04.pdf",
abstract = "
- Axiom is a very powerful computer algebra system which combines two
- language paradigms (functional and OOP). Mathematical world is complex
- and mathematicians use abstraction to design it. This paper presents
- some aspects of the object oriented development in Axiom. The Axiom
- programming is based on several new tools for object oriented
- development, it uses two levels of class and some operations such that
- {\sl coerce}, {\sl retract}, or {\sl convert} which permit the type
- evolution. These notions introduce the concept of multi-view."
+ We discuss the multiple definitions of multivalued functions and their
+ suitability for computer algebra systems. We focus the discussion by
+ taking one specific problem and considering how it is solved using
+ different definitions. Our example problem is the classical one of
+ calculating the roots of a cubic polynomial from the Cardano formulae,
+ which contains fractional powers. We show that some definitions of
+ these functions result in formulae that are correct only in the sense
+ that they give candidates for solutions; these candidates must then be
+ tested. Formulae that are based on single-valued functions, in
+ contract, are efficient and direct."
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Bronstein 87]{Bro87}
- author = "Bronstein, Manuel",
- title = "Integration of Algebraic and Mixed Functions",
- year = "1987",
-in [Wit87], p18
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@inproceedings{Kaha86,
+ author = "Kahan, W.",
+ title = "Branch cuts for complex elementary functions",
+ booktitle = "The State of the Art in Numerical Analysis",
+ year = "1986",
+ month = "April",
+ editor = "Powell, M.J.D and Iserles, A.",
+ publisher = "Oxford University Press"
+}
-\end{chunk}
+\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Bronstein 89]{Bro89}
- author= "Bronstein, M.",
- title = "Simplification of real elementary functions",
- year = "1989",
- pages = "207-211",
- isbn = "0-89791-325-6",
-ACM [ACM89] pages LCCN QA76.95.I59 1989
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Rich96,
+ author = "Rich, Albert D. and Jeffrey, David J.",
+ title = "Function Evaluation on Branch Cuts",
+ journal = "SIGSAM Bull.",
+ issue_date = "June 1996",
+ volume = "30",
+ number = "2",
+ month = "June",
+ year = "1996",
+ issn = "0163-5824",
+ pages = "25--27",
+ numpages = "3",
+ url = "http://doi.acm.org/10.1145/235699.235704",
+ doi = "10.1145/235699.235704",
+ acmid = "235704",
+ publisher = "ACM",
+ address = "New York, NY, USA",
abstract = "
- We describe an algorithm, based on Risch's real structure theorem, that
- determines explicitly all the algebraic relations among a given set of
- real elementary functions. We also provide examples from its
- implementation that illustrate the advantages over the use of complex
- logarithms and exponentials."
+ Once it is decided that a CAS will evaluate multivalued functions on
+ their principal branches, questions arise concerning the branch
+ definitions. The first questions concern the standardization of the
+ positions of the branch cuts. These questions have largely been
+ resolved between the various algebra systems and the numerical
+ libraries, although not completely. In contrast to the computer
+ systems, many mathematical textbooks are much further behind: for
+ example, many popular textbooks still specify that the argument of a
+ complex number lies between 0 and $2\pi$. We do not intend to discuss
+ these first questions here, however. Once the positions of the branch
+ cuts have been fixed, a second set of questions arises concerning the
+ evaluation of functions on their branch cuts."
}
\end{chunk}
\begin{chunk}{axiom.bib}
-\bibitem[Bronstein 91a]{Bro91a}
-@inproceedings{Bron91a,
- author = "Bronstein, M.",
- title = "The Risch Differential Equation on an Algebraic Curve",
- booktitle = "Proc. 1991 Int. Symp. on Symbolic and Algebraic Computation",
- series = "ISSAC'91",
- year = "1991",
- pages = "241-246",
- isbn = "0-89791-437-6",
- publisher = "ACM, NY",
- keywords = "axiomref",
- paper = "Bro91a.pdf",
+@article{Patt96,
+ author = "Patton, Charles M.",
+ title = "A Representation of Branch-cut Information",
+ journal = "SIGSAM Bull.",
+ issue_date = "June 1996",
+ volume = "30",
+ number = "2",
+ month = "June",
+ year = "1996",
+ issn = "0163-5824",
+ pages = "21--24",
+ numpages = "4",
+ url = "http://doi.acm.org/10.1145/235699.235703",
+ doi = "10.1145/235699.235703",
+ acmid = "235703",
+ publisher = "ACM",
+ address = "New York, NY, USA",
+ paper = "Patt96.pdf",
abstract = "
- We present a new rational algorithm for solving Risch differential
- equations over algebraic curves. This algorithm can also be used to
- solve $n^{th}$-order linear ordinary differential equations with
- coefficients in an algebraic extension of the rational functions. In
- the general (``mixed function'') case, this algorithm finds the
- denominator of any solution of the equation."
+ Handling (possibly) multi-valued functions is a problem in all current
+ computer algebra systems. The problem is not an issue of technology.
+ Its solution, however, is tied to a uniform handling of the issues by
+ the mathematics community."
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Bronstein 91c]{Bro91c}
- author = "Bronstein, Manuel",
- title = "Computer Algebra and Indefinite Integrals",
+\begin{chunk}{axiom.bib}
+@article{Squi91,
+ author = "Squire, Jon S.",
+ title = "Rationale for the Proposed Standard for a Generic Package of
+ Complex Elementary Functions",
+ journal = "Ada Lett.",
+ issue_date = "Fall 1991",
+ volume = "XI",
+ number = "7",
+ month = "September",
year = "1991",
- paper = "Bro91c.pdf",
-in Computer Aided Proofs in Analysis, K.R. Meyers et al. (eds)
-Springer-Verlag, NY (1991)
- keywords = "axiomref",
+ issn = "1094-3641",
+ pages = "166--179",
+ numpages = "14",
+ url = "http://doi.acm.org/10.1145/123533.123545",
+ doi = "10.1145/123533.123545",
+ acmid = "123545",
+ publisher = "ACM",
+ address = "New York, NY, USA",
+ paper = "Squi91.pdf",
abstract = "
- We give an overview, from an analytical point of view, of decision
- procedures for determining whether an elementary function has an
- elementary function has an elementary antiderivative. We give examples
- of algebraic functions which are integrable and non-integrable in
- closed form, and mention the current implementation of various computer
- algebra systems."
+ This document provides the background on decisions that were made
+ during the development of the specification for Generic Complex
+ Elementary fuctions. It also rovides some information that was used to
+ develop error bounds, range, domain and definitions of complex
+ elementary functions."
}
-\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Bronstein 92]{Bro92}
- author = "Bronstein, M.",
- title = "Linear Ordinary Differential Equations: Breaking Through the
- Order 2 Barrier",
- year = "1992",
- url =
- "http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac92.ps.gz",
- paper = "Bro92.pdf",
- keywords = "axiomref",
- abstract = "
- A major subproblem for algorithms that either factor ordinary linear
- differential equations or compute their closed form solutions is to
- find their solutions $y$ which satisfy $y^{'}/y \in \overline{K}(x)$
- where $K$ is the constant field for the coefficients of the equation.
- While a decision procedure for this subproblem was known in the
- $19^{th}$ century, it requires factoring polynomials over
- $\overline{K}$ and has not been implemented in full generality. We
- present here an efficient algorithm for this subproblem, which has
- been implemented in the AXIOM computer algebra system for equations of
- arbitrary order over arbitrary fields of characteristic 0. This
- algorithm never needs to compute with the individual complex
- singularities of the equation, and algebraic numbers are added only
- when they appear in the potential solutions. Implementation of the
- complete Singer algorithm for $n=2,3$ based on this building block is
- in progress."
-}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Bronstein 93]{Bro93}
- author = "Bronstein, Manuel (ed)",
- year = "1993",
- month = "July"
- isbn = "0-89791-604-2",
-ISSAC'93: proceedings of the 1993 International Symposium on Symbolic
-and Algebraic Computation, Kiev, Ukraine,
-ACM Press New York, NY 10036, USA, ISBN
-LCCN QA76.95 I59 1993 ACM order number 505930
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Squi91a,
+ editor = "Squire, Jon S.",
+ title = "Proposed Standard for a Generic Package of Complex
+ Elementary Functions",
+ journal = "Ada Lett.",
+ issue_date = "Fall 1991",
+ volume = "XI",
+ number = "7",
+ month = "September",
+ year = "1991",
+ issn = "1094-3641",
+ pages = "140--165",
+ numpages = "26",
+ url = "http://doi.acm.org/10.1145/123533.123544",
+ doi = "10.1145/123533.123544",
+ acmid = "123544",
+ publisher = "ACM",
+ address = "New York, NY, USA",
+ abstract = "
+ This document defines the specification of a generic package of
+ complex elementary functions called Generic Complex Elementary
+ Functions. It does not provide the body of the package."
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Brunelli 08]{Brun08}
- author = "Brunelli, J.C.",
- title = "Streams and Lazy Evaluation Applied to Integrable Models",
- year = "2008",
- url = "http://arxiv.org/PS_cache/nlin/pdf/0408/0408058v1.pdf",
- paper = "Brun08.pdf",
+\section{Square-free Decomposition } %%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@article{Bern97,
+ author = "Bernardin, Laurent",
+ title = "On square-free factorization of multivariate polynomials over a
+ finite field",
+ journal = "Theoretical Computer Science",
+ volume = "187",
+ number = "1-2",
+ year = "1997",
+ month = "November",
+ pages = "105-116",
keywords = "axiomref",
+ paper = "Bern97.pdf",
abstract = "
- Computer algebra procedures to manipulate pseudo-differential
- operators are implemented to perform calculations with integrable
- models. We use lazy evaluation and streams to represent and operate
- with pseudo-differential operators. No order of truncation is needed
- since terms are produced on demand. We give a series of concrete
- examples using the computer algebra language MAPLE."
+ In this paper we present a new deterministic algorithm for computing
+ the square-free decomposition of multivariate polynomials with
+ coefficients from a finite field.
-\end{chunk}
+ Our algorithm is based on Yun's square-free factorization algorithm
+ for characteristic 0. The new algorithm is more efficient than
+ existing, deterministic algorithms based on Musser's squarefree
+ algorithm
-\begin{chunk}{ignore}
-\bibitem[Bronstein 93]{BS93}
- author = "Bronstein, Manuel and Salvy, Bruno",
- title = "Full Partial Fraction Decomposition of Rational Functions",
- year = "1993",
- pages = "157-160",
- isbn = "0-89791-604-2",
-In Bronstein [Bro93] LCCN QA76.95 I59 1993
- keywords = "axiomref",
+ We will show that the modular approach presented by Yun has no
+ significant performance advantage over our algorithm. The new
+ algorithm is also simpler to implement and it can rely on any existing
+ GCD algorithm without having to worry about choosing ``good'' evaluation
+ points.
+
+ To demonstrate this, we present some timings using implementations in
+ Maple (Char et al. 1991), where the new algorithm is used for Release
+ 4 onwards, and Axiom (Jenks and Sutor, 1992) which is the only system
+ known to the author to use and implementation of Yun's modular
+ algorithm mentioned above."
+}
\end{chunk}
\begin{chunk}{axiom.bib}
-@misc{Bro92a,
- author = "Bronstein, Manuel",
- title = "Integration and Differential Equations in Computer Algebra",
- year = "1992",
- url = "http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.576",
- paper = "Bro92a.pdf",
- keywords = "axiomref",
+@article{Chez07,
+ author = "Ch\'eze, Guillaume and Lecerf, Gr\'egoire",
+ title = "Lifting and recombination techniques for absolute factorization",
+ journal = "Journal of Complexity",
+ volume = "23",
+ number = "3",
+ year = "2007",
+ month = "June",
+ pages = "380-420",
+ paper = "Chez07.pdf",
abstract = "
- We describe in this paper how the problems of computing indefinite
- integrals and solving linear ordinary differential equations in closed
- form are now solved by computer algebra systems. After a brief review
- of the mathematical history of those problems, we outline the two
- major algorithms for them (respectively the Risch and Singer
- algorithms) and the recent improvements on those algorithms which has
- allowed them to be implemented."
+ In the vein of recent algorithmic advances in polynomial factorization
+ based on lifting and recombination techniques, we present new faster
+ algorithms for computing the absolute factorization of a bivariate
+ polynomial. The running time of our probabilistic algorithm is less
+ than quadratic in the dense size of the polynomial to be factored."
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Beneke 94]{BS94}
- author = "Beneke, T. and Schwippert, W.",
- title = "Double-track into the future: MathCAD will gain new users with
- Standard and Plus versions",
- year = "1994",
- month = "July",
- pages = "107-110",
- keywords = "axiomref",
-Elektronik, 43(15) CODEN EKRKAR ISSN 0013-5658
-
-\end{chunk}
-
-\begin{chunk}{ignore}
-\bibitem[Bronstein 97a]{Bro97a}
- author = "Bronstein, Manuel and Weil, Jacques-Arthur",
- title = "On Symmetric Powers of Differential Operators",
- series = "ISSAC'97",
- year = "1997",
- pages = "156-163",
- keywords = "axiomref",
- url =
- "http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html"
- paper = "Bro97a.pdf",
- publisher = "ACM, NY",
+\begin{chunk}{axiom.bib}
+@article{Lece07,
+ author = "Lecerf, Gr\'egoire",
+ title = "Improved dense multivariate polynomial factorization algorithms",
+ journal = "Journal of Symbolic Computation",
+ volume = "42",
+ number = "4",
+ year = "2007",
+ month = "April",
+ pages = "477-494",
+ paper = "Lece07.pdf",
abstract = "
- We present alternative algorithms for computing symmetric powers of
- linear ordinary differential operators. Our algorithms are applicable
- to operators with coefficients in arbitrary integral domains and
- become faster than the traditional methods for symmetric powers of
- sufficiently large order, or over sufficiently complicated coefficient
- domains. The basic ideas are also applicable to other computations
- involving cyclic vector techniques, such as exterior powers of
- differential or difference operators."
+ We present new deterministic and probabilistic algorithms that reduce
+ the factorization of dense polynomials from several variables to one
+ variable. The deterministic algorithm runs in sub-quadratic time in
+ the dense size of the input polynomial, and the probabilistic
+ algorithm is softly optimal when the number of variables is at least
+ three. We also investigate the reduction from several to two variables
+ and improve the quantitative versions of Bertini's irreducibility theorem."
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Borwein 00]{Bor00}
- author = "Borwein, Jonathan",
- title = "Multimedia tools for communicating mathematics",
- year = "2000",
- pages = "58",
- isbn = "3-540-42450-4",
- publisher = "Springer-Verlag",
- keywords = "axiomref"
+\begin{chunk}{axiom.bib}
+@article{Wang77,
+ author = "Wang, Paul S.",
+ title = "An efficient squarefree decomposition algorithm",
+ journal = "ACM SIGSAM Bulletin",
+ volume = "11",
+ number = "2",
+ year = "1977",
+ month = "May",
+ pages = "4-6",
+ paper = "Wang77.pdf",
+ abstract = "
+ The concept of polynomial squarefree decomposition is an important one
+ in algebraic computation. The squarefree decomposition process has
+ many uses in computer symbolic computation. A recent survey by D. Yun
+ [3] describes many useful algorithms for this purpose. All of these
+ methods depend on computing the greated common divisor (gcd) of the
+ polynomial to be decomposed and its first derivative (with repect to
+ some variable). In the multivariate case, this gcd computation is
+ non-trivial and dominates the cost for the squarefree decompostion."
+}
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{BT94,
- author = "Brown, R. and Tonks, A.",
- title = "Calculations with simplicial and cubical groups in AXIOM",
- journal = "Journal of Symbolic Computation",
- volume = "17",
- number = "2",
- pages = "159-179",
- year = "1994",
- month = "February",
- misc = "CODEN JSYCEH ISSN 0747-7171",
- keywords = "axiomref"
+@article{Wang79,
+ author = "Wang, Paul S. and Trager, Barry M.",
+ title = "New Algorithms for Polynomial Square-Free Decomposition
+ over the Integers",
+ journal = "SIAM Journal on Computing",
+ volume = "8",
+ number = "3",
+ year = "1979",
+ publisher = "Society for Industrial and Applied Mathematics",
+ issn = "00975397",
+ paper = "Wang79.pdf",
+ abstract = "
+ Previously known algorithms for polynomial square-free decomposition
+ rely on greatest common divisor (gcd) computations over the same
+ coefficient domain where the decomposition is to be performed. In
+ particular, gcd of the given polynomial and its first derivative (with
+ respect to some variable) is obtained to begin with. Application of
+ modular homomorphism and $p$-adic construction (multivariate case) or
+ the Chinese remainder algorithm (univariate case) results in new
+ square-free decomposition algorithms which, generally speaking, take
+ less time than a single gcd between the given polynomial and its first
+ derivative. The key idea is to obtain one or several ``correct''
+ homomorphic images of the desired square-free decomposition
+ first. This provides information as to how many different square-free
+ factors there are, their multiplicities and their homomorphic
+ images. Since the multiplicities are known, only the square-free
+ factors need to be constructed. Thus, these new algorithms are
+ relatively insensitive to the multiplicities of the square-free factors."
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@misc{Brow95,
- author = "Brown, Ronald and Dreckmann, Winfried",
- title = "Domains of data and domains of terms in AXIOM",
- year = "1995",
- keywords = "axiomref",
- paper = "DB95.pdf",
- abstract = "
- The main new concept we wish to illustrate in this paper is a
- distinction between ``domains of data'' and ``domains of terms'', and
- its use in the programming of certain mathematical structures.
- Although this distinction is implicit in much of the programming work
- that has gone into the construction of Axiom categories and domains,
- we believe that a formalisation of this is new, that standards and
- conventions are necessary and will be useful in various other
- contexts. We shall show how this concept may be used for the coding of
- free categories and groupoids on directed graphs."
+@inproceedings{Yun76,
+ author = "Yun, D.Y.Y",
+ title = "On square-free decomposition algorithms",
+ booktitle = "Proceedings of SYMSAC'76",
+ year = "1976",
+ keywords = "survey",
+ pages = "26-35"
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Buchberger 85]{BC85} Buchberger, Bruno and Caviness, Bob F. (eds)
-EUROCAL '85: European Conference on Computer Algebra, Linz, Austria,
-LLCN QA155.7.E4 E86
- isbn = "0-387-15983-5, 0-387-15984-3",
- year = "1985",
- month = "April",
- publisher = "Springer-Verlag, Berlin, Germany",
- keywords = "axiomref",
- misc = "Lecture Notes in Computer Science, Vol 204",
+\section{To Be Classified} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt83,
+ author = "Kaltofen, E.",
+ title = "On the complexity of finding short vectors in integer lattices",
+ booktitle = "Proc. EUROCAL '83",
+ series = "Lect. Notes Comput. Sci.",
+ year = "1983",
+ volume = "162",
+ pages = "236--244",
+ publisher = "Springer-Verlag",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/83/Ka83_eurocal.pdf",
+ paper = "Kalt83.pdf"
+}
\end{chunk}
\begin{chunk}{axiom.bib}
-@misc{Buh05,
- author = "Buhl, Soren L.",
- title = "Some Reflections on Integrating a Computer Algebra System in R",
- year = "2005",
- keywords = "axiomref"
+@InProceedings{Kalt85,
+ author = "Kaltofen, E.",
+ title = "Effective {Hilbert} Irreducibility",
+ booktitle = "Proc. EUROSAM '84",
+ pages = "275--284",
+ crossref = "EUROSAM84",
+ year = "1985",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_infcontr.ps.gz",
+ paper = "Kalt85.ps"
}
-
-\end{chunk}
-
-\begin{chunk}{ignore}
-\bibitem[Burge 91]{Burg91}
- author = "Burge, W.H.",
- title = "Scratchpad and the Rogers-Ramanujan identities",
- year = "1991",
- pages = "189-190",
- isbn = "0-89791-437-6",
- keywords = "axiomref",
- abstract = "
- This note sketches the part played by Scratchpad in obtaining new
- proofs of Euler's theorem and the Rogers-Ramanujan Identities."
-
+
\end{chunk}
\begin{chunk}{axiom.bib}
-@techreport{BW87,
- author = "Burge, W. and Watt, S.",
- title = "Infinite structures in SCRATCHPAD II",
- year = "1987",
- institution = "IBM Research",
- type = "Technical Report",
- number = "RC 12794",
- keywords = "axiomref"
+@TechReport{Kalt85c,
+ author = "Kaltofen, E.",
+ title = "Sparse Hensel lifting",
+ institution = "RPI",
+ address = "Dept. Comput. Sci., Troy, N. Y.",
+ year = "1985",
+ number = "85-12",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_techrep.pdf",
+ paper = "Kalt85c.pdf"
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Burge 87a]{BWM87}
- author = "Burge, William H. and Watt, Stephen M. and Morrison, Scott C.",
- title = "Streams and Power Series",
- year = "1987",
- pages = "9-12",
- keywords = "axiomref",
-in [Wit87], pp9-12
-
-\end{chunk}
-
-\begin{chunk}{ignore}
-\bibitem[Burge 89]{BW89}
- author = "Burge, W. H. and Watt, S. M.",
- title = "Infinite structures in Scratchpad II",
- year = "1989",
- pages = "138-148",
- isbn = "3-540-51517-8",
- keywords = "axiomref",
-in Davenport [Dav89], LCCN QA155.7.E4E86 1987
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt85d,
+ author = "Kaltofen, E.",
+ title = "Sparse Hensel lifting",
+ booktitle = "EUROCAL 85 European Conf. Comput. Algebra Proc. Vol. 2",
+ crossref = "EUROCAL85",
+ pages = "4--17",
+ year = "1985",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_eurocal.pdf",
+ paper = "Kalt85d.pdf"
+}
\end{chunk}
-\subsection{C} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{chunk}{ignore}
-\bibitem[Calmet 94]{Cal94} Calmet, J. (ed)
-Rhine Workshop on Computer Algebra, Proceedings.
-Universit{\"a}t Karsruhe, Karlsruhe, Germany 1994
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@Article{Mill88,
+ author = "Miller, G.L. and Ramachandran, V. and Kaltofen, E.",
+ title = "Efficient parallel evaluation of straight-line code and
+ arithmetic circuits",
+ journal = "SIAM J. Comput.",
+ year = "1988",
+ volume = "17",
+ number = "4",
+ pages = "687--695",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/MRK88.pdf",
+ paper = "Mill88.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Camion 92]{CCM92}
- author = "Camion, Paul and Courteau, Bernard and Montpetit, Andre",
- title = "A combinatorial problem in Hamming Graphs and its solution
- in Scratchpad",
- year = "1992",
- month = "January",
- keywords = "axiomref",
-Rapports de recherche 1586, Institut National de Recherche en
-Informatique et en Automatique, Le Chesnay, France, 12pp
+\begin{chunk}{axiom.bib}
+@Article{Greg88,
+ author = "Gregory, B.; Kaltofen, E.",
+ title = "Analysis of the binary complexity of asymptotically fast
+ algorithms for linear system solving",
+ journal = "SIGSAM Bulletin",
+ year = "1988",
+ month = "April",
+ volume = "22",
+ number = "2",
+ pages = "41--49",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/GrKa88.pdf",
+ paper = "Grey88.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Caprotti 00]{CCR00}
- author = "Caprotti, Olga and Cohen, Arjeh M. and Riem, Manfred",
- title = "Java Phrasebooks for Computer Algebra and Automated Deduction",
- url = "http://www.sigsam.org/bulletin/articles/132/paper8.pdf",
- paper = "CCR00.pdf",
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@Article{Kalt89a,
+ author = "Kaltofen, E.; Rolletschek, H.",
+ title = "Computing greatest common divisors and factorizations in
+ quadratic number fields",
+ journal = "Math. Comput.",
+ year = "1989",
+ volume = "53",
+ number = "188",
+ pages = "697--720",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/KaRo89.pdf",
+ paper = "Kalt89a.pdf"
+}
\end{chunk}
\begin{chunk}{axiom.bib}
-@misc{CC99,
- author = "Capriotti, O. and Carlisle, D.",
- title = "OpenMath and MathML: Semantic Mark Up for Mathematics",
- year = "1999",
- url = "http://www.acm.org/crossroads/xrds6-2/openmath.html",
- keywords = "axiomref"
+@Unpublished{Kalt89b,
+ author = "Kaltofen, E.",
+ title = "Processor efficient parallel computation of polynomial greatest
+ common divisors",
+ year = "1989",
+ month = "July",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_gcd.ps.gz",
+ paper = "Kalt89b.ps"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@misc{Capr99,
- author = "Capriotti, Olga and Cohen, Arjeh M. and Cuypers, Hans and
- Sterk, Hans",
- title = "OpenMath Technology for Interactive Mathematical Documents",
- year = "2002",
- pages = "51-66",
- publisher = "Springer-Verlag, Berlin, Germany",
- url = "http://www.win.tue.nl/~hansc/lisbon.pdf",
- paper = "Capr99.pdf",
- misc = "in Multimedia Tools for Communicating Mathematics",
- keywords = "axiomref"
+@TechReport{Kalt89c,
+ author = "Kaltofen, E.",
+ title = "Parallel Algebraic Algorithm Design",
+ institution = "RPI",
+ address = "Dept. Comput. Sci., Troy, New York",
+ year = "1989",
+ month = "July",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_parallel.ps.gz",
+ paper = "Kalt89c.ps"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@misc{Carp04,
- author = "Carpent, Quentin and Conil, Christophe",
- title = "Utilisation de logiciels libres pour la r\'ealisation de TP MT26",
- year = "2004",
- paper = "Carp04.pdf",
- keywords = "axiomref"
+@InProceedings{Cann89,
+ author = "Canny, J. and Kaltofen, E. and Yagati, Lakshman",
+ title = "Solving systems of non-linear polynomial equations faster",
+ booktitle = "Proc. 1989 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC89",
+ pages = "121--128",
+ year = "1989",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/CKL89.pdf",
+ paper = "Cann89.pdf"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@misc{Chu85,
- author = "Chudnovsky, D.V and Chudnovsky, G.V.",
- title = "Elliptic Curve Calculations in Scratchpad II",
- year = "1985",
- institution = "Mathematics Dept., IBM Research",
- type = "Scratchpad II Newsletter 1 (1)",
- keywords = "axiomref"
+@Article{Kalt90b,
+ author = "Kaltofen, E.",
+ title = "Computing the irreducible real factors and components of an
+ algebraic curve",
+ journal = "Applic. Algebra Engin. Commun. Comput.",
+ year = "1990",
+ volume = "1",
+ number = "2",
+ pages = "135--148",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/Ka90_aaecc.pdf",
+ paper = "Kalt90b.pdf"
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Chudnovsky 87]{Chu87}
- author = "Chudnovsky, D.V and Chudnovsky, G.V.",
- title = "New Analytic Methods of Polynomial Root Finding",
- year = "1987",
- pages = "2",
- keywords = "axiomref",
-in [Wit87]
+\begin{chunk}{axiom.bib}
+@Article{Kalt90d,
+ author = "Kaltofen, E.; Trager, B.",
+ title = "Computing with polynomials given by black boxes for their
+ evaluations: Greatest common divisors, factorization, separation of
+ numerators and denominators",
+ journal = "J. Symbolic Comput.",
+ year = "1990",
+ volume = "9",
+ number = "3",
+ pages = "301--320",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/KaTr90.pdf",
+ paper = "Kalt90d.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Chudnovsky 89]{Chu89}
- author = "Chudnovsky, D.V. and Chudnovsky, G.V.",
- title = "The computation of classical constants",
- year = "1989",
- month = "November",
- pages = "8178-8182",
- keywords = "axiomref",
-Proc. Natl. Acad. Sci. USA Vol 86
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt91a,
+ author = "Kaltofen, E. and Singer, M.F.",
+ editor = "D. V. Shirkov and V. A. Rostovtsev and V. P. Gerdt",
+ title = "Size efficient parallel algebraic circuits for partial derivatives",
+ booktitle =
+ "IV International Conference on Computer Algebra in Physical Research",
+ pages = "133--145",
+ publisher = "World Scientific Publ. Co.",
+ year = "1991",
+ address = "Singapore",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaSi91.pdf",
+ paper = "Kalt91a.pdf"
+}
\end{chunk}
\begin{chunk}{axiom.bib}
-@proceedings{CJ86,
- editor = "Chudnovsky, David and Jenks, Richard",
- title = "Computers in Mathematics",
- year = "1986",
- month = "July",
- isbn = "0-8247-8341-7",
- note = "International Conference on Computers and Mathematics",
- publisher = "Marcel Dekker, Inc",
- keywords = "axiomref"
+@InProceedings{Kalt93,
+ author = "Kaltofen, E.",
+ title = "Computational Differentiation and Algebraic Complexity Theory",
+ booktitle = "Workshop Report on First Theory Institute on Computational
+ Differentiation",
+ editor = "C. H. Bischof and A. Griewank and P. M. Khademi",
+ publisher = "Argonne National Laboratory",
+ address = "Argonne, Illinois",
+ series = "Tech. Rep.",
+ volume = "ANL/MCS-TM-183",
+ month = "December",
+ year = "1993",
+ pages = "28--30",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_diff.pdf",
+ paper = "Kalt93.pdf"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@misc{Cohe03,
- author = "Cohen, Arjeh and Cuypers, M. and Barreiro, Hans and
- Reinaldo, Ernesto and Sterk, Hans",
- title = "Interactive Mathematical Documents on the Web",
- year = "2003",
- pages = "289-306",
- editor = "Joswig, M. and Takayma, N.",
- publisher = "Springer-Verlag, Berlin, Germany",
- keywords = "axiomref",
- misc = "in Algebra, Geometry and Software Systems"
+@Article{Kalt93b,
+ author = "Kaltofen, E.",
+ title = "Direct proof of a theorem by Kalkbrener, Sweedler, and Taylor",
+ journal = "SIGSAM Bulletin",
+ year = "1993",
+ volume = "27",
+ number = "4",
+ pages = "2",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_sambull.ps.gz",
+ paper = "Kalt93b.ps"
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Cohen 91]{CC91} Cohen, G.; Charpin, P.; (ed)
-EUROCODE '90 International Symposium on
-Coding Theory and Applications Proceedings. Springer-Verlag, Berlin, Germany
-/ Heidelberg, Germany / London, UK / etc., 1991 ISBN 0-387-54303-1
-(New York), 3-540-54303-1 (Berlin), LCCN QA268.E95 1990
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt94,
+ author = "Kaltofen, E. and Pan, V.",
+ title = "Parallel solution of Toeplitz and Toeplitz-like linear
+ systems over fields of small positive characteristic",
+ booktitle = "Proc. First Internat. Symp. Parallel Symbolic Comput.",
+ crossref = "PASCO94",
+ pages = "225--233",
+ year = "1994",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/KaPa94.pdf",
+ paper = "Kalt94.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Conrad (a)]{CFMPxxa}
- author = "Conrad, Marc and French, Tim and Maple, Carsten and Pott, Sandra",
- title = "Approaching Inheritance from a Natural Mathematical Perspective
- and from a Java Driven Viewpoint: a Comparative Review",
- keywords = "axiomref",
- paper = "CFMPxxa.pdf",
- abstract = "
- It is well-known that few object-oriented programming languages allow
- objects to change their nature at run-time. There have been a number
- of reasons presented for this, but it appears that there is a real
- need for matters to change. In this paper we discuss the need for
- object-oriented programming languages to reflect the dynamic nature of
- problems, particularly those arising in a mathematical context. It is
- from this context that we present a framework that realistically
- represents the dynamic and evolving characteristic of problems and
- algorithms."
+\begin{chunk}{axiom.bib}
+@InProceedings{Sama95,
+ author = "Samadani, M. and Kaltofen, E.",
+ title = "Prediction based task scheduling in distributed computing",
+ booktitle = "Languages, Compilers and Run-Time Systems for Scalable
+ Computers",
+ editor = "B. K. Szymanski and B. Sinharoy",
+ publisher = "Kluwer Academic Publ.",
+ address = "Boston",
+ pages = "317--320",
+ year = "1996",
+ url =
+ "http://www.math.ncsu.edu/~kaltofen/bibliography/95/SaKa95_poster.ps.gz",
+ paper = "Sama95.ps"
+}
\end{chunk}
\begin{chunk}{axiom.bib}
-@misc{CFMPxxb,
- author = "Conrad, Marc and French, Tim and Maple, Carsten and Pott, Sandra",
- title = "Mathematical Use Cases lead naturally to non-standard Inheritance
- Relationships: How to make them accessible in a mainstream language?",
- paper = "CFMPxxb.pdf",
- keywords = "axiomref",
- abstract = "
- Conceptually there is a strong correspondence between Mathematical
- Reasoning and Object-Oriented techniques. We investigate how the ideas
- of Method Renaming, Dynamic Inheritance and Interclassing can be used
- to strengthen this relationship. A discussion is initiated concerning
- the feasibility of each of these features."
+@InProceedings{Erli96,
+ author = "Erlingsson, U. and Kaltofen, E. and Musser, D.",
+ title = "Generic {Gram}-{Schmidt} Orthogonalization by Exact Division",
+ booktitle = "Proc. 1996 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC96",
+ year = "1996",
+ pages = "275--282",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/96/EKM96.pdf",
+ paper = "Erli96.pdf"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@misc{Cuyp10,
- author = "Cuypers, Hans and Hendriks, Maxim and Knopper, Jan Willem",
- title = "Interactive Geometry inside MathDox",
- year = "2010",
- url = "http://www.win.tue.nl/~hansc/MathDox_and_InterGeo_paper.pdf",
- paper = "Cuyp10",
- keywords = "axiomref"
+@InProceedings{Kalt96,
+ author = "Kaltofen, E. and Lobo, A.",
+ title = "On rank properties of {Toeplitz} matrices over finite fields",
+ booktitle = "Proc. 1996 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC96",
+ year = "1996",
+ pages = "241--249",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/96/KaLo96_issac.pdf",
+ paper = "Kalt96.pdf"
}
\end{chunk}
-\subsection{D} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{axiom.bib}
-@inproceedings{Dalm97,
- author = {Dalmas, St\'ephane and Ga\"etano, Marc and Watt, Stephen},
- title = "An OpenMath 1.0 Implementation",
- booktitle = "Proc. 1997 Int. Symp. on Symbolic and Algebraic Computation",
- series = "ISSAC'97",
+@Article{Kalt97,
+ author = "Kaltofen, E.",
+ title = "Teaching Computational Abstract Algebra",
+ journal = "Journal of Symbolic Computation",
+ volume = "23",
+ number = "5-6",
+ pages = "503--515",
year = "1997",
- isbn = "0-89791-875-4",
- location = "Kihei, Maui, Hawaii, USA",
- pages = "241-248",
- numpages = "8",
- url = "http://doi.acm.org/10.1145/258726.258794",
- doi = "10.1145/258726.258794",
- acmid = "258794",
- publisher = "ACM, New York, NY USA",
- keywords = "axiomref"
+ note = "Special issue on education, L. Lambe, editor.",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/Ka97_jsc.pdf",
+ keywords = "axiomref,read",
+ paper = "Kalt97.pdf",
+ abstract = "
+ We report on the contents and pedagogy of a course in abstract algebra
+ that was taught with the aid of educational software developed within
+ the Mathematica system. We describe the topics covered and the
+ didactical use of the corresponding Mathematica packages, as well as
+ draw conclusions for future such courses from the students' comments
+ and our own experience."
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Dalmas 92]{Dal92} Dalmas, S.
-``A polymorphic functional language applied to symbolic computation''
-In Wang [Wan92] pp369-375 ISBN 0-89791-489-9 (soft cover) 0-89791-490-2
-(hard cover) LCCN QA76.95.I59 1992
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@Unpublished{Hitz97,
+ author = "Hitz, M. A. and Kaltofen, E.",
+ title = "The {Kharitonov} theorem and its applications in symbolic
+ mathematical computation",
+ year = "1997",
+ month = "May",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/HiKa97_kharit.pdf",
+ paper = "Hitz97.pdf"
+}
\end{chunk}
\begin{chunk}{axiom.bib}
-@misc{Daly88,
- author = "Daly, Timothy",
- title = "Axiom in an Educational Setting, Axiom course slide deck",
- year = "1988",
- month = "January",
- keywords = "axiomref"
+@InProceedings{Bern99,
+ author = "Bernardin, L. and Char, B. and Kaltofen, E.",
+ title = "Symbolic Computation in {Java}: an Appraisement",
+ booktitle = "Proc. 1999 Internat. Symp. Symbolic Algebraic Comput.",
+ crossref = "ISSAC99",
+ year = "1999",
+ pages = "237--244",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/99/BCK99.pdf",
+ paper = "Bern99.pdf"
}
\end{chunk}
-\begin{chunk}{ignore}TPDHERE
-\bibitem[Daly 02]{Dal02} Daly, Timothy
-``Axiom as open source''
-SIGSAM Bulletin (ACM Special Interest Group
-on Symbolic and Algebraic Manipulation) 36(1) pp28-?? March 2002
-CODEN SIGSBZ ISSN 0163-5824
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt02,
+ author = "Kaltofen, Erich and McLean, Michael and Norris, Larry",
+ title = "`{Using} {Maple} to Grade {Maple}' Assessment Software from
+ {North Carolina State University}",
+ booktitle = "Proceedings 2002 Maple Workshop",
+ year = "2002",
+ publisher = "Waterloo Maple Inc.",
+ address = "Waterloo, Canada",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/02/KMN02.pdf",
+ paper = "Kalt02.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Daly 03]{Dal03} Daly, Timothy
-``The Axiom Wiki Website''
-\verb|axiom.axiom-developer.org|
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@Book{Grab03,
+ editor = "Grabmeier, J. and Kaltofen, E. and Weispfenning, V.",
+ title = "Computer Algebra Handbook",
+ publisher = "Springer-Verlag",
+ year = "2003",
+ note = "637 + xx~pages + CD-ROM. Includes E. Kaltofen and V. Weispfenning
+ \S1.4 Computer algebra -- impact on research, pages 4--6;
+ E. Kaltofen
+ \S2.2.3 Absolute factorization of polynomials, page 26;
+ E. Kaltofen and B. D. Saunders
+ \S2.3.1 Linear systems, pages 36--38;
+ R. M. Corless, E. Kaltofen and S. M. Watt
+ \S2.12.3 Hybrid methods, pages 112--125;
+ E. Kaltofen
+ \S4.2.17 FoxBox and other blackbox systems, pages 383--385.",
+ isbn = "3-540-65466-6",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/01/symnum.pdf",
+ paper = "Grab03.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Daly 06]{Dal06} Daly, Timothy
-``Axiom Volume 1: Tutorial''
-Lulu, Inc. 860 Aviation Parkway,
-Suite 300, Morrisville, NC 27560 USA, 2006 ISBN 141166597X 287pp
-\verb|www.lulu.com/content/190827|
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt07,
+ author = "Kaltofen, Erich and Li, Bin and Sivaramakrishnan, Kartik and
+ Yang, Zhengfeng and Zhi, Lihong",
+ title = "Lower bounds for approximate factorizations via semidefinite
+ programming (extended abstract)",
+ year = "2007",
+ booktitle =
+ "SNC'07 Proc. 2007 Internat. Workshop on Symbolic-Numeric Comput.",
+ crossref = "SNC07",
+ pages = "203--204",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/07/KLSYZ07.pdf",
+ paper = "Kalt07.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Daly 09]{Dal09} Daly, Timothy
-``The Axiom Literate Documentation''
-\verb|axiom-developer.org/axiom-website/documentation.html|
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@Article{Borw07,
+ author = "Borwein, Peter and Kaltofen, Erich and Mossinghoff, Michael J.",
+ title = "Irreducible Polynomials and {Barker} Sequences",
+ journal = "{ACM} Communications in Computer Algebra",
+ volume = "162",
+ number = "4",
+ year = "2007",
+ pages = "118--121",
+ month = "December",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/07/BKM07.pdf",
+ paper = "Borw07.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Daly 13]{Dal13} Daly, Timothy
-``Literate Programming in the Large''
-April 8-9, 2013 Portland Oregon
-\verb|conf.writethedocs.org|
-\verb|daly.axiom-developer.org|
-\verb|www.youtube.com/watch?v=Av0PQDVTP4A|
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@Article{Kalt10,
+ author = "Kaltofen, Erich and Lavin, Mark",
+ title = "Efficiently Certifying Non-Integer Powers",
+ journal = "Computational Complexity",
+ year = "2010",
+ volume = "19",
+ number = "3",
+ month = "September",
+ pages = "355--366",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/09/KaLa09.pdf",
+ paper = "Kalt10.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Davenport 79a]{Dav79a} Davenport, J.H.
-``What can SCRATCHPAD/370 do?''
-VM/370 SPAD.SCRIPTS August 24, 1979 SPAD.SCRIPT
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt11,
+ author = "Kaltofen, Erich L. and Nehring, Michael",
+ title = "Supersparse black box rational function interpolation",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'11",
+ crossref = "ISSAC11",
+ month = "June",
+ year = "2011",
+ pages = "177--185",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/11/KaNe11.pdf",
+ paper = "Kalt11.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Davenport 80]{Dav80} Davenport, J.H.; Jenks, R.D.
-``MODLISP -- an Introduction''
-Proc LISP80, 1980, and IBM RC8357 Oct 1980
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@InProceedings{Gren11a,
+ author = "Grenet, Bruno and Kaltofen, Erich L. and Koiran, Pascal
+ and Portier, Natacha",
+ title = "Symmetric Determinantal Representation of Weakly Skew Circuits",
+ booktitle = "Proc. 28th Internat. Symp. on Theoretical Aspects of Computer
+ Science, STACS 2011",
+ crossref = "STACS11",
+ pages = "543--554",
+ year = "2011",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/11/GKKP11.pdf",
+ paper = "Gren11a.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Davenport 84]{DGJ84} Davenport, J.; Gianni, P.; Jenks, R.;
-Miller, V.; Morrison, S.; Rothstein, M.; Sundaresan, C.; Sutor, R.;
-Trager, B.
-``Scratchpad''
-Mathematical Sciences Department, IBM Thomas Watson Research Center 1984
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt11a,
+ author = "Kaltofen, Erich L. and Nehring, Michael and Saunders, David B.",
+ title = "Quadratic-Time Certificates in Linear Algebra",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'11",
+ crossref = "ISSAC11",
+ month = "June",
+ year = "2011",
+ pages = "171--176",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/11/KNS11.pdf",
+ paper = "Kalt11a.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Davenport 84a]{Dav84a} Davenport, James H.
-``A New Algebra System''
-%\verb|axiom-developer.org/axiom-website/papers/Dav84a.pdf|
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt11b,
+ author = "Kaltofen, Erich L. and Lee, Wen-shin and Yang, Zhengfeng",
+ title = "Fast estimates of {Hankel} matrix condition numbers
+ and numeric sparse interpolation",
+ booktitle = "Proc. 2011 Internat. Workshop on Symbolic-Numeric Comput.",
+ month = "June",
+ crossref = "SNC11",
+ year = "2011",
+ pages = "130--136",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/11/KLY11.pdf",
+ paper = "Kalt11b.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Davenport 85]{Dav85} Davenport, James H.
-``The LISP/VM Foundation of Scratchpad II''
-The Scratchpad II Newsletter, Volume 1, Number 1, September 1, 1985
-IBM Corporation, Yorktown Heights, NY
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@InProceedings{Guo12,
+ author = "Guo, Feng and Kaltofen, Erich L. and Zhi, Lihong",
+ title = "Certificates of Impossibility of {Hilbert}-{Artin} Representations
+ of a Given Degree for Definite Polynomials and Functions",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'12",
+ crossref = "ISSAC12",
+ month = "July",
+ year = "2012",
+ pages = "195--202",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/12/GKZ12.pdf",
+ paper = "Guo12.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Davenport 88]{DST88} Davenport, J.H.; Siret, Y.; Tournier, E.
-Computer Algebra: Systems and Algorithms for Algebraic Computation.
-Academic Press, New York, NY, USA, 1988, ISBN 0-12-204232-9
-\verb|staff.bath.ac.uk/masjhd/masternew.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/DST88.pdf|
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@InProceedings{Come12a,
+ author = "Comer, Matthew T. and Kaltofen, Erich L. and Pernet, Cl{\'e}ment",
+ title = "Sparse Polynomial Interpolation and {Berlekamp}/\allowbreak
+ {Massey} Algorithms That Correct Outlier Errors in Input Values",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'12",
+ crossref = "ISSAC12",
+ month = "July",
+ year = "2012",
+ pages = "138--145",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/12/CKP12.pdf",
+ paper = "Come12a.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Davenport 14]{Dav14} Davenport, James H.
-``Computer Algebra textbook''
-\verb|staff.bath.ac.uk/masjhd/JHD-CA.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Dav14.pdf|
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@InProceedings{Boye13,
+ author = "Boyer, Brice and Comer, Matthew T. and Kaltofen, Erich L.",
+ title = "Sparse Polynomial Interpolation by Variable Shift in
+ the Presence of Noise and Outliers in the Evaluations",
+ booktitle =
+ "Proc. Tenth Asian Symposium on Computer Mathematics (ASCM 2012)",
+ year = "2013",
+ month = "October",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/13/BCK13.pdf",
+ paper = "Boye13.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Davenport 89]{Dav89} Davenport, J.H. (ed)
-EUROCAL '87 European Conference on Computer Algebra Proceedings
-Springer-Verlag, Berlin, Germany / Heidelberg, Germany / London,
-UK / etc., 1989 ISBN 3-540-51517-8 LCCN QA155.7.E4E86 1987
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt13b,
+ author = "Kaltofen, Erich and Yang, Zhengfeng",
+ title = "Sparse multivariate function recovery from values with noise and
+ outlier errors",
+ year = "2013",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'13",
+ crossref = "ISSAC13",
+ pages = "219--226",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/13/KaYa13.pdf",
+ paper = "Kalt13b.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Davenport 90]{DT90} Davenport, J. H.; Trager, B. M.
-``Scratchpad's view of algebra I: Basic commutative algebra''
-In Miola [Mio90], pp40-54. ISBN 0-387-52531-9 (New York),
-3-540-52531-9 (Berlin). LCCN QA76.9.S88I576 1990 also in AXIOM Technical
-Report, ATR/1, NAG Ltd., Oxford, 1992
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt13c,
+ author = "Kaltofen, Erich L.",
+ title = "Symbolic Computation and Complexity Theory Transcript of My Talk",
+ booktitle =
+ "Proc. Tenth Asian Symposium on Computer Mathematics (ASCM 2012)",
+ year = "2013",
+ month = "October",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/13/Ka13.pdf",
+ paper = "Kalt13c.pdf"
+}
\end{chunk}
\begin{chunk}{axiom.bib}
-@inproceedings{Dave91,
- author = "Davenport, J. H. and Gianni, P. and Trager, B. M.",
- title = "Scratchpad's View of Algebra II:
- A Categorical View of Factorization",
- booktitle = "Proc. 1991 Int. Symp. on Symbolic and Algebraic Computation",
- series = "ISSAC '91",
- year = "1991",
- isbn = "0-89791-437-6",
- location = "Bonn, West Germany",
- pages = "32--38",
- numpages = "7",
- url = "http://doi.acm.org/10.1145/120694.120699",
- doi = "10.1145/120694.120699",
- acmid = "120699",
- publisher = "ACM",
- address = "New York, NY, USA",
- keywords = "axiomref",
- paper = "Dave91.pdf",
- abstract = "
- This paper explains how Scratchpad solves the problem of presenting a
- categorical view of factorization in unique factorization domains,
- i.e. a view which can be propagated by functors such as
- SparseUnivariatePolynomial or Fraction. This is not easy, as the
- constructive version of the classical concept of
- UniqueFactorizationDomain cannot be so propagated. The solution
- adopted is based largely on Seidenberg's conditions (F) and (P), but
- there are several additional points that have to be borne in mind to
- produce reasonably efficient algorithms in the required generality.
-
- The consequence of the algorithms and interfaces presented in this
- paper is that Scratchpad can factorize in any extension of the
- integers or finite fields by any combination of polynomial, fraction
- and algebraic extensions: a capability far more general than any other
- computer algebra system possesses. The solution is not perfect: for
- example we cannot use these general constructions to factorize
- polyinmoals in $\overline{Z[\sqrt{-5}]}[x]$ since the domain
- $Z[\sqrt{-5}]$ is not a unique factorization domain, even though
- $\overline{Z[\sqrt{-5}]}$ is, since it is a field. Of course, we can
- factor polynomials in $\overline{Z}[\sqrt{-5}][x]$"
+@InProceedings{Kalt14,
+ author = "Kaltofen, Erich L. and Yang, Zhengfeng",
+ title = "Sparse Multivariate Function Recovery With a High Error Rate
+ in Evaluations",
+ year = "2014",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'14",
+ crossref = "ISSAC14",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/14/KaYa14.pdf",
+ paper = "Kalt14.pdf"
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Davenport 92]{DGT92} Davenport, J. H.;, Gianni, P.; Trager, B. M.
-``Scratchpad's view of algebra II: A categorical view of factorization''
-Technical Report TR4/92 (ATR/2) (NP2491), Numerical Algorithms Group, Inc.,
-Downer's Grove, IL, USA and Oxford, UK, December 1992
-\verb|www.nag.co.uk/doc/TechRep/axiomtr.html|
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@InProceedings{Kalt14a,
+ author = "Kaltofen, Erich L. and Pernet, Cl{\'e}ment",
+ title = "Sparse Polynomial Interpolation Codes and Their Decoding
+ Beyond Half the Minimal Distance",
+ year = "2014",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'14",
+ crossref = "ISSAC14",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/14/KaPe14.pdf",
+ paper = "Kalt14a.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Davenport 92a]{Dav92a} Davenport, J. H.
-``The AXIOM system''
-AXIOM Technical Report TR5/92 (ATR/3)
-(NP2492) Numerical Algorithms Group, Inc., Downer's Grove, IL, USA and
-Oxford, UK, December 1992
-\verb|www.nag.co.uk/doc/TechRep/axiomtr.html|
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@InProceedings{Duma14,
+ author = "Dumas, Jean-Guillaume and Kaltofen, Erich L.",
+ title = "Essentially Optimal Interactive Certificates In Linear Algebra",
+ year = "2014",
+ booktitle = "Internat. Symp. Symbolic Algebraic Comput. ISSAC'14",
+ crossref = "ISSAC14",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/14/DuKa14.pdf",
+ paper = "Duma14.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Davenport 92b]{Dav92b} Davenport, J. H.
-``How does one program in the AXIOM system?''
-AXIOM Technical Report TR6/92 (ATR/4)(NP2493)
-Numerical Algorithms Group, Inc., Downer's
-Grove, IL, USA and Oxford, UK December 1992
-\verb|www.nag.co.uk/doc/TechRep/axiomtr.html|
-%\verb|axiom-developer.org/axiom-website/papers/Dav92b.pdf|
- keywords = "axiomref",
- abstract = "
- Axiom is a computer algebra system superficially like many others, but
- fundamentally different in its internal construction, and therefore in
- the possibilities it offers to its users and programmers. In these
- lecture notes, we will explain, by example, the methodology that the
- author uses for programming substantial bits of mathematics in Axiom."
+\begin{chunk}{axiom.bib}
+@InProceedings{Boye14,
+ author = "Boyer, Brice B. and Kaltofen, Erich L.",
+ title = "Numerical Linear System Solving With Parametric Entries By
+ Error Correction",
+ year = "2014",
+ booktitle = "SNC'14 Proc. 2014 Int. Workshop on Symbolic-Numeric Comput.",
+ crossref = "SNC14",
+ url = "http://www.math.ncsu.edu/~kaltofen/bibliography/14/BoKa14.pdf",
+ paper = "Boye14.pdf"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Davenport 92c]{DT92} Davenport, J. H.; Trager, B. M.
-``Scratchpad's view of algebra I: Basic commutative algebra''
-DISCO 90 Capri, Italy April 1990 ISBN 0-387-52531-9 pp40-54
-Technical Report TR3/92 (ATR/1)(NP2490), Numerical
-Algorithms Group, Inc., Downer's Grove, IL, USA and Oxford, UK,
-December 1992.
-\verb|www.nag.co.uk/doc/TechRep/axiomtr.html|
- keywords = "axiomref",
+\section{Axiom Citations in the Literature}
-\end{chunk}
+\subsection{A} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
-\bibitem[Davenport 93]{Dav93} Davenport, J. H.
-``Primality testing revisited''
-Technical Report TR2/93 (ATR/6)(NP2556) Numerical Algorithms Group, Inc.,
-Downer's Grove, IL, USA and Oxford, UK, August 1993
-\verb|www.nag.co.uk/doc/TechRep/axiomtr.html|
+\bibitem[ACM 89]{ACM89} ACM, editor
+Proceedings of the ACM-SIGSAM 1989 International
+Symposium on Symbolic and Algebraic Computation, ISSAC '89 ACM Press,
+New York, NY 10036, USA, 1989, , LCCN QA76.95.I59
+ year = "1989",
+ isbn = "0-89791-325-6",
keywords = "axiomref",
-
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Davenport (a)]{DFxx} Davenport, James; Faure, Christ\'ele
-``The Unknown in Computer Algebra''
-\verb|axiom-wiki.newsynthesis.org/public/refs/TheUnknownInComputerAlgebra.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/DFxx.pdf|
+\bibitem[ACM 94]{ACM94} ACM, editor
+ISSAC '94. Proceedings of the International
+Symposium on Symbolic and Algebraic Computation. ACM Press, New York, NY,
+10036, USA, 1994, . LCCN QA76.95.I59
+ year = "1994",
+ isbn = "0-89791-638-7",
keywords = "axiomref",
- abstract = "
- Computer algebra systems have to deal with the confusion between
- ``programming variables'' and ``mathematical symbols''. We claim that
- they should also deal with ``unknowns'', i.e. elements whose values
- are unknown, but whose type is known. For examples $x^p \ne x$ if $x$
- is a symbol, but $x^p = x$ if $x \in GF(p)$. We show how we have
- extended Axiom to deal with this concept."
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Davenport 00]{Dav00} Davenport, James
-``13th OpenMath Meeting''
-James H. Davenport
-``A New Algebra System''
-May 1984
-\verb|xml.coverpages.org/openmath13.html|
-%\verb|axiom-developer.org/axiom-website/papers/Dav00.pdf|
+\begin{chunk}{axiom.bib}
+@article{Augo91,
+ author = "Augot, D. and Charpin, P. and Sendrier, N.",
+ title = "The miniumum distance of some binary codes via the
+ Newton's identities",
+ journal = "Cohen and Charping [CC91]",
+ year = "1991",
+ pages = "65-73",
+ isbn = "0-387-54303-1",
+ misc = "3-540-54303-1 (Berlin). LCCN QA268.E95 1990",
keywords = "axiomref",
+ paper = "Augo91.pdf"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Davenport 12]{Dav12} Davenport, J.H.
-``Computer Algebra''
-\verb|staff.bath.ac.uk/masjhd/JHD-CA.pdf|
+\bibitem[Adams 94]{AL94}
+ author = "Adams, William W. and Loustaunau, Philippe",
+ title = "An Introduction to Gr\"obner Bases",
+ year = "1994",
+American Mathematical Society (1994)
+ isbn = "0-8218-3804-0",
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Davenport (b)]{DSTxx} Davenport, J. H.; Siret; Tournier
-``Computer Algebra'' \hfill
-\verb|staff.bath.ac.uk/masjhd/masternew.pdf|
+\bibitem[Andrews 84]{And84}
+ author = "Andrews, George E.",
+ title = "Ramanujan and SCRATCHPAD",
+ year = "1984",
+ pages = "383-??",
keywords = "axiomref",
+In Golden and Hussain [GH84]
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Dewar 94]{Dew94} Dewar, M. C.
-``Manipulating Fortran Code in AXIOM and the AXIOM-NAG Link''
-Proceedings of the Workshop on Symbolic and Numeric Computing, ed by Apiola, H.
-and Laine, M. and Valkeila, E. pp1-12 University of Helsinki, Finland (1994)
+\bibitem[Andrews 88]{And88}
+ author = "Andrews, G. E.",
+ title = "Application of Scratchpad to problems in special functions and
+ combinatorics",
+ year = "1988"
+ pages = "158-??",
+ isbn = "3-540-18928-9",
keywords = "axiomref",
+In Janssen [Jan88], pages 158-?? ISBN
+0-387-18928-9 LCCN QA155.7.E4T74
\end{chunk}
-\begin{chunk}{axiom.bib}
-@misc{Dewa,
- author = "Dewar, Mike",
- title = "OpenMath: An Overview",
- url = "http://www.sigsam.org/bulletin/articles/132/paper1.pdf",
- paper = "Dewa.pdf",
- keywords = "axiomref"
-}
+\begin{chunk}{ignore}
+\bibitem[Anon 91]{Ano91}
+ author = "Anonymous",
+ year = "1991,
+ keywords = "axiomref",
+Proceedings 1991 Annual Conference, American Society for
+Engineering Education. Challenges of a Changing World. ASEE, Washington, DC
+ 2 vol.
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Dicrescenzo 89]{DD89} Dicrescenzo, C.; Duval, D.
-``Algebraic extensions and algebraic closure in Scratchpad II''
-In Gianni [Gia89], pp440-446 ISBN 3-540-51084-2
-LCCN QA76.95.I57 1998 Conference held jointly with AAECC-6
+\bibitem[Anon 92]{Ano92}
+ author = "Anonymous",
+ year = "1992",
keywords = "axiomref",
+Programming environments for high-level scientific problem solving.
+IFIP TC2/WG 2.5 working conference. IFIP Transactions. A Computer Science
+and Technology, A-2:??, CODEN ITATEC. ISSN 0926-5473
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Dingle 94]{Din94} Dingle, Adam; Fateman, Richard
-``Branch Cuts in Computer Algebra''
-1994 ISSAC, Oxford (UK), July 1994
-\verb|www.cs.berkeley.edu/~fateman/papers/ding.ps|
-%\verb|axiom-developer.org/axiom-website/papers/Din94.pdf|
+\bibitem[Anono 95]{Ano95}
+ author =Anonymous
+ keywords = "axiomref",
+ year = "1995",
+GAMM 94 annual meeting. Zeitschrift fur Angewandte Mathematik und
+Physik, 75 (suppl. 2), CODEN ZAMMAX, ISSN 0044-2267
+
+\end{chunk}
+
+\subsection{B} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@article{Bacl14,
+ author = "Baclawski, Krystian",
+ title = "SPAD language type checker",
+ journal = "unknown",
+ year = "2014",
+ url = "http://github.com/cahirwpz/phd",
keywords = "axiomref",
abstract = "
- Many standard functions, such as the logarithms and square root
- functions, cannot be defined continuously on the complex
- plane. Mistaken assumptions about the properties of these functions
- lead computer algebra systems into various conundrums. We discuss how
- they can manipulate such functions in a useful fashion."
+ The project aims to deliver a new type checker for SPAD language.
+ Several improvements over current type checker are planned.
+ \begin{itemize}
+ \item introduce better type inference
+ \item introduce modern language constructs
+ \item produce understandable diagnostic messages
+ \item eliminate well known bugs in the type system
+ \item find new type errors
+ \end{itemize}"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[DLMF]{DLMF}.
-``Digital Library of Mathematical Functions''
-\verb|dlmf.nist.gov/software/#T1|
+\bibitem[Blair 70]{BGJ70}
+ author = "Blair, Fred W and Griesmer, James H. and Jenks, Richard D.",
+ title = "An interactive facility for symbolic mathematics",
+ year = "1970",
+ pages = "394-419",
keywords = "axiomref",
+Proc. International Computing Symposium, Bonn, Germany,
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Dooley 99]{Doo99} Dooley, Sam editor.
-ISSAC 99: July 29-31, 1999, Simon Fraser University,
-Vancouver, BC, Canada: proceedings of the 1999 International Symposium on
-Symbolic and Algebraic Computation. ACM Press, New York, NY 10036, USA, 1999.
-ISBN 1-58113-073-2 LCCN QA76.95.I57 1999
+\bibitem[Blair 70a]{BJ70}
+ author = "Blair, Fred W. and Jenks, Richard D.",
+ title = "LPL: LISP programming language",
+ year = "1970",
keywords = "axiomref",
+IBM Research Report, RC3062 Sept
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Dos Reis 12]{DR12} Dos Reis, Gabriel
-``A System for Axiomatic Programming''
-Proc. Conf. on Intelligent Computer Mathematics, Springer (2012)
-\verb|www.axiomatics.org/~gdr/liz/cicm-2012.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/DR12.pdf|
+\begin{chunk}{axiom.bib}
+\bibitem[Broadbery 95]{BGDW95}
+ author = "Broadbery, P. A. and G{\'o}mez-D{\'\i}az, T. and Watt, S. M.",
+ title = "On the Implementation of Dynamic Evaluation",
+ year = "1995",
+ pages = "77-84",
keywords = "axiomref",
+ isbn = "0-89791-699-9",
+ url = "http://pdf.aminer.org/000/449/014/on_the_implementation_of_dynamic_evaluation.pdf",
+ paper = "BGDW95.pdf",
abstract = "
- We present the design and implementation of a system for axiomatic
- programming, and its application to mathematical software
- construction. Key novelties include a direct support for user-defined
- axioms establishing local equality between types, and overload
- resolution based on equational theories and user-defined local
- axioms. We illustrate uses of axioms, and their organization into
- concepts, in structured generic programming as practiced in
- computational mathematical systems."
+ Dynamic evaluation is a technique for producing multiple results
+ according to a decision tree which evolves with program execution.
+ Sometimes it is desired to produce results for all possible branches
+ in the decision tree, while on other occasions, it may be sufficient
+ to compute a single result which satisfies certain properties. This
+ techinique finds use in computer algebra where computing the correct
+ result depends on recognizing and properly handling special cases of
+ parameters. In previous work, programs using dynamic evaluation have
+ explored all branches of decision trees by repeating the computations
+ prior to decision points.
+
+ This paper presents two new implementations of dynamic evaluation
+ which avoid recomputing intermediate results. The first approach uses
+ Scheme ``continuations'' to record state for resuming program
+ execution. The second implementation uses the Unix ``fork'' operation
+ to form new processes to explore alternative branches in parallel."
}
+
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Doye 97]{Doy97} Doye, Nicolas James
-``Order Sorted Computer Algebra and Coercions''
-Ph.D. Thesis University of Bath 1997
-%\verb|axiom-developer.org/axiom-website/papers/Doy97.pdf|
+\begin{chunk}{axiom.bib}
+\bibitem[Boehm 89]{Boe89}
+@inproceedings{Boe89,
+ author = "Boehm, Hans-J.",
+ title = "Type Inference in the Presence of Type Abstraction",
+ year = "1989",
+ pages = "192-206",
keywords = "axiomref",
+ url = "http://www.acm.org/pubs/citations/proceedings/pldi/73141/p192-boehm",
+ paper = "Boe89.pdf",
+ booktitle = "ACM SIGPLAN Notices",
+ volume = "24",
+ number = "7",
+ month = "July",
abstract = "
- Computer algebra systems are large collections of routines for solving
- mathematical problems algorithmically, efficiently and above all,
- symbolically. The more advanced and rigorous computer algebra systems
- (for example, Axiom) use the concept of strong types based on
- order-sorted algebra and category theory to ensure that operations are
- only applied to expressions when they ``make sense''.
-
- In cases where Axiom uses notions which are not covered by current
- mathematics we shall present new mathematics which will allow us to
- prove that all such cases are reducible to cases covered by the
- current theory. On the other hand, we shall also point out all the
- cases where Axiom deviates undesirably from the mathematical ideal.
- Furthermore we shall propose solutions to these deviations.
+ A number of recent programming language designs incorporate a type
+ checking system based on the Girard-Reynolds polymorphic
+ $\lambda$-calculus. This allows the construction of general purpose,
+ reusable software without sacrificing compile-time type checking. A
+ major factor constraining the implementation of these languages is the
+ difficulty of automatically inferring the lengthy type information
+ that is otherwise required if full use is made of these
+ languages. There is no known algorithm to solve any natural and fully
+ general formulation of the ``type inference'' problem. One very
+ reasonable formulation of the problem is known to be undecidable.
- Strongly typed systems (especially of mathematics) become unusable
- unless the system can change the type in a way a user expects. We wish
- any change expected by a user to be automated, ``natural'', and
- unique. ``Coercions'' are normally viewed as ``natural type changing
- maps''. This thesis shall rigorously define the word ``coercion'' in
- the context of computer algebra systems.
+ Here we define a restricted version of the type inference problem and
+ present an efficient algorithm for its solution. We argue that the
+ restriction is sufficiently weak to be unobtrusive in practice."
+}
- We shall list some assumptions so that we may prove new results so
- that all coercions are unique. This concept is called ``coherence''.
+\end{chunk}
- We shall give an algorithm for automatically creating all coercions in
- type system which adheres to a set of assumptions. We shall prove that
- this is an algorithm and that it always returns a coercion when one
- exists. Finally, we present a demonstration implementation of this
- automated coerion algorithm in Axiom."
+\begin{chunk}{axiom.bib}
+@inproceedings{BHGM04,
+ author = "Boulton, Richard and Hardy, Ruth and Gottliebsen, Hanne
+ and Martin, Ursula",
+ title = "Design verification for control engineering",
+ year = "2004",
+ month = "April",
+ booktitle = "Proc 4th Int. Conf. on Integrated Formal Methods",
+ keywords = "axiomref",
+ abstract = "
+ We introduce control engineering as a new domain of application for
+ formal methods. We discuss design verification, drawing attention to
+ the role played by diagrammatic evaluation criteria involving numeric
+ plots of a design, such as Nichols and Bode plots. We show that
+ symbolic computation and computational logic can be used to discharge
+ these criteria and provide symbolic, automated, and very general
+ alternatives to these standard numeric tests. We illustrate our work
+ with reference to a standard reference model drawn from military
+ avionics."
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Doye 99]{Doy99} Doye, Nicolas J.
-``Automated coercion for Axiom''
-In Dooley [Doo99], pp229-235
-ISBN 1-58113-073-2 LCCN QA76.95.I57 1999 ACM Press
-\verb|www.acm.org/citation.cfm?id=309944|
+\bibitem[Boulanger 91]{Bou91}
+ author = "Boulanger, Jean-Louis",
+ title = "Etude de la compilation de scratchpad 2",
+ year = "1991",
+ month = "September",
+Rapport de DEA Universite dl lille 1
keywords = "axiomref",
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Dominguez 01]{DR01} Dom\'inguez, C\'esar; Rubio, Julio
-``Modeling Inheritance as Coercion in a Symbolic Computation System''
-ISSAC 2001 ACM 1-58113-417-7/01/0007
-%\verb|axiom-developer.org/axiom-website/papers/DR01.pdf|
- keywords = "axiomref",
- abstract = "
- In this paper the analysis of the data structures used in a symbolic
- computation system, called Kenzo, is undertaken. We deal with the
- specification of the inheritance relationship since Kenzo is an
- object-oriented system, written in CLOS, the Common Lisp Object
- System. We focus on a particular case, namely the relationship between
- simplicial sets and chain complexes, showing how the order-sorted
- algebraic specifications formalisms can be adapted, through the
- ``inheritance as coercion'' metaphor, in order to model this Kenzo
- fragment."
+\begin{chunk}{axiom.bib}
+@misc{Bou93a,
+ author = "Boulanger, Jean-Louis",
+ title = "Axiom, language fonctionnel \`a d\'evelopement objet",
+ year = "1993",
+ month = "October",
+ paper = "Bou93a.pdf",
+ keywords = "axiomref"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Dunstan 97]{Dun97} Dunstan, Martin and Ursula, Martin and
- Linton, Steve
-``Embedded Verification Techniques for Computer Algebra Systems''
-Grant citation GR/L48256 Nov 1, 1997-Feb 28, 2001
-\verb|www.cs.st-andrews.ac.uk/research/output/detail?output=ML97.php|
+\begin{chunk}{axiom.bib}
+@misc{Bou93b,
+ author = "Boulanger, Jean-Louis",
+ title = "AXIOM, A Functional Language with Object Oriented Development",
+ year = "1993",
+ paper = "Bou93b.pdf",
keywords = "axiomref",
+ abstract = "
+ We present in this paper, a study about the computer algebra system
+ Axiom, which gives us many very interesting Software engineering
+ concepts. This language is a functional language with an Object
+ Oriented Development. This feature is very important for modeling the
+ mathematical world (Hierarchy) and provides a running with
+ mathematical sense. (All objects are functions). We present many
+ problems of running and development in Axiom. We can note that Aiom is
+ the only system of this category."
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Adams 01]{DGKM01} Adams, Andrew; Dunstan, Martin; Gottliebsen, Hanne;
-Kelsey, Tom; Martin, Ursula; Owre, Sam
-``Computer Algebra meets Automated Theorem Proving: Integrating Maple and PVS''
-TPHOLS 2001, Edinburgh
-\verb|www.csl.sri.com/~owre/papers/tphols01/tphols01.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/DGKM01.pdf|
+\bibitem[Boulanger 94]{Bou94}
+ author = "Boulanger, J.L.",
+ title = "Object Oriented Method for Axiom",
+ year = "1995",
+ month = "February",
+ pages = "33-41",
+ paper = "Bou94.pdf",
+ACM SIGPLAN Notices, 30(2) CODEN SINODQ ISSN 0362-1340
keywords = "axiomref",
abstract = "
- We describe an interface between version 6 of the Maple computer
- algebra system with the PVS automated theorem prover. The interface is
- designed to allow Maple users access to the robust and checkable proof
- environment of PVS. We also extend this environment by the provision
- of a library of proof strategies for use in real analysis. We
- demonstrate examples using the interface and the real analysis
- library. These examples provide proofs which are both illustrative and
- applicable to genuine symbolic computation problems."
+ Axiom is a very powerful computer algebra system which combines two
+ language paradigms (functional and OOP). Mathematical world is complex
+ and mathematicians use abstraction to design it. This paper presents
+ some aspects of the object oriented development in Axiom. The Axiom
+ programming is based on several new tools for object oriented
+ development, it uses two levels of class and some operations such that
+ {\sl coerce}, {\sl retract}, or {\sl convert} which permit the type
+ evolution. These notions introduce the concept of multi-view."
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Duval 92]{DJ92} Duval D.; Jung, F.
-``Examples of problem solving using computer algebra''
-IFIP Transactions. A. Computer Science and Technology, A-2 pp133-141, 143 1992
-CODEN ITATEC. ISSN 0926-5473
+\bibitem[Bronstein 87]{Bro87}
+ author = "Bronstein, Manuel",
+ title = "Integration of Algebraic and Mixed Functions",
+ year = "1987",
+in [Wit87], p18
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Duval 94]{Duv94} Duval, Dominique
-``Symbolic or algebraic computation?''
-Madrid Spain, NAG conference (private copy of paper)
+\bibitem[Bronstein 89]{Bro89}
+ author= "Bronstein, M.",
+ title = "Simplification of real elementary functions",
+ year = "1989",
+ pages = "207-211",
+ isbn = "0-89791-325-6",
+ACM [ACM89] pages LCCN QA76.95.I59 1989
keywords = "axiomref",
+ abstract = "
+ We describe an algorithm, based on Risch's real structure theorem, that
+ determines explicitly all the algebraic relations among a given set of
+ real elementary functions. We also provide examples from its
+ implementation that illustrate the advantages over the use of complex
+ logarithms and exponentials."
+}
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Duva95,
- author = "Duval, D.",
- title = "Evaluation dynamique et cl\^oture alg\'ebrique en Axiom",
- journal = "Journal of Pure and Applied Algebra",
- volume = "99",
- year = "1995",
- pages = "267--295.",
- keywords = "axiomref"
+\bibitem[Bronstein 91a]{Bro91a}
+@inproceedings{Bron91a,
+ author = "Bronstein, M.",
+ title = "The Risch Differential Equation on an Algebraic Curve",
+ booktitle = "Proc. 1991 Int. Symp. on Symbolic and Algebraic Computation",
+ series = "ISSAC'91",
+ year = "1991",
+ pages = "241-246",
+ isbn = "0-89791-437-6",
+ publisher = "ACM, NY",
+ keywords = "axiomref",
+ paper = "Bro91a.pdf",
+ abstract = "
+ We present a new rational algorithm for solving Risch differential
+ equations over algebraic curves. This algorithm can also be used to
+ solve $n^{th}$-order linear ordinary differential equations with
+ coefficients in an algebraic extension of the rational functions. In
+ the general (``mixed function'') case, this algorithm finds the
+ denominator of any solution of the equation."
}
\end{chunk}
-\subsection{E} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Erocal 10]{ES10} Er\"ocal, Burcin; Stein, William
-``The Sage Project''
-\verb|wstein.org/papers/icms/icms_2010.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/ES10.pdf|
+\bibitem[Bronstein 91c]{Bro91c}
+ author = "Bronstein, Manuel",
+ title = "Computer Algebra and Indefinite Integrals",
+ year = "1991",
+ paper = "Bro91c.pdf",
+in Computer Aided Proofs in Analysis, K.R. Meyers et al. (eds)
+Springer-Verlag, NY (1991)
keywords = "axiomref",
abstract = "
- Sage is a free, open source, self-contained distribution of
- mathematical software, including a large library that provides a
- unified interface to the components of this distribution. This library
- also builds on the components of Sage to implement novel algorithms
- covering a broad range of mathematical functionality from algebraic
- combinatorics to number theory and arithmetic geometry."
-
+ We give an overview, from an analytical point of view, of decision
+ procedures for determining whether an elementary function has an
+ elementary function has an elementary antiderivative. We give examples
+ of algebraic functions which are integrable and non-integrable in
+ closed form, and mention the current implementation of various computer
+ algebra systems."
+}
\end{chunk}
-\subsection{F} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Fateman 90]{Fat90} Fateman, R. J.
-``Advances and trends in the design and construction of algebraic
-manipulation systems''
-In Watanabe and Nagata [WN90], pp60-67 ISBN 0-89791-401-5 LCCN QA76.95.I57 1990
+\bibitem[Bronstein 92]{Bro92}
+ author = "Bronstein, M.",
+ title = "Linear Ordinary Differential Equations: Breaking Through the
+ Order 2 Barrier",
+ year = "1992",
+ url =
+ "http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac92.ps.gz",
+ paper = "Bro92.pdf",
keywords = "axiomref",
-
+ abstract = "
+ A major subproblem for algorithms that either factor ordinary linear
+ differential equations or compute their closed form solutions is to
+ find their solutions $y$ which satisfy $y^{'}/y \in \overline{K}(x)$
+ where $K$ is the constant field for the coefficients of the equation.
+ While a decision procedure for this subproblem was known in the
+ $19^{th}$ century, it requires factoring polynomials over
+ $\overline{K}$ and has not been implemented in full generality. We
+ present here an efficient algorithm for this subproblem, which has
+ been implemented in the AXIOM computer algebra system for equations of
+ arbitrary order over arbitrary fields of characteristic 0. This
+ algorithm never needs to compute with the individual complex
+ singularities of the equation, and algebraic numbers are added only
+ when they appear in the potential solutions. Implementation of the
+ complete Singer algorithm for $n=2,3$ based on this building block is
+ in progress."
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Fateman 05]{Fat05} Fateman, R. J.
-``An incremental approach to building a mathematical expert out of software''
-4/19/2005\hfill
-\verb|www.cs.berkeley.edu/~fateman/papers/axiom.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Fat05.pdf|
+\bibitem[Bronstein 93]{Bro93}
+ author = "Bronstein, Manuel (ed)",
+ year = "1993",
+ month = "July"
+ isbn = "0-89791-604-2",
+ISSAC'93: proceedings of the 1993 International Symposium on Symbolic
+and Algebraic Computation, Kiev, Ukraine,
+ACM Press New York, NY 10036, USA, ISBN
+LCCN QA76.95 I59 1993 ACM order number 505930
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Fateman 06]{Fat06} Fateman, R. J.
-``Building Algebra Systems by Overloading Lisp''
-\verb|www.cs.berkeley.edu/~fateman/generic/overload-small.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Fat06.pdf|
+\bibitem[Brunelli 08]{Brun08}
+ author = "Brunelli, J.C.",
+ title = "Streams and Lazy Evaluation Applied to Integrable Models",
+ year = "2008",
+ url = "http://arxiv.org/PS_cache/nlin/pdf/0408/0408058v1.pdf",
+ paper = "Brun08.pdf",
keywords = "axiomref",
abstract = "
- Some of the earliest computer algebra systems (CAS) looked like
- overloaded languages of the same era. FORMAC, PL/I FORMAC, Formula
- Algol, and others each took advantage of a pre-existing language base
- and expanded the notion of a numeric value to include mathematical
- expressions. Much more recently, perhaps encouraged by the growth in
- popularity of C++, we have seen a renewal of the use of overloading to
- implement a CAS.
-
- This paper makes three points. 1. It is easy to do overloading in
- Common Lisp, and show how to do it in detail. 2. Overloading per se
- provides an easy solution to some simple programming problems. We show
- how it can be used for a ``demonstration'' CAS. Other simple and
- plausible overloadings interact nicely with this basic system. 3. Not
- all goes so smoothly: we can view overloading as a case study and
- perhaps an object lesson since it fails to solve a number of
- fairly-well articulated and difficult design issues in CAS for which
- other approaches are preferable."
+ Computer algebra procedures to manipulate pseudo-differential
+ operators are implemented to perform calculations with integrable
+ models. We use lazy evaluation and streams to represent and operate
+ with pseudo-differential operators. No order of truncation is needed
+ since terms are produced on demand. We give a series of concrete
+ examples using the computer algebra language MAPLE."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Faure 00a]{FDN00a} Faure, Christ\'ele; Davenport, James
-``Parameters in Computer Algebra''
+\bibitem[Bronstein 93]{BS93}
+ author = "Bronstein, Manuel and Salvy, Bruno",
+ title = "Full Partial Fraction Decomposition of Rational Functions",
+ year = "1993",
+ pages = "157-160",
+ isbn = "0-89791-604-2",
+In Bronstein [Bro93] LCCN QA76.95 I59 1993
keywords = "axiomref",
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Faure 00b]{FDN00b} Faure, Christ\'ele; Davenport, James;
-Naciri, Hanane
-``Multi-values Computer Algebra''
-ISSN 0249-6399 Institut National De Recherche en Informatique et en
-Automatique Sept. 2000 No. 4001
-\verb|hal.inria.fr/inria-00072643/PDF/RR-4401.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/FDN00b.pdf|
+\begin{chunk}{axiom.bib}
+@misc{Bro92a,
+ author = "Bronstein, Manuel",
+ title = "Integration and Differential Equations in Computer Algebra",
+ year = "1992",
+ url = "http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.576",
+ paper = "Bro92a.pdf",
keywords = "axiomref",
abstract = "
- One of the main strengths of computer algebra is being able to solve a
- family of problems with one computation. In order to express not only
- one problem but a family of problems, one introduces some symbols
- which are in fact the parameters common to all the problems of the
- family.
-
- The user must be able to understand in which way these parameters
- affect the result when he looks at the answer. Otherwise it may lead
- to completely wrong calculations, which when used for numerical
- applications bring nonsensical answers. This is the case in most
- current Computer Algebra Systems we know because the form of the
- answer is never explicitly conditioned by the values of the
- parameters. The user is not even informed that the given answer may be
- wrong in some cases then computer algebra systems can not be entirely
- trustworthy. We have introduced multi-valued expressions called {\sl
- conditional} expressions, in which each potential value is associated
- with a condition on some parameters. This is used, in particular, to
- capture the situation in integration, where the form of the answer can
- depend on whether certain quantities are positive, negative or
- zero. We show that it is also necessary when solving modular linear
- equations or deducing congruence conditions from complex expressions."
-
-\end{chunk}
-
-\begin{chunk}{ignore}
-\bibitem[Fitch 84]{Fit84} Fitch, J. P. (ed)
-EUROSAM '84: International Symposium on Symbolic and
-Algebraic Computation, Cambridge, England, July 9-11, 1984, volume 174 of
-Lecture Notes in Computer Science. Springer-Verlag, Berlin, Germany /
-Heildelberg, Germany / London, UK / etc., 1984 ISBN 0-387-13350-X
-LCCN QA155.7.E4 I57 1984
- keywords = "axiomref",
+ We describe in this paper how the problems of computing indefinite
+ integrals and solving linear ordinary differential equations in closed
+ form are now solved by computer algebra systems. After a brief review
+ of the mathematical history of those problems, we outline the two
+ major algorithms for them (respectively the Risch and Singer
+ algorithms) and the recent improvements on those algorithms which has
+ allowed them to be implemented."
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Fitch 93]{Fit93} Fitch, J. (ed)
-Design and Implementation of Symbolic Computation Systems
-International Symposium DISCO '92 Proceedings. Springer-Verlag, Berlin,
-Germany / Heildelberg, Germany / London, UK / etc., 1993. ISBN 0-387-57272-4
-(New York), 3-540-57272-4 (Berlin). LCCN QA76.9.S88I576 1992
+\bibitem[Beneke 94]{BS94}
+ author = "Beneke, T. and Schwippert, W.",
+ title = "Double-track into the future: MathCAD will gain new users with
+ Standard and Plus versions",
+ year = "1994",
+ month = "July",
+ pages = "107-110",
keywords = "axiomref",
+Elektronik, 43(15) CODEN EKRKAR ISSN 0013-5658
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Fogus 11]{Fog11} Fogus, Michael
-``UnConj''
-\verb|clojure.com/blog/2011/11/22/unconj.html|
+\bibitem[Bronstein 97a]{Bro97a}
+ author = "Bronstein, Manuel and Weil, Jacques-Arthur",
+ title = "On Symmetric Powers of Differential Operators",
+ series = "ISSAC'97",
+ year = "1997",
+ pages = "156-163",
keywords = "axiomref",
+ url =
+ "http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/mb_papers.html"
+ paper = "Bro97a.pdf",
+ publisher = "ACM, NY",
+ abstract = "
+ We present alternative algorithms for computing symmetric powers of
+ linear ordinary differential operators. Our algorithms are applicable
+ to operators with coefficients in arbitrary integral domains and
+ become faster than the traditional methods for symmetric powers of
+ sufficiently large order, or over sufficiently complicated coefficient
+ domains. The basic ideas are also applicable to other computations
+ involving cyclic vector techniques, such as exterior powers of
+ differential or difference operators."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Fortenbacher 90]{For90} Fortenbacher, A.
-``Efficient type inference and coercion in computer algebra''
-In Miola [Mio90], pp56-60. ISBN 0-387-52531-9 (New York), 3-540-52531-9
-(Berlin). LCCN QA76.9.S88I576 1990
- keywords = "axiomref",
+\bibitem[Borwein 00]{Bor00}
+ author = "Borwein, Jonathan",
+ title = "Multimedia tools for communicating mathematics",
+ year = "2000",
+ pages = "58",
+ isbn = "3-540-42450-4",
+ publisher = "Springer-Verlag",
+ keywords = "axiomref"
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Fouche 90]{Fou90} Fouche, Francois
-``Une implantation de l'algorithme de Kovacic en Scratchpad''
-Technical report, Institut de Recherche Math{\'{e}}matique Avanc{\'{e}}e''
-Strasbourg, France, 1990 31pp
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{BT94,
+ author = "Brown, R. and Tonks, A.",
+ title = "Calculations with simplicial and cubical groups in AXIOM",
+ journal = "Journal of Symbolic Computation",
+ volume = "17",
+ number = "2",
+ pages = "159-179",
+ year = "1994",
+ month = "February",
+ misc = "CODEN JSYCEH ISSN 0747-7171",
+ keywords = "axiomref"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[FSF 14]{FSF14} FSF
-``Free Software Directory''
-\verb|directory.fsf.org/wiki/Axiom|
+\begin{chunk}{axiom.bib}
+@misc{Brow95,
+ author = "Brown, Ronald and Dreckmann, Winfried",
+ title = "Domains of data and domains of terms in AXIOM",
+ year = "1995",
keywords = "axiomref",
+ paper = "DB95.pdf",
+ abstract = "
+ The main new concept we wish to illustrate in this paper is a
+ distinction between ``domains of data'' and ``domains of terms'', and
+ its use in the programming of certain mathematical structures.
+ Although this distinction is implicit in much of the programming work
+ that has gone into the construction of Axiom categories and domains,
+ we believe that a formalisation of this is new, that standards and
+ conventions are necessary and will be useful in various other
+ contexts. We shall show how this concept may be used for the coding of
+ free categories and groupoids on directed graphs."
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Frisco ]{Fris} Frisco
-``Objectives and Results''
-\verb|www.nag.co.uk/projects/frisco/frisco/node3.htm|
+\bibitem[Buchberger 85]{BC85} Buchberger, Bruno and Caviness, Bob F. (eds)
+EUROCAL '85: European Conference on Computer Algebra, Linz, Austria,
+LLCN QA155.7.E4 E86
+ isbn = "0-387-15983-5, 0-387-15984-3",
+ year = "1985",
+ month = "April",
+ publisher = "Springer-Verlag, Berlin, Germany",
keywords = "axiomref",
+ misc = "Lecture Notes in Computer Science, Vol 204",
\end{chunk}
-\subsection{G} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{chunk}{ignore}
-\bibitem[Gebauer 86]{GM86} Gebauer, R{\"u}diger; M{\"o}ller, H. Michael
-``Buchberger's algorithm and staggered linear bases''
-In Bruce W. Char, editor. Proceedings of the 1986
-Symposium on Symbolic and Algebraic Computation: SYMSAC '86, July 21-23, 1986
-Waterloo, Ontario, pp218-221 ACM Press, New York, NY 10036, USA, 1986.
-ISBN 0-89791-199-7 LCCN QA155.7.E4 A281 1986 ACM order number 505860
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@misc{Buh05,
+ author = "Buhl, Soren L.",
+ title = "Some Reflections on Integrating a Computer Algebra System in R",
+ year = "2005",
+ keywords = "axiomref"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gebauer 88]{GM88} Gebauer, R.; M{\"o}ller, H. M.
-``On an installation of Buchberger's algorithm''
-Journal of Symbolic Computation, 6(2-3) pp275-286 1988
-CODEN JSYCEH ISSN 0747-7171
-\verb|www.sciencedirect.com/science/article/pii/S0747717188800488/pdf|
-\verb|?md5=f6ccf63002ef3bc58aaa92e12ef18980&|
-\verb|pid=1-s2.0-S0747717188800488-main.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/GM88.pdf|
+\bibitem[Burge 91]{Burg91}
+ author = "Burge, W.H.",
+ title = "Scratchpad and the Rogers-Ramanujan identities",
+ year = "1991",
+ pages = "189-190",
+ isbn = "0-89791-437-6",
keywords = "axiomref",
abstract = "
- Buchberger's algorithm calculates Groebner bases of polynomial
- ideals. Its efficiency depends strongly on practical criteria for
- detecting superfluous reductions. Buchberger recommends two
- criteria. The more important one is interpreted in this paper as a
- criterion for detecting redundant elements in a basis of a module of
- syzygies. We present a method for obtaining a reduced, nearly minimal
- basis of that module. The simple procedure for detecting (redundant
- syzygies and )superfluous reductions is incorporated now in our
- installation of Buchberger's algorithm in SCRATCHPAD II and REDUCE
- 3.3. The paper concludes with statistics stressing the good
- computational properties of these installations."
+ This note sketches the part played by Scratchpad in obtaining new
+ proofs of Euler's theorem and the Rogers-Ramanujan Identities."
\end{chunk}
\begin{chunk}{axiom.bib}
-@book{Gedd92,
- author = "Geddes, Keith and Czapor, O. and Stephen R. and Labahn, George",
- title = "Algorithms For Computer Algebra",
- publisher = "Kluwer Academic Publishers",
- isbn = "0-7923-9259-0",
- month = "September",
- year = "1992",
+@techreport{BW87,
+ author = "Burge, W. and Watt, S.",
+ title = "Infinite structures in SCRATCHPAD II",
+ year = "1987",
+ institution = "IBM Research",
+ type = "Technical Report",
+ number = "RC 12794",
keywords = "axiomref"
}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gianni 87]{Gia87} Gianni, Patrizia
-``Primary Decomposition of Ideals''
-in [Wit87], pp12-13
+\bibitem[Burge 87a]{BWM87}
+ author = "Burge, William H. and Watt, Stephen M. and Morrison, Scott C.",
+ title = "Streams and Power Series",
+ year = "1987",
+ pages = "9-12",
keywords = "axiomref",
+in [Wit87], pp9-12
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gianni 88]{Gia88} Gianni, Patrizia.; Trager, Barry.;
-Zacharias, Gail.
-``Gr\"obner Bases and Primary Decomposition of Polynomial Ideals''
-J. Symbolic Computation 6, 149-167 (1988)
-\verb|www.sciencedirect.com/science/article/pii/S0747717188800403/pdf|
-\verb|?md5=40c29b67947035884904fd4597ddf710&|
-\verb|pid=1-s2.0-S0747717188800403-main.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Gia88.pdf|
+\bibitem[Burge 89]{BW89}
+ author = "Burge, W. H. and Watt, S. M.",
+ title = "Infinite structures in Scratchpad II",
+ year = "1989",
+ pages = "138-148",
+ isbn = "3-540-51517-8",
keywords = "axiomref",
+in Davenport [Dav89], LCCN QA155.7.E4E86 1987
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Gianni 89a]{Gia89} Gianni, P. (Patrizia) (ed)
-Symbolic and Algebraic Computation.
-International Symposium ISSAC '88, Rome, Italy, July 4-8, 1988. Proceedings,
-volume 358 of Lecture Notes in Computer Science. Springer-Verlag, Berlin,
-Germany / Heildelberg, Germany / London, UK / etc., 1989. ISBN 3-540-51084-2
-LCCN QA76.95.I57 1988 Conference held jointly with AAECC-6
- keywords = "axiomref",
-
-\end{chunk}
+\subsection{C} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
-\bibitem[Gianni 89b]{GM89} Gianni, P.; Mora, T.
-``Algebraic solution of systems of polynomial equations using
-Gr{\"o}bner bases.''
-In Huguet and Poli [HP89], pp247-257 ISBN 3-540-51082-6 LCCN QA268.A35 1987
+\bibitem[Calmet 94]{Cal94} Calmet, J. (ed)
+Rhine Workshop on Computer Algebra, Proceedings.
+Universit{\"a}t Karsruhe, Karlsruhe, Germany 1994
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gil 92]{Gil92} Gil, I.
-``Computation of the Jordan canonical form of a square matrix (using
-the Axiom programming language)''
-In Wang [Wan92], pp138-145.
-ISBN 0-89791-489-9 (soft cover), 0-89791-490-2 (hard cover)
-LCCN QA76.95.I59 1992
+\bibitem[Camion 92]{CCM92}
+ author = "Camion, Paul and Courteau, Bernard and Montpetit, Andre",
+ title = "A combinatorial problem in Hamming Graphs and its solution
+ in Scratchpad",
+ year = "1992",
+ month = "January",
keywords = "axiomref",
+Rapports de recherche 1586, Institut National de Recherche en
+Informatique et en Automatique, Le Chesnay, France, 12pp
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gomez-Diaz 92]{Gom92} G\'omez-D'iaz, Teresa
-``Quelques applications de l`\'evaluation dynamique''
-Ph.D. Thesis L'Universite De Limoges March 1992
+\bibitem[Caprotti 00]{CCR00}
+ author = "Caprotti, Olga and Cohen, Arjeh M. and Riem, Manfred",
+ title = "Java Phrasebooks for Computer Algebra and Automated Deduction",
+ url = "http://www.sigsam.org/bulletin/articles/132/paper8.pdf",
+ paper = "CCR00.pdf",
keywords = "axiomref",
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Gomez-Diaz 93]{Gom93} G\'omez-D\'iaz, Teresa
-``Examples of using Dynamic Constructible Closure''
-IMACS Symposium SC-1993
-%\verb|axiom-developer.org/axiom-website/papers/Gom93.pdf|
- keywords = "axiomref",
- abstract = "
- We present here some examples of using the ``Dynamic Constructible
- Closure'' program, which performs automatic case distinction in
- computations involving parameters over a base field $K$. This program
- is an application of the ``Dynamic Evaluation'' principle, which
- generalizes traditional evaluation and was first used to deal with
- algebraic numbers."
+\begin{chunk}{axiom.bib}
+@misc{CC99,
+ author = "Capriotti, O. and Carlisle, D.",
+ title = "OpenMath and MathML: Semantic Mark Up for Mathematics",
+ year = "1999",
+ url = "http://www.acm.org/crossroads/xrds6-2/openmath.html",
+ keywords = "axiomref"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Goodwin 91]{GBL91} Goodwin, B. M.; Buonopane, R. A.; Lee, A.
-``Using MathCAD in teaching material and energy balance concepts''
-In Anonymous [Ano91], pp345-349 (vol. 1) 2 vols.
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@misc{Capr99,
+ author = "Capriotti, Olga and Cohen, Arjeh M. and Cuypers, Hans and
+ Sterk, Hans",
+ title = "OpenMath Technology for Interactive Mathematical Documents",
+ year = "2002",
+ pages = "51-66",
+ publisher = "Springer-Verlag, Berlin, Germany",
+ url = "http://www.win.tue.nl/~hansc/lisbon.pdf",
+ paper = "Capr99.pdf",
+ misc = "in Multimedia Tools for Communicating Mathematics",
+ keywords = "axiomref"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Golden 4]{GH84} Golden, V. Ellen; Hussain, M. A. (eds)
-Proceedings of the 1984 MACSYMA Users' Conference:
-Schenectady, New York, July 23-25, 1984, General Electric,
-Schenectady, NY, USA, 1984
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@misc{Carp04,
+ author = "Carpent, Quentin and Conil, Christophe",
+ title = "Utilisation de logiciels libres pour la r\'ealisation de TP MT26",
+ year = "2004",
+ paper = "Carp04.pdf",
+ keywords = "axiomref"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Gonnet 96]{Gon96} Gonnet, Gaston H.
-``Official verion 1.0 of the Meta Content Dictionary''
-\verb|www.inf.ethz.ch/personal/gonnet/ContDict/Meta|
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@misc{Chu85,
+ author = "Chudnovsky, D.V and Chudnovsky, G.V.",
+ title = "Elliptic Curve Calculations in Scratchpad II",
+ year = "1985",
+ institution = "Mathematics Dept., IBM Research",
+ type = "Scratchpad II Newsletter 1 (1)",
+ keywords = "axiomref"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Goodloe 93]{GL93} Goodloe, A.; Loustaunau, P.
-``An abstract data type development of graded rings''
-In Fitch [Fit93], pp193-202. ISBN 0-387-57272-4 (New York),
-3-540-57272-4 (Berlin). LCCN QA76.9.S88I576 1992
+\bibitem[Chudnovsky 87]{Chu87}
+ author = "Chudnovsky, D.V and Chudnovsky, G.V.",
+ title = "New Analytic Methods of Polynomial Root Finding",
+ year = "1987",
+ pages = "2",
keywords = "axiomref",
+in [Wit87]
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gottliebsen 05]{GKM05} Gottliebsen, Hanne; Kelsey, Tom;
-Martin, Ursula
-``Hidden verification for computational mathematics''
-Journal of Symbolic Computation, Vol39, Num 5, pp539-567 (2005)
-\verb|www.sciencedirect.com/science/article/pii/S0747717105000295|
-%\verb|axiom-developer.org/axiom-website/papers/GKM05.pdf|
+\bibitem[Chudnovsky 89]{Chu89}
+ author = "Chudnovsky, D.V. and Chudnovsky, G.V.",
+ title = "The computation of classical constants",
+ year = "1989",
+ month = "November",
+ pages = "8178-8182",
keywords = "axiomref",
- abstract = "
- We present hidden verification as a means to make the power of
- computational logic available to users of computer algebra systems
- while shielding them from its complexity. We have implemented in PVS a
- library of facts about elementary and transcendental function, and
- automatic procedures to attempt proofs of continuity, convergence and
- differentiability for functions in this class. These are called
- directly from Maple by a simple pipe-lined interface. Hence we are
- able to support the analysis of differential equations in Maple by
- direct calls to PVS for: result refinement and verification, discharge
- of verification conditions, harnesses to ensure more reliable
- differential equation solvers, and verifiable look-up tables."
+Proc. Natl. Acad. Sci. USA Vol 86
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Grabe 98]{Gra98} Gr\"abe, Hans-Gert
-``About the Polynomial System Solve Facility of Axiom, Macyma, Maple
-Mathematica, MuPAD, and Reduce''
-%\verb|axiom-developer.org/axiom-website/papers/Gra98.pdf|
- keywords = "axiomref",
- abstract = "
- We report on some experiences with the general purpose Computer
- Algebra Systems (CAS) Axiom, Macsyma, Maple, Mathematica, MuPAD, and
- Reduce solving systems of polynomial equations and the way they
- present their solutions. This snapshot (taken in the spring of 1996)
- of the current power of the different systems in a special area
- concentrates on both CPU-times and the quality of the output."
+\begin{chunk}{axiom.bib}
+@proceedings{CJ86,
+ editor = "Chudnovsky, David and Jenks, Richard",
+ title = "Computers in Mathematics",
+ year = "1986",
+ month = "July",
+ isbn = "0-8247-8341-7",
+ note = "International Conference on Computers and Mathematics",
+ publisher = "Marcel Dekker, Inc",
+ keywords = "axiomref"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Grabmeier 91]{GHK91} Grabmeier, J.; Huber, K.; Krieger, U.
-``Das ComputeralgebraSystem AXIOM bei kryptologischen und
-verkehrstheoretischen Untersuchungen des
-Forschunginstituts der Deutschen Bundespost TELEKOM''
-Technischer Report TR 75.91.20, IBM Wissenschaftliches
-Zentrum, Heidelberg, Germany, 1991
+\begin{chunk}{axiom.bib}
+@misc{Cohe03,
+ author = "Cohen, Arjeh and Cuypers, M. and Barreiro, Hans and
+ Reinaldo, Ernesto and Sterk, Hans",
+ title = "Interactive Mathematical Documents on the Web",
+ year = "2003",
+ pages = "289-306",
+ editor = "Joswig, M. and Takayma, N.",
+ publisher = "Springer-Verlag, Berlin, Germany",
keywords = "axiomref",
+ misc = "in Algebra, Geometry and Software Systems"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Grabmeier 92]{GS92} Grabmeier, J.; Scheerhorn, A.
-``Finite fields in Axiom''
-AXIOM Technical Report TR7/92 (ATR/5)(NP2522),
-Numerical Algorithms Group, Inc., Downer's
-Grove, IL, USA and Oxford, UK, 1992
-\verb|www.nag.co.uk/doc/TechRep/axiomtr.html|
-and Technical Report, IBM Heidelberg Scientific Center, 1992
+\bibitem[Cohen 91]{CC91} Cohen, G.; Charpin, P.; (ed)
+EUROCODE '90 International Symposium on
+Coding Theory and Applications Proceedings. Springer-Verlag, Berlin, Germany
+/ Heidelberg, Germany / London, UK / etc., 1991 ISBN 0-387-54303-1
+(New York), 3-540-54303-1 (Berlin), LCCN QA268.E95 1990
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Grabmeier 03]{GKW03} Grabmeier, Johannes; Kaltofen, Erich;
-Weispfenning, Volker (eds)
-Computer algebra handbook: foundations, applications, systems.
-Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
-2003. ISBN 3-540-65466-6 637pp Includes CDROM
-\verb|www.springer.com/sgw/cda/frontpage/|
-\verb|0,11855,1-102-22-1477871-0,00.html|
+\bibitem[Conrad (a)]{CFMPxxa}
+ author = "Conrad, Marc and French, Tim and Maple, Carsten and Pott, Sandra",
+ title = "Approaching Inheritance from a Natural Mathematical Perspective
+ and from a Java Driven Viewpoint: a Comparative Review",
keywords = "axiomref",
+ paper = "CFMPxxa.pdf",
+ abstract = "
+ It is well-known that few object-oriented programming languages allow
+ objects to change their nature at run-time. There have been a number
+ of reasons presented for this, but it appears that there is a real
+ need for matters to change. In this paper we discuss the need for
+ object-oriented programming languages to reflect the dynamic nature of
+ problems, particularly those arising in a mathematical context. It is
+ from this context that we present a framework that realistically
+ represents the dynamic and evolving characteristic of problems and
+ algorithms."
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Griesmer 71]{GJ71} Griesmer, J. H.; Jenks, R.D.
-``SCRATCHPAD/1 -- an interactive facility for symbolic mathematics''
-In Petrick [Pet71], pp42-58. LCCN QA76.5.S94 1971
-\verb|delivery.acm.org/10.1145/810000/806266/p42-griesmer.pdf|
-SYMSAC'71 Proc. second ACM Symposium on Symbolic and Algebraic
-Manipulation pp45-48
-%\verb|axiom-developer.org/axiom-website/papers/GJ71.pdf| REF:00027
+\begin{chunk}{axiom.bib}
+@misc{CFMPxxb,
+ author = "Conrad, Marc and French, Tim and Maple, Carsten and Pott, Sandra",
+ title = "Mathematical Use Cases lead naturally to non-standard Inheritance
+ Relationships: How to make them accessible in a mainstream language?",
+ paper = "CFMPxxb.pdf",
keywords = "axiomref",
abstract = "
- The SCRATCHPAD/1 system is designed to provide an interactive symbolic
- computational facility for the mathematician user. The system features
- a user language designed to capture the style and succinctness of
- mathematical notation, together with a facility for conveniently
- introducing new notations into the language. A comprehensive system
- library incorporates symbolic capabilities provided by such systems as
- SIN, MATHLAB, and REDUCE."
+ Conceptually there is a strong correspondence between Mathematical
+ Reasoning and Object-Oriented techniques. We investigate how the ideas
+ of Method Renaming, Dynamic Inheritance and Interclassing can be used
+ to strengthen this relationship. A discussion is initiated concerning
+ the feasibility of each of these features."
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Griesmer 72a]{GJ72a} Griesmer, J.; Jenks, R.
-``Experience with an online symbolic math system SCRATCHPAD''
-in Online'72 [Onl72] ISBN 0-903796-02-3 LCCN QA76.55.O54 1972 Two volumes
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@misc{Cuyp10,
+ author = "Cuypers, Hans and Hendriks, Maxim and Knopper, Jan Willem",
+ title = "Interactive Geometry inside MathDox",
+ year = "2010",
+ url = "http://www.win.tue.nl/~hansc/MathDox_and_InterGeo_paper.pdf",
+ paper = "Cuyp10",
+ keywords = "axiomref"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Griesmer 72b]{GJ72b} Griesmer, James H.; Jenks, Richard D.
-``SCRATCHPAD: A capsule view''
-ACM SIGPLAN Notices, 7(10) pp93-102, 1972. Proceedings of the symposium
-on Two-dimensional man-machine communications. Mark B. Wells and
-James B. Morris (eds.).
- keywords = "axiomref",
-
-\end{chunk}
+\subsection{D} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{chunk}{ignore}
-\bibitem[Griesmer 75]{GJY75} Griesmer, J.H.; Jenks, R.D.; Yun, D.Y.Y
-``SCRATCHPAD User's Manual''
-IBM Research Publication RA70 June 1975
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@inproceedings{Dalm97,
+ author = {Dalmas, St\'ephane and Ga\"etano, Marc and Watt, Stephen},
+ title = "An OpenMath 1.0 Implementation",
+ booktitle = "Proc. 1997 Int. Symp. on Symbolic and Algebraic Computation",
+ series = "ISSAC'97",
+ year = "1997",
+ isbn = "0-89791-875-4",
+ location = "Kihei, Maui, Hawaii, USA",
+ pages = "241-248",
+ numpages = "8",
+ url = "http://doi.acm.org/10.1145/258726.258794",
+ doi = "10.1145/258726.258794",
+ acmid = "258794",
+ publisher = "ACM, New York, NY USA",
+ keywords = "axiomref"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Griesmer 76]{GJY76} Griesmer, J.H.; Jenks, R.D.; Yun, D.Y.Y
-``A Set of SCRATCHPAD Examples''
-April 1976 (private copy)
+\bibitem[Dalmas 92]{Dal92} Dalmas, S.
+``A polymorphic functional language applied to symbolic computation''
+In Wang [Wan92] pp369-375 ISBN 0-89791-489-9 (soft cover) 0-89791-490-2
+(hard cover) LCCN QA76.95.I59 1992
keywords = "axiomref",
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Gruntz 94]{GM94} Gruntz, D.; Monagan, M.
-``Introduction to Gauss''
-SIGSAM Bulletin (ACM Special Interest Group on Symbolic and Algebraic
-Manipulation), 28(3) pp3-19 August 1994 CODEN SIGSBZ ISSN 0163-5824
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@misc{Daly88,
+ author = "Daly, Timothy",
+ title = "Axiom in an Educational Setting, Axiom course slide deck",
+ year = "1988",
+ month = "January",
+ keywords = "axiomref"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Gruntz 96]{Gru96} Gruntz, Dominik
-``On Computing Limits in a Symbolic Manipulation System''
-Thesis, Swiss Federal Institute of Technology Z\"urich 1996
-Diss. ETH No. 11432
-\verb|www.cybertester.com/data/gruntz.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Gru96.pdf|
- keywords = "axiomref",
- abstract = "
- This thesis presents an algorithm for computing (one-sided) limits
- within a symbolic manipulation system. Computing limtis is an
- important facility, as limits are used both by other functions such as
- the definite integrator and to get directly some qualitative
- information about a given function.
-
- The algorithm we present is very compact, easy to understand and easy
- to implement. It overcomes the cancellation problem other algorithms
- suffer from. These goals were achieved using a uniform method, namely
- by expanding the whole function into a series in terms of its most
- rapidly varying subexpression instead of a recursive bottom up
- expansion of the function. In the latter approach exact error terms
- have to be kept with each approximation in order to resolve the
- cancellation problem, and this may lead to an intermediate expression
- swell. Our algorithm avoids this problem and is thus suited to be
- implemented in a symbolic manipulation system."
+\begin{chunk}{ignore}TPDHERE
+\bibitem[Daly 02]{Dal02} Daly, Timothy
+``Axiom as open source''
+SIGSAM Bulletin (ACM Special Interest Group
+on Symbolic and Algebraic Manipulation) 36(1) pp28-?? March 2002
+CODEN SIGSBZ ISSN 0163-5824
+ keywords = "axiomref",
\end{chunk}
-\subsection{H} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Boyle 88]{Boyl88} Boyle, Ann
-``Future Directions for Research in Symbolic Computation''
-Soc. for Industrial and Applied Mathematics, Philadelphia (1990)
-\verb|www.eecis.udel.edu/~caviness/wsreport.pdf|
-%\verb|axiom-developer.org/axiom-website/Boyl88.pdf|
+\bibitem[Daly 03]{Dal03} Daly, Timothy
+``The Axiom Wiki Website''
+\verb|axiom.axiom-developer.org|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Hassner 87]{HBW87} Hassner, Martin; Burge, William H.;
-Watt, Stephen M.
-``Construction of Algebraic Error Control Codes (ECC) on the Elliptic
-Riemann Surface''
-in [Wit87], pp5-8
+\bibitem[Daly 06]{Dal06} Daly, Timothy
+``Axiom Volume 1: Tutorial''
+Lulu, Inc. 860 Aviation Parkway,
+Suite 300, Morrisville, NC 27560 USA, 2006 ISBN 141166597X 287pp
+\verb|www.lulu.com/content/190827|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Heck 01]{Hec01} Heck, A.
-``Variables in computer algebra, mathematics and science''
-The International Journal of Computer Algebra in Mathematics Education
-Vol. 8 No. 3 pp195-210 (2001)
+\bibitem[Daly 09]{Dal09} Daly, Timothy
+``The Axiom Literate Documentation''
+\verb|axiom-developer.org/axiom-website/documentation.html|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Huguet 89]{HP89} Huguet, L.; Poli, A. (eds).
-Applied Algebra, Algebraic Algorithms and Error-Correcting Codes.
-5th International Conference AAECC-5 Proceedings.
-Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
-1989. ISBN 3-540-51082-6. LCCN QA268.A35 1987
+\bibitem[Daly 13]{Dal13} Daly, Timothy
+``Literate Programming in the Large''
+April 8-9, 2013 Portland Oregon
+\verb|conf.writethedocs.org|
+\verb|daly.axiom-developer.org|
+\verb|www.youtube.com/watch?v=Av0PQDVTP4A|
keywords = "axiomref",
\end{chunk}
-\subsection{J} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Jacob 93]{JOS93} Jacob, G.; Oussous, N. E.; Steinberg, S. (eds)
-Proceedings SC 93
-International IMACS Symposium on Symbolic Computation. New Trends and
-Developments. LIFL Univ. Lille, Lille France, 1993
+\bibitem[Davenport 79a]{Dav79a} Davenport, J.H.
+``What can SCRATCHPAD/370 do?''
+VM/370 SPAD.SCRIPTS August 24, 1979 SPAD.SCRIPT
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Janssen 88]{Jan88} Jan{\ss}en, R. (ed)
-Trends in Computer Algebra, International Symposium
-Bad Neuenahr, May 19-21, 1987, Proceedings, volume 296 of Lecture Notes in
-Computer Science.
-Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
-1988 ISBN 3-540-18928-9, 0-387-18928-9 LCCN QA155.7.E4T74 1988
+\bibitem[Davenport 80]{Dav80} Davenport, J.H.; Jenks, R.D.
+``MODLISP -- an Introduction''
+Proc LISP80, 1980, and IBM RC8357 Oct 1980
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jenks 69]{Jen69} Jenks, R. D.
-``META/LISP: An interactive translator writing system''
-Research Report International Business Machines, Inc., Thomas J.
-Watson Research Center, Yorktown Heights, NY, USA, 1969 RC2968 July 1970
+\bibitem[Davenport 84]{DGJ84} Davenport, J.; Gianni, P.; Jenks, R.;
+Miller, V.; Morrison, S.; Rothstein, M.; Sundaresan, C.; Sutor, R.;
+Trager, B.
+``Scratchpad''
+Mathematical Sciences Department, IBM Thomas Watson Research Center 1984
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jenks 71]{Jen71} Jenks, R. D.
-``META/PLUS: The syntax extension facility for SCRATCHPAD''
-Research Report RC 3259, International Business Machines, Inc., Thomas J.
-Watson Research Center, Yorktown Heights, NY, USA, 1971
-% REF:00040
+\bibitem[Davenport 84a]{Dav84a} Davenport, James H.
+``A New Algebra System''
+%\verb|axiom-developer.org/axiom-website/papers/Dav84a.pdf|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jenks 74]{Jen74} Jenks, R. D.
-``The SCRATCHPAD language''
-ACM SIGPLAN Notices, 9(4) pp101-111 1974 CODEN SINODQ. ISSN 0362-1340
+\bibitem[Davenport 85]{Dav85} Davenport, James H.
+``The LISP/VM Foundation of Scratchpad II''
+The Scratchpad II Newsletter, Volume 1, Number 1, September 1, 1985
+IBM Corporation, Yorktown Heights, NY
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jen76]{Jen76} Jenks, Richard D.
-``A pattern compiler''
-In Richard D. Jenks, editor,
-SYMSAC '76: proceedings of the 1976 ACM Symposium on Symbolic and Algebraic
-Computation, August 10-12, 1976, Yorktown Heights, New York, pp60-65,
-ACM Press, New York, NY 10036, USA, 1976. LCCN QA155.7.EA .A15 1976
-QA9.58.A11 1976
+\bibitem[Davenport 88]{DST88} Davenport, J.H.; Siret, Y.; Tournier, E.
+Computer Algebra: Systems and Algorithms for Algebraic Computation.
+Academic Press, New York, NY, USA, 1988, ISBN 0-12-204232-9
+\verb|staff.bath.ac.uk/masjhd/masternew.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/DST88.pdf|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jenks 79]{Jen79} Jenks, R. D.
-``MODLISP: An Introduction''
-Proc EUROSAM 79, pp466-480, 1979 and IBMRC8073 Jan 1980
+\bibitem[Davenport 14]{Dav14} Davenport, James H.
+``Computer Algebra textbook''
+\verb|staff.bath.ac.uk/masjhd/JHD-CA.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Dav14.pdf|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jenks 81]{JT81} Jenks, R.D.; Trager, B.M.
-``A Language for Computational Algebra''
-Proceedings of SYMSAC81, Symposium on Symbolic and Algebraic Manipulation,
-Snowbird, Utah August, 1981
+\bibitem[Davenport 89]{Dav89} Davenport, J.H. (ed)
+EUROCAL '87 European Conference on Computer Algebra Proceedings
+Springer-Verlag, Berlin, Germany / Heidelberg, Germany / London,
+UK / etc., 1989 ISBN 3-540-51517-8 LCCN QA155.7.E4E86 1987
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jenks 81a]{JT81a} Jenks, R.D.; Trager, B.M.
-``A Language for Computational Algebra''
-SIGPLAN Notices, New York: Association for Computing Machiner, Nov 1981
+\bibitem[Davenport 90]{DT90} Davenport, J. H.; Trager, B. M.
+``Scratchpad's view of algebra I: Basic commutative algebra''
+In Miola [Mio90], pp40-54. ISBN 0-387-52531-9 (New York),
+3-540-52531-9 (Berlin). LCCN QA76.9.S88I576 1990 also in AXIOM Technical
+Report, ATR/1, NAG Ltd., Oxford, 1992
keywords = "axiomref",
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Jenks 81b]{JT81b} Jenks, R.D.; Trager, B.M.
-``A Language for Computational Algebra''
-IBM Research Report RC8930 IBM Yorktown Heights, NY
+\begin{chunk}{axiom.bib}
+@inproceedings{Dave91,
+ author = "Davenport, J. H. and Gianni, P. and Trager, B. M.",
+ title = "Scratchpad's View of Algebra II:
+ A Categorical View of Factorization",
+ booktitle = "Proc. 1991 Int. Symp. on Symbolic and Algebraic Computation",
+ series = "ISSAC '91",
+ year = "1991",
+ isbn = "0-89791-437-6",
+ location = "Bonn, West Germany",
+ pages = "32--38",
+ numpages = "7",
+ url = "http://doi.acm.org/10.1145/120694.120699",
+ doi = "10.1145/120694.120699",
+ acmid = "120699",
+ publisher = "ACM",
+ address = "New York, NY, USA",
keywords = "axiomref",
+ paper = "Dave91.pdf",
+ abstract = "
+ This paper explains how Scratchpad solves the problem of presenting a
+ categorical view of factorization in unique factorization domains,
+ i.e. a view which can be propagated by functors such as
+ SparseUnivariatePolynomial or Fraction. This is not easy, as the
+ constructive version of the classical concept of
+ UniqueFactorizationDomain cannot be so propagated. The solution
+ adopted is based largely on Seidenberg's conditions (F) and (P), but
+ there are several additional points that have to be borne in mind to
+ produce reasonably efficient algorithms in the required generality.
+
+ The consequence of the algorithms and interfaces presented in this
+ paper is that Scratchpad can factorize in any extension of the
+ integers or finite fields by any combination of polynomial, fraction
+ and algebraic extensions: a capability far more general than any other
+ computer algebra system possesses. The solution is not perfect: for
+ example we cannot use these general constructions to factorize
+ polyinmoals in $\overline{Z[\sqrt{-5}]}[x]$ since the domain
+ $Z[\sqrt{-5}]$ is not a unique factorization domain, even though
+ $\overline{Z[\sqrt{-5}]}$ is, since it is a field. Of course, we can
+ factor polynomials in $\overline{Z}[\sqrt{-5}][x]$"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jenks 84a]{Jen84a} Jenks, Richard D.
-``The new SCRATCHPAD language and system for computer algebra''
-In Golden and Hussain [GH84], pp409-??
+\bibitem[Davenport 92]{DGT92} Davenport, J. H.;, Gianni, P.; Trager, B. M.
+``Scratchpad's view of algebra II: A categorical view of factorization''
+Technical Report TR4/92 (ATR/2) (NP2491), Numerical Algorithms Group, Inc.,
+Downer's Grove, IL, USA and Oxford, UK, December 1992
+\verb|www.nag.co.uk/doc/TechRep/axiomtr.html|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jenks 84b]{Jen84b} Jenks, Richard D.
-``A primer: 11 keys to New Scratchpad''
-In Fitch [Fit84], pp123-147. ISBN 0-387-13350-X LCCN QA155.7.E4 I57 1984
+\bibitem[Davenport 92a]{Dav92a} Davenport, J. H.
+``The AXIOM system''
+AXIOM Technical Report TR5/92 (ATR/3)
+(NP2492) Numerical Algorithms Group, Inc., Downer's Grove, IL, USA and
+Oxford, UK, December 1992
+\verb|www.nag.co.uk/doc/TechRep/axiomtr.html|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jenks 86]{JWS86} Jenks, Richard D.; Sutor, Robert S.;
-Watt, Stephen M.
-``Scratchpad II: An Abstract Datatype System for Mathematical Computation''
-Research Report RC 12327 (\#55257), Iinternational Business Machines, Inc.,
-Thomas J. Watson Research Center, Yorktown Heights, NY, USA, 1986 23pp
-\verb|www.csd.uwo.ca/~watt/pub/reprints/1987-ima-spadadt.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/JWS86.pdf|
+\bibitem[Davenport 92b]{Dav92b} Davenport, J. H.
+``How does one program in the AXIOM system?''
+AXIOM Technical Report TR6/92 (ATR/4)(NP2493)
+Numerical Algorithms Group, Inc., Downer's
+Grove, IL, USA and Oxford, UK December 1992
+\verb|www.nag.co.uk/doc/TechRep/axiomtr.html|
+%\verb|axiom-developer.org/axiom-website/papers/Dav92b.pdf|
keywords = "axiomref",
abstract = "
- Scratchpad II is an abstract datatype language and system that is
- under development in the Computer Algebra Group, Mathematical Sciences
- Department, at the IBM Thomas J. Watson Research Center. Some features
- of APL that made computation particularly elegant have been borrowed.
- Many different kinds of computational objects and data structures are
- provided. Facilities for computation include symbolic integration,
- differentiation, factorization, solution of equations and linear
- algebra. Code economy and modularity is achieved by having
- polymorphic packages of functions that may create datatypes. The use
- of categories makes these facilities as general as possible."
+ Axiom is a computer algebra system superficially like many others, but
+ fundamentally different in its internal construction, and therefore in
+ the possibilities it offers to its users and programmers. In these
+ lecture notes, we will explain, by example, the methodology that the
+ author uses for programming substantial bits of mathematics in Axiom."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jenks 87]{JWS87} Jenks, Richard D.; Sutor, Robert S.;
-Watt, Stephen M.
-``Scratchpad II: an Abstract Datatype System for Mathematical Computation''
-Proceedings Trends in Computer Algebra, Bad Neuenahr, LNCS 296,
-Springer Verlag, (1987)
+\bibitem[Davenport 92c]{DT92} Davenport, J. H.; Trager, B. M.
+``Scratchpad's view of algebra I: Basic commutative algebra''
+DISCO 90 Capri, Italy April 1990 ISBN 0-387-52531-9 pp40-54
+Technical Report TR3/92 (ATR/1)(NP2490), Numerical
+Algorithms Group, Inc., Downer's Grove, IL, USA and Oxford, UK,
+December 1992.
+\verb|www.nag.co.uk/doc/TechRep/axiomtr.html|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jenks 88]{JSW88} Jenks, R. D.; Sutor, R. S.; Watt, S. M.
-``Scratchpad II: An abstract datatype system for mathematical computation''
-In Jan{\ss}en [Jan88],
-pp12-?? ISBN 3-540-18928-9, 0-387-18928-9 LCCN QA155.7.E4T74 1988
+\bibitem[Davenport 93]{Dav93} Davenport, J. H.
+``Primality testing revisited''
+Technical Report TR2/93 (ATR/6)(NP2556) Numerical Algorithms Group, Inc.,
+Downer's Grove, IL, USA and Oxford, UK, August 1993
+\verb|www.nag.co.uk/doc/TechRep/axiomtr.html|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jenks 88a]{Jen88a} Jenks, R. D.
-``A Guide to Programming in BOOT''
-Computer Algebra Group, Mathematical Sciences Department, IBM Research
-Draft September 5, 1988
+\bibitem[Davenport (a)]{DFxx} Davenport, James; Faure, Christ\'ele
+``The Unknown in Computer Algebra''
+\verb|axiom-wiki.newsynthesis.org/public/refs/TheUnknownInComputerAlgebra.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/DFxx.pdf|
+ keywords = "axiomref",
+ abstract = "
+ Computer algebra systems have to deal with the confusion between
+ ``programming variables'' and ``mathematical symbols''. We claim that
+ they should also deal with ``unknowns'', i.e. elements whose values
+ are unknown, but whose type is known. For examples $x^p \ne x$ if $x$
+ is a symbol, but $x^p = x$ if $x \in GF(p)$. We show how we have
+ extended Axiom to deal with this concept."
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Davenport 00]{Dav00} Davenport, James
+``13th OpenMath Meeting''
+James H. Davenport
+``A New Algebra System''
+May 1984
+\verb|xml.coverpages.org/openmath13.html|
+%\verb|axiom-developer.org/axiom-website/papers/Dav00.pdf|
+ keywords = "axiomref",
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Davenport 12]{Dav12} Davenport, J.H.
+``Computer Algebra''
+\verb|staff.bath.ac.uk/masjhd/JHD-CA.pdf|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jenks 88b]{Jen88b} Jenks, Richard
-``The Scratchpad II Computer Algebra System Interactive Environment Users
-Guide''
- Spring 1988
+\bibitem[Davenport (b)]{DSTxx} Davenport, J. H.; Siret; Tournier
+``Computer Algebra'' \hfill
+\verb|staff.bath.ac.uk/masjhd/masternew.pdf|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jenks 88c]{JWS88} Jenks, R. D.; Sutor, R. S.; Watt, S. M.
-``Scratchpad II: an abstract datatype system for mathematical computation''
-In Jan{\ss}en
-[Jan88], pp12-37. ISBN 3-540-18928-9, 0-387-18928-9 LCCN QA155.7.E4T74 1988
+\bibitem[Dewar 94]{Dew94} Dewar, M. C.
+``Manipulating Fortran Code in AXIOM and the AXIOM-NAG Link''
+Proceedings of the Workshop on Symbolic and Numeric Computing, ed by Apiola, H.
+and Laine, M. and Valkeila, E. pp1-12 University of Helsinki, Finland (1994)
keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
-@book{Jenk92,
- author = "Jenks, Richard D. and Sutor, Robert S.",
- title = "AXIOM: The Scientific Computation System",
- publisher = "Springer-Verlag, Berlin, Germany",
- year = "1992",
- isbn = "0-387-97855-0",
- keywords = "axiomref"
+@misc{Dewa,
+ author = "Dewar, Mike",
+ title = "OpenMath: An Overview",
+ url = "http://www.sigsam.org/bulletin/articles/132/paper1.pdf",
+ paper = "Dewa.pdf",
+ keywords = "axiomref"
}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jenks 94]{JT94} Jenks, R. D.; Trager, B. M.
-``How to make AXIOM into a Scratchpad''
-In ACM [ACM94], pp32-40 ISBN 0-89791-638-7 LCCN QA76.95.I59 1994
-%\verb|axiom-developer.org/axiom-website/papers/JT94.pdf|
+\bibitem[Dicrescenzo 89]{DD89} Dicrescenzo, C.; Duval, D.
+``Algebraic extensions and algebraic closure in Scratchpad II''
+In Gianni [Gia89], pp440-446 ISBN 3-540-51084-2
+LCCN QA76.95.I57 1998 Conference held jointly with AAECC-6
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Joswig 03]{JT03} Joswig, Michael; Takayama, Nobuki
-``Algebra, geometry, and software systems''
-Springer-Verlag ISBN 3-540-00256-1 p291
+\bibitem[Dingle 94]{Din94} Dingle, Adam; Fateman, Richard
+``Branch Cuts in Computer Algebra''
+1994 ISSAC, Oxford (UK), July 1994
+\verb|www.cs.berkeley.edu/~fateman/papers/ding.ps|
+%\verb|axiom-developer.org/axiom-website/papers/Din94.pdf|
keywords = "axiomref",
+ abstract = "
+ Many standard functions, such as the logarithms and square root
+ functions, cannot be defined continuously on the complex
+ plane. Mistaken assumptions about the properties of these functions
+ lead computer algebra systems into various conundrums. We discuss how
+ they can manipulate such functions in a useful fashion."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Joyner 06]{J006} Joyner, David
-``OSCAS - Maxima''
-SIGSAM Communications in Computer Algebra, 157 2006
-\verb|sage.math.washington.edu/home/wdj/sigsam/oscas-cca1.pdf|
+\bibitem[DLMF]{DLMF}.
+``Digital Library of Mathematical Functions''
+\verb|dlmf.nist.gov/software/#T1|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Joyner 14]{JO14} Joyner, David
-``Links to some open source mathematical programs''
-\verb|www.opensourcemath.org/opensource_math.html|
+\bibitem[Dooley 99]{Doo99} Dooley, Sam editor.
+ISSAC 99: July 29-31, 1999, Simon Fraser University,
+Vancouver, BC, Canada: proceedings of the 1999 International Symposium on
+Symbolic and Algebraic Computation. ACM Press, New York, NY 10036, USA, 1999.
+ISBN 1-58113-073-2 LCCN QA76.95.I57 1999
keywords = "axiomref",
\end{chunk}
-\subsection{K} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Kauers 08]{Kau08} Kauers, Manuel
-``Integration of Algebraic Functions: A Simple Heuristic for Finding
-the Logarithmic Part''
-ISSAC July 2008 ACM 978-1-59593-904 pp133-140
-\verb|www.risc.jku.at/publications/download/risc_3427/Ka01.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Kau08.pdf|
+\bibitem[Dos Reis 12]{DR12} Dos Reis, Gabriel
+``A System for Axiomatic Programming''
+Proc. Conf. on Intelligent Computer Mathematics, Springer (2012)
+\verb|www.axiomatics.org/~gdr/liz/cicm-2012.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/DR12.pdf|
keywords = "axiomref",
abstract = "
- A new method is proposed for finding the logarithmic part of an
- integral over an algebraic function. The method uses Gr{\"o}bner bases
- and is easy to implement. It does not have the feature of finding a
- closed form of an integral whenever there is one. But it very often
- does, as we will show by a comparison with the built-in integrators of
- some computer algebra systems."
-
+ We present the design and implementation of a system for axiomatic
+ programming, and its application to mathematical software
+ construction. Key novelties include a direct support for user-defined
+ axioms establishing local equality between types, and overload
+ resolution based on equational theories and user-defined local
+ axioms. We illustrate uses of axioms, and their organization into
+ concepts, in structured generic programming as practiced in
+ computational mathematical systems."
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Keady 94]{KN94} Keady, G.; Nolan, G.
-``Production of Argument SubPrograms in the AXIOM -- NAG
-link: examples involving nonleanr systems''
-Technical Report TR1/94
-ATR/7 (NP2680), Numerical Algorithms Group, Inc., Downer's Grove, IL, USA and
-Oxford, UK, 1994
-\verb|www.nag.co.uk/doc/TechRep/axiomtr.html|
+\bibitem[Doye 97]{Doy97} Doye, Nicolas James
+``Order Sorted Computer Algebra and Coercions''
+Ph.D. Thesis University of Bath 1997
+%\verb|axiom-developer.org/axiom-website/papers/Doy97.pdf|
keywords = "axiomref",
+ abstract = "
+ Computer algebra systems are large collections of routines for solving
+ mathematical problems algorithmically, efficiently and above all,
+ symbolically. The more advanced and rigorous computer algebra systems
+ (for example, Axiom) use the concept of strong types based on
+ order-sorted algebra and category theory to ensure that operations are
+ only applied to expressions when they ``make sense''.
+
+ In cases where Axiom uses notions which are not covered by current
+ mathematics we shall present new mathematics which will allow us to
+ prove that all such cases are reducible to cases covered by the
+ current theory. On the other hand, we shall also point out all the
+ cases where Axiom deviates undesirably from the mathematical ideal.
+ Furthermore we shall propose solutions to these deviations.
+
+ Strongly typed systems (especially of mathematics) become unusable
+ unless the system can change the type in a way a user expects. We wish
+ any change expected by a user to be automated, ``natural'', and
+ unique. ``Coercions'' are normally viewed as ``natural type changing
+ maps''. This thesis shall rigorously define the word ``coercion'' in
+ the context of computer algebra systems.
+
+ We shall list some assumptions so that we may prove new results so
+ that all coercions are unique. This concept is called ``coherence''.
+
+ We shall give an algorithm for automatically creating all coercions in
+ type system which adheres to a set of assumptions. We shall prove that
+ this is an algorithm and that it always returns a coercion when one
+ exists. Finally, we present a demonstration implementation of this
+ automated coerion algorithm in Axiom."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Kelsey 99]{Kel99} Kelsey, Tom
-``Formal Methods and Computer Algebra: A Larch Specification of AXIOM
-Categories and Functors''
-Ph.D. Thesis, University of St Andrews, 1999
+\bibitem[Doye 99]{Doy99} Doye, Nicolas J.
+``Automated coercion for Axiom''
+In Dooley [Doo99], pp229-235
+ISBN 1-58113-073-2 LCCN QA76.95.I57 1999 ACM Press
+\verb|www.acm.org/citation.cfm?id=309944|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Kelsey 00a]{Kel00a} Kelsey, Tom
-``Formal specification of computer algebra''
-University of St Andrews, 6th April 2000
-\verb|www.cs.st-andrews.cs.uk/~tom/pub/fscbs.ps|
-%\verb|axiom-developer.org/axiom-website/papers/Kel00a.pdf|
+\bibitem[Dominguez 01]{DR01} Dom\'inguez, C\'esar; Rubio, Julio
+``Modeling Inheritance as Coercion in a Symbolic Computation System''
+ISSAC 2001 ACM 1-58113-417-7/01/0007
+%\verb|axiom-developer.org/axiom-website/papers/DR01.pdf|
keywords = "axiomref",
abstract = "
- We investigate the use of formal methods languages and tools in the
- design and development of computer algebra systems (henceforth CAS).
- We demonstrate that errors in CAS design can be identified and
- corrected by the use of (i) abstract specifications of types and
- procedures, (ii) automated proofs of properties of the specifications,
- and (iii) interface specifications which assist the verification of
- pre- and post conditions of implemented code."
+ In this paper the analysis of the data structures used in a symbolic
+ computation system, called Kenzo, is undertaken. We deal with the
+ specification of the inheritance relationship since Kenzo is an
+ object-oriented system, written in CLOS, the Common Lisp Object
+ System. We focus on a particular case, namely the relationship between
+ simplicial sets and chain complexes, showing how the order-sorted
+ algebraic specifications formalisms can be adapted, through the
+ ``inheritance as coercion'' metaphor, in order to model this Kenzo
+ fragment."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Kelsey 00b]{Kel00b} Kelsey, Tom
-``Formal specification of computer algebra''
-(slides) University of St Andrews, Sept 21, 2000
-\verb|www.cs.st-andrews.cs.uk/~tom/pub/fscbstalk.ps|
+\bibitem[Dunstan 97]{Dun97} Dunstan, Martin and Ursula, Martin and
+ Linton, Steve
+``Embedded Verification Techniques for Computer Algebra Systems''
+Grant citation GR/L48256 Nov 1, 1997-Feb 28, 2001
+\verb|www.cs.st-andrews.ac.uk/research/output/detail?output=ML97.php|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Kendall 99a]{Ken99a} Kendall, W.S.
-``Itovsn3 in AXIOM: modules, algebras and stochastic differentials''
-\verb|www2.warwick.ac.uk/fac/sci/statistics/staff/academic-research/|
-\verb|kendall/personal/ppt/328.ps.gz|
+\bibitem[Adams 01]{DGKM01} Adams, Andrew; Dunstan, Martin; Gottliebsen, Hanne;
+Kelsey, Tom; Martin, Ursula; Owre, Sam
+``Computer Algebra meets Automated Theorem Proving: Integrating Maple and PVS''
+TPHOLS 2001, Edinburgh
+\verb|www.csl.sri.com/~owre/papers/tphols01/tphols01.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/DGKM01.pdf|
keywords = "axiomref",
+ abstract = "
+ We describe an interface between version 6 of the Maple computer
+ algebra system with the PVS automated theorem prover. The interface is
+ designed to allow Maple users access to the robust and checkable proof
+ environment of PVS. We also extend this environment by the provision
+ of a library of proof strategies for use in real analysis. We
+ demonstrate examples using the interface and the real analysis
+ library. These examples provide proofs which are both illustrative and
+ applicable to genuine symbolic computation problems."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Kendall 99b]{Ken99b} Kendall, W.S.
-``Symbolic It\^o calculus in AXIOM: an ongoing story
-\verb|www2.warwick.ac.uk/fac/sci/statistics/staff/academic-research/|
-\verb|kendall/personal/ppt/327.ps.gz|
+\bibitem[Duval 92]{DJ92} Duval D.; Jung, F.
+``Examples of problem solving using computer algebra''
+IFIP Transactions. A. Computer Science and Technology, A-2 pp133-141, 143 1992
+CODEN ITATEC. ISSN 0926-5473
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Kosleff 91]{Kos91} P.-V. Koseleff
-``Word games in free Lie algebras: several bases and formulas''
-Theoretical Computer Science 79(1) pp241-256 Feb. 1991 CODEN TCSCDI
-ISSN 0304-3975
+\bibitem[Duval 94]{Duv94} Duval, Dominique
+``Symbolic or algebraic computation?''
+Madrid Spain, NAG conference (private copy of paper)
keywords = "axiomref",
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Kusche 89]{KKM89} Kusche, K.; Kutzler, B.; Mayr, H.
-``Implementation of a geometry theorem proving package in SCRATCHPAD II''
-In Davenport [Dav89] pp246-257 ISBN 3-540-51517-8 LCCN QA155.7.E4E86 1987
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@article{Duva95,
+ author = "Duval, D.",
+ title = "Evaluation dynamique et cl\^oture alg\'ebrique en Axiom",
+ journal = "Journal of Pure and Applied Algebra",
+ volume = "99",
+ year = "1995",
+ pages = "267--295.",
+ keywords = "axiomref"
+}
\end{chunk}
-\subsection{L} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{E} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
-\bibitem[Lahey 08]{Lah08} Lahey, Tim
-``Sage Integration Testing''
-\verb|github.com/tjl/sage_int_testing| Dec. 2008
+\bibitem[Erocal 10]{ES10} Er\"ocal, Burcin; Stein, William
+``The Sage Project''
+\verb|wstein.org/papers/icms/icms_2010.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/ES10.pdf|
keywords = "axiomref",
+ abstract = "
+ Sage is a free, open source, self-contained distribution of
+ mathematical software, including a large library that provides a
+ unified interface to the components of this distribution. This library
+ also builds on the components of Sage to implement novel algorithms
+ covering a broad range of mathematical functionality from algebraic
+ combinatorics to number theory and arithmetic geometry."
\end{chunk}
+\subsection{F} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Lambe 89]{Lam89} Lambe, L. A.
-``Scratchpad II as a tool for mathematical research''
-Notices of the AMS, February 1928 pp143-147
+\bibitem[Fateman 90]{Fat90} Fateman, R. J.
+``Advances and trends in the design and construction of algebraic
+manipulation systems''
+In Watanabe and Nagata [WN90], pp60-67 ISBN 0-89791-401-5 LCCN QA76.95.I57 1990
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lambe 91]{Lam91} Lambe, L. A.
-``Resolutions via homological perturbation''
-Journal of Symbolic Computation 12(1) pp71-87 July 1991
-CODEN JSYCEH ISSN 0747-7171
+\bibitem[Fateman 05]{Fat05} Fateman, R. J.
+``An incremental approach to building a mathematical expert out of software''
+4/19/2005\hfill
+\verb|www.cs.berkeley.edu/~fateman/papers/axiom.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Fat05.pdf|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lambe 92]{Lam92} Lambe, Larry
-``Next Generation Computer Algebra Systems AXIOM and the Scratchpad
-Concept: Applications to Research in Algebra''
-$21^{st}$ Nordic Congress of Mathematicians 1992
-%\verb|axiom-developer.org/axiom-website/papers/Lam92.pdf|
+\bibitem[Fateman 06]{Fat06} Fateman, R. J.
+``Building Algebra Systems by Overloading Lisp''
+\verb|www.cs.berkeley.edu/~fateman/generic/overload-small.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Fat06.pdf|
keywords = "axiomref",
abstract = "
- One way in which mathematicians deal with infinite amounts of data is
- symbolic representation. A simple example is the quadratic equation
- \[x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
- a formula which uses symbolic representation to describe the solutions
- to an infinite class of equations. Most computer algebra systems can
- deal with polynomials with symbolic coefficients, but what if symbolic
- exponents are called for (e.g. $1+t^i$)? What if symbolic limits on
- summations are also called for, for example
- \[1+t+\ldots+t^i=\sum_j{t^j}\]
+ Some of the earliest computer algebra systems (CAS) looked like
+ overloaded languages of the same era. FORMAC, PL/I FORMAC, Formula
+ Algol, and others each took advantage of a pre-existing language base
+ and expanded the notion of a numeric value to include mathematical
+ expressions. Much more recently, perhaps encouraged by the growth in
+ popularity of C++, we have seen a renewal of the use of overloading to
+ implement a CAS.
- The ``Scratchpad Concept'' is a theoretical ideal which allows the
- implementation of objects at this level of abstraction and beyond in a
- mathematically consistent way. The Axiom computer algebra system is an
- implementation of a major part of the Scratchpad Concept. Axiom
- (formerly called Scratchpad) is a language with extensible
- parameterized types and generic operators which is based on the
- notions of domains and categories. By examining some aspects of the
- Axiom system, the Scratchpad Concept will be illustrated. It will be
- shown how some complex problems in homologicial algebra were solved
- through the use of this system."
+ This paper makes three points. 1. It is easy to do overloading in
+ Common Lisp, and show how to do it in detail. 2. Overloading per se
+ provides an easy solution to some simple programming problems. We show
+ how it can be used for a ``demonstration'' CAS. Other simple and
+ plausible overloadings interact nicely with this basic system. 3. Not
+ all goes so smoothly: we can view overloading as a case study and
+ perhaps an object lesson since it fails to solve a number of
+ fairly-well articulated and difficult design issues in CAS for which
+ other approaches are preferable."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lambe 93]{Lam93} Lambe, Larry
-``On Using Axiom to Generate Code''
-(preprint) 1993
+\bibitem[Faure 00a]{FDN00a} Faure, Christ\'ele; Davenport, James
+``Parameters in Computer Algebra''
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lambe 93a]{LL93} Lambe, Larry; Luczak, Richard
-``Object-Oriented Mathematical Programming and Symbolic/Numeric Interface''
-$3^{rd}$ International Conf. on Expert Systems in Numerical Computing 1993
-%\verb|axiom-developer.org/axiom-website/papers/LL93.pdf|
+\bibitem[Faure 00b]{FDN00b} Faure, Christ\'ele; Davenport, James;
+Naciri, Hanane
+``Multi-values Computer Algebra''
+ISSN 0249-6399 Institut National De Recherche en Informatique et en
+Automatique Sept. 2000 No. 4001
+\verb|hal.inria.fr/inria-00072643/PDF/RR-4401.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/FDN00b.pdf|
keywords = "axiomref",
abstract = "
- The Axiom language is based on the notions of ``categories'',
- ``domains'', and ``packages''. These concepts are used to build an
- interface between symbolic and numeric calculations. In particular, an
- interface to the NAG Fortran Library and Axiom's algebra and graphics
- facilities is presented. Some examples of numerical calculations in a
- symbolic computational environment are also included using the finite
- element method. While the examples are elementary, we believe that
- they point to very powerful methods for combining numeric and symbolic
- computational techniques."
-
-\end{chunk}
+ One of the main strengths of computer algebra is being able to solve a
+ family of problems with one computation. In order to express not only
+ one problem but a family of problems, one introduces some symbols
+ which are in fact the parameters common to all the problems of the
+ family.
-\begin{chunk}{ignore}
-\bibitem[Lebedev 08]{Leb08} Lebedev, Yuri
-``OpenMath Library for Computing on Riemann Surfaces''
-PhD thesis, Nov 2008 Florida State University
-\verb|www.math.fsu.edu/~ylebedev/research/HyperbolicGeometry.html|
- keywords = "axiomref",
+ The user must be able to understand in which way these parameters
+ affect the result when he looks at the answer. Otherwise it may lead
+ to completely wrong calculations, which when used for numerical
+ applications bring nonsensical answers. This is the case in most
+ current Computer Algebra Systems we know because the form of the
+ answer is never explicitly conditioned by the values of the
+ parameters. The user is not even informed that the given answer may be
+ wrong in some cases then computer algebra systems can not be entirely
+ trustworthy. We have introduced multi-valued expressions called {\sl
+ conditional} expressions, in which each potential value is associated
+ with a condition on some parameters. This is used, in particular, to
+ capture the situation in integration, where the form of the answer can
+ depend on whether certain quantities are positive, negative or
+ zero. We show that it is also necessary when solving modular linear
+ equations or deducing congruence conditions from complex expressions."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[LeBlanc 91]{LeB91} LeBlanc, S.E.
-``The use of MathCAD and Theorist in the ChE classroom''
-In Anonymous [Ano91], pp287-299 (vol. 1) 2 vols.
+\bibitem[Fitch 84]{Fit84} Fitch, J. P. (ed)
+EUROSAM '84: International Symposium on Symbolic and
+Algebraic Computation, Cambridge, England, July 9-11, 1984, volume 174 of
+Lecture Notes in Computer Science. Springer-Verlag, Berlin, Germany /
+Heildelberg, Germany / London, UK / etc., 1984 ISBN 0-387-13350-X
+LCCN QA155.7.E4 I57 1984
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lecerf 96]{Le96} Lecerf, Gr\'egoire
-``Dynamic Evaluation and Real Closure Implementation in Axiom''
-June 29, 1996
-\verb|lecerf.perso.math.cnrs.fr/software/drc/drc.ps|
-%\verb|axiom-developer.org/axiom-website/papers/Le96.ps|
+\bibitem[Fitch 93]{Fit93} Fitch, J. (ed)
+Design and Implementation of Symbolic Computation Systems
+International Symposium DISCO '92 Proceedings. Springer-Verlag, Berlin,
+Germany / Heildelberg, Germany / London, UK / etc., 1993. ISBN 0-387-57272-4
+(New York), 3-540-57272-4 (Berlin). LCCN QA76.9.S88I576 1992
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lecerf 96a]{Le96a} Lecerf, Gr\'egoire
-``The Dynamic Real Closure implemented in Axiom''
-\verb|lecerf.perso.math.cnrs.fr/software/drc/drc.ps|
+\bibitem[Fogus 11]{Fog11} Fogus, Michael
+``UnConj''
+\verb|clojure.com/blog/2011/11/22/unconj.html|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Levelt 95]{Lev95} Levelt, A. H. M. (ed)
-ISSAC '95: Proceedings of the 1995 International
-Symposium on Symbolic and Algebraic Computation: July 10-12, 1995, Montreal,
-Canada ISSAC-PROCEEDINGS-1995. ACM Press, New York, NY 10036, USA, 1995
-ISBN 0-89791-699-9 LCCN QA76.95 I59 1995 ACM order number 505950
+\bibitem[Fortenbacher 90]{For90} Fortenbacher, A.
+``Efficient type inference and coercion in computer algebra''
+In Miola [Mio90], pp56-60. ISBN 0-387-52531-9 (New York), 3-540-52531-9
+(Berlin). LCCN QA76.9.S88I576 1990
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Li 06]{LM06} Li, Xin; Maza, Moreno
-``Efficient Implementation of Polynomial Arithmetic in a Multiple-Level
-Programming Environment''
-Lecture Notes in
-Computer Science Springer Vol 4151/2006 ISBN 978-3-540-38084-9 pp12-23
-Proceedings of International Congress of Mathematical Software ICMS 2006
-\verb|www.csd.uwo.ca/~moreno//Publications/Li-MorenoMaza-ICMS-06.pdf|
+\bibitem[Fouche 90]{Fou90} Fouche, Francois
+``Une implantation de l'algorithme de Kovacic en Scratchpad''
+Technical report, Institut de Recherche Math{\'{e}}matique Avanc{\'{e}}e''
+Strasbourg, France, 1990 31pp
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Li 10]{YL10} Li, Yue; Dos Reis, Gabriel
-``A Quantitative Study of Reductions in Algebraic Libraries''
-PASCO 2010
-\verb|www.axiomatics.org/~gdr/concurrency/quant-pasco10.pdf|
+\bibitem[FSF 14]{FSF14} FSF
+``Free Software Directory''
+\verb|directory.fsf.org/wiki/Axiom|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Li 11]{YL11} Li, Yue; Dos Reis, Gabriel
-``An Automatic Parallelization Framework for Algebraic Computation
-Systems''
-ISSAC 2011
-\verb|www.axiomatics.org/~gdr/concurrency/oa-conc-issac11.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/YL11.pdf|
+\bibitem[Frisco ]{Fris} Frisco
+``Objectives and Results''
+\verb|www.nag.co.uk/projects/frisco/frisco/node3.htm|
keywords = "axiomref",
- abstract = "
- This paper proposes a non-intrusive automatic parallelization
- framework for typeful and property-aware computer algebra systems."
\end{chunk}
+\subsection{G} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Ligatsikas 96]{Liga96} Ligatsikas, Zenon; Rioboo, Renaud;
-Roy, Marie Francoise
-``Generic computation of the real closure of an ordered field''
-Math. and Computers in Simulation 42 pp 541-549 (1996)
-%\verb|axiom-developer.org/axiom-website/papers/Liga96.pdf|
+\bibitem[Gebauer 86]{GM86} Gebauer, R{\"u}diger; M{\"o}ller, H. Michael
+``Buchberger's algorithm and staggered linear bases''
+In Bruce W. Char, editor. Proceedings of the 1986
+Symposium on Symbolic and Algebraic Computation: SYMSAC '86, July 21-23, 1986
+Waterloo, Ontario, pp218-221 ACM Press, New York, NY 10036, USA, 1986.
+ISBN 0-89791-199-7 LCCN QA155.7.E4 A281 1986 ACM order number 505860
keywords = "axiomref",
- abstract = "
- This paper describes a generalization of the real closure computation
- of an ordered field (Rioboo, 1991) enabling to use different technques
- to code a single real algebraic number."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Linton 93]{Lin93} Linton, Steve
-``Vector Enumeration Programs, version 3.04''
-\verb|www.cs.st-andrews.ac.uk/~sal/nme/nme_toc.html#SEC1|
+\bibitem[Gebauer 88]{GM88} Gebauer, R.; M{\"o}ller, H. M.
+``On an installation of Buchberger's algorithm''
+Journal of Symbolic Computation, 6(2-3) pp275-286 1988
+CODEN JSYCEH ISSN 0747-7171
+\verb|www.sciencedirect.com/science/article/pii/S0747717188800488/pdf|
+\verb|?md5=f6ccf63002ef3bc58aaa92e12ef18980&|
+\verb|pid=1-s2.0-S0747717188800488-main.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/GM88.pdf|
keywords = "axiomref",
+ abstract = "
+ Buchberger's algorithm calculates Groebner bases of polynomial
+ ideals. Its efficiency depends strongly on practical criteria for
+ detecting superfluous reductions. Buchberger recommends two
+ criteria. The more important one is interpreted in this paper as a
+ criterion for detecting redundant elements in a basis of a module of
+ syzygies. We present a method for obtaining a reduced, nearly minimal
+ basis of that module. The simple procedure for detecting (redundant
+ syzygies and )superfluous reductions is incorporated now in our
+ installation of Buchberger's algorithm in SCRATCHPAD II and REDUCE
+ 3.3. The paper concludes with statistics stressing the good
+ computational properties of these installations."
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Liska 97]{LD97} Liska, Richard; Drska, Ladislav; Limpouch, Jiri;
-Sinor, Milan; Wester, Michael; Winkler, Franz
-``Computer Algebra - algorithms, systems and applications''
-June 2, 1997
-\verb|kfe.fjfi.cvut.cz/~liska/ca/all.html|
- keywords = "axiomref",
+\begin{chunk}{axiom.bib}
+@book{Gedd92,
+ author = "Geddes, Keith and Czapor, O. and Stephen R. and Labahn, George",
+ title = "Algorithms For Computer Algebra",
+ publisher = "Kluwer Academic Publishers",
+ isbn = "0-7923-9259-0",
+ month = "September",
+ year = "1992",
+ keywords = "axiomref"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lucks 86]{Luc86} Lucks, Michael
-``A fast implementation of polynomial factorization''
-In Bruce W. Char, editor, Proceedings of the 1986 Symposium on Symbolic
-and Algebraic Computation: SYMSAC '86, July 21-23, 1986, Waterloo, Ontario,
-pp228-232 ACM Press, New York, NY 10036, USA, 1986. ISBN 0-89791-199-7
-LCCN QA155.7.E4 A281 1986 ACM order number 505860
+\bibitem[Gianni 87]{Gia87} Gianni, Patrizia
+``Primary Decomposition of Ideals''
+in [Wit87], pp12-13
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lueken 77]{Lue77} Lueken, E.
-``Ueberlegungen zur Implementierung eines Formelmanipulationssystems''
-Master's thesis, Technischen Universit{\"{a}}t Carolo-Wilhelmina zu
-Braunschweig. Braunschweig, Germany, 1977
+\bibitem[Gianni 88]{Gia88} Gianni, Patrizia.; Trager, Barry.;
+Zacharias, Gail.
+``Gr\"obner Bases and Primary Decomposition of Polynomial Ideals''
+J. Symbolic Computation 6, 149-167 (1988)
+\verb|www.sciencedirect.com/science/article/pii/S0747717188800403/pdf|
+\verb|?md5=40c29b67947035884904fd4597ddf710&|
+\verb|pid=1-s2.0-S0747717188800403-main.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Gia88.pdf|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lynch 91]{LM91} Lynch, R.; Mavromatis, H. A.
-``New quantum mechanical perturbation technique
-using an 'electronic scratchpad' on an inexpensive computer''
-American Journal of Pyhsics, 59(3) pp270-273, March 1991.
-CODEN AJPIAS ISSN 0002-9505
+\bibitem[Gianni 89a]{Gia89} Gianni, P. (Patrizia) (ed)
+Symbolic and Algebraic Computation.
+International Symposium ISSAC '88, Rome, Italy, July 4-8, 1988. Proceedings,
+volume 358 of Lecture Notes in Computer Science. Springer-Verlag, Berlin,
+Germany / Heildelberg, Germany / London, UK / etc., 1989. ISBN 3-540-51084-2
+LCCN QA76.95.I57 1988 Conference held jointly with AAECC-6
keywords = "axiomref",
\end{chunk}
-\subsection{M} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Mahboubi 05]{Mah05} Mahboubi, Assia
-``Programming and certifying the CAD algorithm inside the coq system''
-Mathematics, Algorithms, Proofs, volume 05021 of Dagstuhl
-Seminar Proceedings, Schloss Dagstuhl (2005)
+\bibitem[Gianni 89b]{GM89} Gianni, P.; Mora, T.
+``Algebraic solution of systems of polynomial equations using
+Gr{\"o}bner bases.''
+In Huguet and Poli [HP89], pp247-257 ISBN 3-540-51082-6 LCCN QA268.A35 1987
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Mathews 89]{Mat89} Mathews, J.
-``Symbolic computational algebra applied to Picard iteration''
-Mathematics and computer education, 23(2) pp117-122 Spring 1989 CODEN MCEDDA,
-ISSN 0730-8639
+\bibitem[Gil 92]{Gil92} Gil, I.
+``Computation of the Jordan canonical form of a square matrix (using
+the Axiom programming language)''
+In Wang [Wan92], pp138-145.
+ISBN 0-89791-489-9 (soft cover), 0-89791-490-2 (hard cover)
+LCCN QA76.95.I59 1992
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[McJones 11]{McJ11} McJones, Paul
-``Software Presentation Group -- Common Lisp family''
-\verb|www.softwarepreservation.org/projects/LISP/common_lisp_family|
+\bibitem[Gomez-Diaz 92]{Gom92} G\'omez-D'iaz, Teresa
+``Quelques applications de l`\'evaluation dynamique''
+Ph.D. Thesis L'Universite De Limoges March 1992
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Melachrinoudis 90]{MR90} Melachrinoudis, E.; Rumpf, D. L.
-``Teaching advantages of transparent computer software -- MathCAD''
-CoED, 10(1) pp71-76, January-March 1990 CODEN CWLJDP ISSN 0736-8607
+\bibitem[Gomez-Diaz 93]{Gom93} G\'omez-D\'iaz, Teresa
+``Examples of using Dynamic Constructible Closure''
+IMACS Symposium SC-1993
+%\verb|axiom-developer.org/axiom-website/papers/Gom93.pdf|
keywords = "axiomref",
+ abstract = "
+ We present here some examples of using the ``Dynamic Constructible
+ Closure'' program, which performs automatic case distinction in
+ computations involving parameters over a base field $K$. This program
+ is an application of the ``Dynamic Evaluation'' principle, which
+ generalizes traditional evaluation and was first used to deal with
+ algebraic numbers."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Miola 90]{Mio90} Miola, A. (ed)
-``Design and Implementation of Symbolic Computation Systems''
-International Symposium DISCO '90, Capri, Italy, April 10-12, 1990, Proceedings
-volume 429 of Lecture Notes in Cmputer Science,
-Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
-1990 ISBN 0-387-52531-9 (New York), 3-540-52531-9 (Berlin) LCCN QA76.9.S88I576
-1990
+\bibitem[Goodwin 91]{GBL91} Goodwin, B. M.; Buonopane, R. A.; Lee, A.
+``Using MathCAD in teaching material and energy balance concepts''
+In Anonymous [Ano91], pp345-349 (vol. 1) 2 vols.
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Miola 93]{Mio93} Miola, A. (ed)
-``Design and Implementation of Symbolic Computation Systems''
-International Symposium DISCO '93 Gmunden, Austria, September 15-17, 1993:
-Proceedings.
-Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
-1993 ISBN 3-540-57235-X LCCN QA76.9.S88I576 1993
+\bibitem[Golden 4]{GH84} Golden, V. Ellen; Hussain, M. A. (eds)
+Proceedings of the 1984 MACSYMA Users' Conference:
+Schenectady, New York, July 23-25, 1984, General Electric,
+Schenectady, NY, USA, 1984
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Missura 94]{Miss94} Missura, Stephan A.; Weber, Andreas
-``Using Commutativity Properties for Controlling Coercions''
-\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/|
-\verb|WeberA/MissuraWeber94a.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Miss94.pdf|
+\bibitem[Gonnet 96]{Gon96} Gonnet, Gaston H.
+``Official verion 1.0 of the Meta Content Dictionary''
+\verb|www.inf.ethz.ch/personal/gonnet/ContDict/Meta|
keywords = "axiomref",
- abstract = "
- This paper investigates some soundness conditions which have to be
- fulfilled in systems with coercions and generic operators. A result of
- Reynolds on unrestricted generic operators is extended to generic
- operators which obey certain constraints. We get natural conditions
- for such operators, which are expressed within the theoretic framework
- of category theory. However, in the context of computer algebra, there
- arise examples of coercions and generic operators which do not fulfil
- these conditions. We describe a framework -- relaxing the above
- conditions -- that allows distinguishing between cases of ambiguities
- which can be resolved in a quite natural sense and those which
- cannot. An algorithm is presented that detects such unresolvable
- ambiguities in expressions."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Monagan 87]{Mon87} Monagan, Michael B.
-``Support for Data Structures in Scratchpad II''
-in [Wit87], pp17-18
+\bibitem[Goodloe 93]{GL93} Goodloe, A.; Loustaunau, P.
+``An abstract data type development of graded rings''
+In Fitch [Fit93], pp193-202. ISBN 0-387-57272-4 (New York),
+3-540-57272-4 (Berlin). LCCN QA76.9.S88I576 1992
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Monagan 93]{Mon93} Monagan, M. B.
-``Gauss: a parameterized domain of computation system with
-support for signature functions''
-In Miola [Mio93], pp81-94 ISBN 3-540-57235-X LCCN QA76.9.S88I576 1993
+\bibitem[Gottliebsen 05]{GKM05} Gottliebsen, Hanne; Kelsey, Tom;
+Martin, Ursula
+``Hidden verification for computational mathematics''
+Journal of Symbolic Computation, Vol39, Num 5, pp539-567 (2005)
+\verb|www.sciencedirect.com/science/article/pii/S0747717105000295|
+%\verb|axiom-developer.org/axiom-website/papers/GKM05.pdf|
keywords = "axiomref",
+ abstract = "
+ We present hidden verification as a means to make the power of
+ computational logic available to users of computer algebra systems
+ while shielding them from its complexity. We have implemented in PVS a
+ library of facts about elementary and transcendental function, and
+ automatic procedures to attempt proofs of continuity, convergence and
+ differentiability for functions in this class. These are called
+ directly from Maple by a simple pipe-lined interface. Hence we are
+ able to support the analysis of differential equations in Maple by
+ direct calls to PVS for: result refinement and verification, discharge
+ of verification conditions, harnesses to ensure more reliable
+ differential equation solvers, and verifiable look-up tables."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Mora 89]{Mor89} Mora, T. (ed)
-Applied Algebra, Algebraic Algorithms and Error-Correcting
-Codes, 6th International Conference, AAECC-6, Rome, Italy, July 4-8, 1998,
-Proceedings, volume 357 of Lecture Notes in Computer Science
-Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
-1989 ISBN 3-540-51083-4, LCCN QA268.A35 1988 Conference held jointly with
-ISSAC '88
+\bibitem[Grabe 98]{Gra98} Gr\"abe, Hans-Gert
+``About the Polynomial System Solve Facility of Axiom, Macyma, Maple
+Mathematica, MuPAD, and Reduce''
+%\verb|axiom-developer.org/axiom-website/papers/Gra98.pdf|
keywords = "axiomref",
+ abstract = "
+ We report on some experiences with the general purpose Computer
+ Algebra Systems (CAS) Axiom, Macsyma, Maple, Mathematica, MuPAD, and
+ Reduce solving systems of polynomial equations and the way they
+ present their solutions. This snapshot (taken in the spring of 1996)
+ of the current power of the different systems in a special area
+ concentrates on both CPU-times and the quality of the output."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Moses 71]{Mos71} Moses, Joel
-``Algebraic Simplification: A Guide for the Perplexed''
-CACM August 1971 Vol 14 No. 8 pp527-537
+\bibitem[Grabmeier 91]{GHK91} Grabmeier, J.; Huber, K.; Krieger, U.
+``Das ComputeralgebraSystem AXIOM bei kryptologischen und
+verkehrstheoretischen Untersuchungen des
+Forschunginstituts der Deutschen Bundespost TELEKOM''
+Technischer Report TR 75.91.20, IBM Wissenschaftliches
+Zentrum, Heidelberg, Germany, 1991
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Moses 08]{Mos08} Moses, Joel
-``Macsyma: A Personal History''
-Invited Presentation in Milestones in Computer Algebra, May 2008, Tobago
-\verb|esd.mit.edu/Faculty_Pages/moses/Macsyma.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Mos08.pdf|
+\bibitem[Grabmeier 92]{GS92} Grabmeier, J.; Scheerhorn, A.
+``Finite fields in Axiom''
+AXIOM Technical Report TR7/92 (ATR/5)(NP2522),
+Numerical Algorithms Group, Inc., Downer's
+Grove, IL, USA and Oxford, UK, 1992
+\verb|www.nag.co.uk/doc/TechRep/axiomtr.html|
+and Technical Report, IBM Heidelberg Scientific Center, 1992
keywords = "axiomref",
- abstract = "
- The Macsyma system arose out of research on mathematical software in
- the AI group at MIT in the 1960's. Algorithm development in symbolic
- integration and simplification arose out of the interest of people,
- such as the author, who were also mathematics students. The later
- development of algorithms for the GCD of sparse polynomials, for
- example, arose out of the needs of our user community. During various
- times in the 1970's the computer on which Macsyma ran was one of the
- most popular notes on the ARPANET. We discuss the attempts in the late
- 70's and the 80's to develop Macsyma systems that ran on popular
- computer architectures. Finally, we discuss the impact of the
- fundamental ideas in Macsyma on current research on large scale
- engineering systems."
\end{chunk}
-\subsection{N} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Naylor]{NPxx} Naylor, William; Padget, Julian
-``From Untyped to Polymorphically Typed Objects in Mathematical Web
-Services''
-%\verb|axiom-developer.org/axiom-website/papers/NPxx.pdf|
+\bibitem[Grabmeier 03]{GKW03} Grabmeier, Johannes; Kaltofen, Erich;
+Weispfenning, Volker (eds)
+Computer algebra handbook: foundations, applications, systems.
+Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
+2003. ISBN 3-540-65466-6 637pp Includes CDROM
+\verb|www.springer.com/sgw/cda/frontpage/|
+\verb|0,11855,1-102-22-1477871-0,00.html|
keywords = "axiomref",
- abstract = "
- OpenMath is a widely recognized approach to the semantic markup of
- mathematics that is often used for communication between OpenMath
- compliant systems. The Aldor language has a sophisticated
- category-based type system that was specifically developed for the
- purpose of modelling mathematical structures, while the system itself
- supports the creation of small-footprint applications suitable for
- deployment as web services. In this paper we present our first results
- of how one may perform translations from generic OpenMath objects into
- values in specific Aldor domains, describing how the Aldor interfae
- domain ExpresstionTree is used to achieve this. We outline our Aldor
- implementation of an OpenMath translator, and describe an efficient
- extention of this to the Parser category. In addition, the Aldor
- service creation and invocation mechanism are explained. Thus we are
- in a position to develop and deploy mathematical web services whose
- descriptions may be directly derived from Aldor's rich type language."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Naylor 95]{N95} Naylor, Bill
-``Symbolic Interface for an advanced hyperbolic PDE solver''
-\verb|www.sci.csd.uwo.ca/~bill/Papers/symbInterface2.ps|
-%\verb|axiom-developer.org/axiom-website/papers/N95.pdf|
+\bibitem[Griesmer 71]{GJ71} Griesmer, J. H.; Jenks, R.D.
+``SCRATCHPAD/1 -- an interactive facility for symbolic mathematics''
+In Petrick [Pet71], pp42-58. LCCN QA76.5.S94 1971
+\verb|delivery.acm.org/10.1145/810000/806266/p42-griesmer.pdf|
+SYMSAC'71 Proc. second ACM Symposium on Symbolic and Algebraic
+Manipulation pp45-48
+%\verb|axiom-developer.org/axiom-website/papers/GJ71.pdf| REF:00027
keywords = "axiomref",
abstract = "
- An Axiom front end is described, which is used to generate
- mathematical objects needed by one of the latest NAG routines, to be
- included in the Mark 17 version of the NAG Numerical library. This
- routine uses powerful techniques to find the solution to Hyperbolic
- Partial Differential Equations in conservation form and in one spatial
- dimension. These mathematical objects are non-trivial, requiring much
- mathematical knowledge on the part of the user, which is otherwise
- irrelvant to the physical problem which is to be solved. We discuss
- the individual mathematical objects, considering the mathematical
- theory which is relevant, and some of the problems which have been
- encountered and solved during the FORTRAN generation necessary to
- realise the object. Finally we display some of our results."
+ The SCRATCHPAD/1 system is designed to provide an interactive symbolic
+ computational facility for the mathematician user. The system features
+ a user language designed to capture the style and succinctness of
+ mathematical notation, together with a facility for conveniently
+ introducing new notations into the language. A comprehensive system
+ library incorporates symbolic capabilities provided by such systems as
+ SIN, MATHLAB, and REDUCE."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Naylor 00b]{ND00} Naylor, W.A.; Davenport, J.H.
-``A Monte-Carlo Extension to a Category-Based Type System''
-\verb|www.sci.csd.uwo.ca/~bill/Papers/monteCarCat3.ps|
-%\verb|axiom-developer.org/axiom-website/papers/ND00.pdf|
+\bibitem[Griesmer 72a]{GJ72a} Griesmer, J.; Jenks, R.
+``Experience with an online symbolic math system SCRATCHPAD''
+in Online'72 [Onl72] ISBN 0-903796-02-3 LCCN QA76.55.O54 1972 Two volumes
keywords = "axiomref",
- abstract = "
- The normal claim for mathematics is that all calculations are 100\%
- accurate and therefore one calculation can rely completely on the
- results of sub-calculations, hoever there exist {\sl Monte-Carlo}
- algorithms which are often much faster than the equivalent
- deterministic ones where the results will have a prescribed
- probability (presumably small) of being incorrect. However there has
- been little discussion of how such algorithms can be used as building
- blocks in Computer Algebra. In this paper we describe how the
- computational category theory which is the basis of the type structure
- used in the Axiom computer algebra system may be extended to cover
- probabilistic algorithms, which use Monte-Carlo techniques. We follow
- this with a specific example which uses Straight Line Program
- representation."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Norman 75]{Nor75} Norman, A. C.
-``Computing with formal power series''
-ACM Transactions on Mathematical Software, 1(4) pp346-356
-Dec. 1975 CODEN ACMSCU ISSN 0098-3500
+\bibitem[Griesmer 72b]{GJ72b} Griesmer, James H.; Jenks, Richard D.
+``SCRATCHPAD: A capsule view''
+ACM SIGPLAN Notices, 7(10) pp93-102, 1972. Proceedings of the symposium
+on Two-dimensional man-machine communications. Mark B. Wells and
+James B. Morris (eds.).
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Norman 75a]{Nor75a} Norman, A.C.
-``The SCRATCHPAD Power Series Package''
-IBM T.J. Watson Research RC4998
+\bibitem[Griesmer 75]{GJY75} Griesmer, J.H.; Jenks, R.D.; Yun, D.Y.Y
+``SCRATCHPAD User's Manual''
+IBM Research Publication RA70 June 1975
keywords = "axiomref",
\end{chunk}
-\subsection{O} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Ollivier 89]{Oll89} Ollivier, F.
-``Inversibility of rational mappings and structural
-identifiablility in automatics''
-In ACM [ACM89], pp43-54 ISBN 0-89791-325-6 LCCN QA76.95.I59 1989
+\bibitem[Griesmer 76]{GJY76} Griesmer, J.H.; Jenks, R.D.; Yun, D.Y.Y
+``A Set of SCRATCHPAD Examples''
+April 1976 (private copy)
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Online 72]{Onl72}.
-Online 72: conference proceedings ... international conference on online
-interactive computing, Brunel University, Uxbridge, England, 4-7 September
-1972 ISBN 0-903796-02-3 LCCN QA76.55.O54 1972 Two volumes.
+\bibitem[Gruntz 94]{GM94} Gruntz, D.; Monagan, M.
+``Introduction to Gauss''
+SIGSAM Bulletin (ACM Special Interest Group on Symbolic and Algebraic
+Manipulation), 28(3) pp3-19 August 1994 CODEN SIGSBZ ISSN 0163-5824
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[OpenMath]{OpenMa}.
-``OpenMath Technical Overview''
-\verb|www.openmath.org/overview/technical.html|
+\bibitem[Gruntz 96]{Gru96} Gruntz, Dominik
+``On Computing Limits in a Symbolic Manipulation System''
+Thesis, Swiss Federal Institute of Technology Z\"urich 1996
+Diss. ETH No. 11432
+\verb|www.cybertester.com/data/gruntz.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Gru96.pdf|
keywords = "axiomref",
+ abstract = "
+ This thesis presents an algorithm for computing (one-sided) limits
+ within a symbolic manipulation system. Computing limtis is an
+ important facility, as limits are used both by other functions such as
+ the definite integrator and to get directly some qualitative
+ information about a given function.
+
+ The algorithm we present is very compact, easy to understand and easy
+ to implement. It overcomes the cancellation problem other algorithms
+ suffer from. These goals were achieved using a uniform method, namely
+ by expanding the whole function into a series in terms of its most
+ rapidly varying subexpression instead of a recursive bottom up
+ expansion of the function. In the latter approach exact error terms
+ have to be kept with each approximation in order to resolve the
+ cancellation problem, and this may lead to an intermediate expression
+ swell. Our algorithm avoids this problem and is thus suited to be
+ implemented in a symbolic manipulation system."
\end{chunk}
-\subsection{P} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{H} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
-\bibitem[Page 07]{Pa07} Page, William S.
-``Axiom - Open Source Computer Algebra System''
-Poster ISSAC 2007 Proceedings Vol 41 No 3 Sept 2007 p114
+\bibitem[Boyle 88]{Boyl88} Boyle, Ann
+``Future Directions for Research in Symbolic Computation''
+Soc. for Industrial and Applied Mathematics, Philadelphia (1990)
+\verb|www.eecis.udel.edu/~caviness/wsreport.pdf|
+%\verb|axiom-developer.org/axiom-website/Boyl88.pdf|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Petitot 90]{Pet90} Petitot, Michel
-``Types r\'ecursifs en scratchpad, application aux polyn\^omes non
-commutatifs''
-LIFL, 1990
+\bibitem[Hassner 87]{HBW87} Hassner, Martin; Burge, William H.;
+Watt, Stephen M.
+``Construction of Algebraic Error Control Codes (ECC) on the Elliptic
+Riemann Surface''
+in [Wit87], pp5-8
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Petitot 93]{Pet93} Petitot, M.
-``Experience with Axiom''
-In Jacob et al. [JOS93], page 240
+\bibitem[Heck 01]{Hec01} Heck, A.
+``Variables in computer algebra, mathematics and science''
+The International Journal of Computer Algebra in Mathematics Education
+Vol. 8 No. 3 pp195-210 (2001)
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Petric 71]{Pet71} Petric, S. R. (ed)
-Proceedings of the second symposium on Symbolic and
-Algebraic Manipulation, March 23-25, 1971, Los Angeles, California, ACM Press,
-New York, NY 10036, USA, 1971. LCCN QA76.5.S94 1971
+\bibitem[Huguet 89]{HP89} Huguet, L.; Poli, A. (eds).
+Applied Algebra, Algebraic Algorithms and Error-Correcting Codes.
+5th International Conference AAECC-5 Proceedings.
+Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
+1989. ISBN 3-540-51082-6. LCCN QA268.A35 1987
keywords = "axiomref",
\end{chunk}
+\subsection{J} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Pinch 93]{Pin93} Pinch, R.G.E.
-``Some Primality Testing Algorithms''
-Devlin, Keith (ed.)
-Computers and Mathematics November 1993, Vol 40, Number 9 pp1203-1210
+\bibitem[Jacob 93]{JOS93} Jacob, G.; Oussous, N. E.; Steinberg, S. (eds)
+Proceedings SC 93
+International IMACS Symposium on Symbolic Computation. New Trends and
+Developments. LIFL Univ. Lille, Lille France, 1993
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Poll (b)]{Polxx} Poll, Erik
-``The type system of Axiom''
-%\verb|axiom-developer.org/axiom-website/papers/Polxx.pdf|
+\bibitem[Janssen 88]{Jan88} Jan{\ss}en, R. (ed)
+Trends in Computer Algebra, International Symposium
+Bad Neuenahr, May 19-21, 1987, Proceedings, volume 296 of Lecture Notes in
+Computer Science.
+Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
+1988 ISBN 3-540-18928-9, 0-387-18928-9 LCCN QA155.7.E4T74 1988
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Purtilo 86]{Pur86} Purtilo, J.
-``Applications of a software interconnection system in mathematical
-problem solving environments'' In Bruce W. Char, editor. Proceedings of the
-1986 Symposium on Symbolic and Algebraic Computation: SYMSAC '86, July 21-23,
-ACM Press, New York, NY 10036, USA, 1986. ISBN 0-89791-199-7 LCCN QA155.7.E4
-A281 1986 ACM order number 505860
+\bibitem[Jenks 69]{Jen69} Jenks, R. D.
+``META/LISP: An interactive translator writing system''
+Research Report International Business Machines, Inc., Thomas J.
+Watson Research Center, Yorktown Heights, NY, USA, 1969 RC2968 July 1970
keywords = "axiomref",
\end{chunk}
-\subsection{R} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Rainer 14]{Rain14} Joswig, Rainer
-``2014: 30+ Years Common Lisp the Language''
-\verb|lispm.de/30ycltl|
+\bibitem[Jenks 71]{Jen71} Jenks, R. D.
+``META/PLUS: The syntax extension facility for SCRATCHPAD''
+Research Report RC 3259, International Business Machines, Inc., Thomas J.
+Watson Research Center, Yorktown Heights, NY, USA, 1971
+% REF:00040
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Rioboo 03a]{Riob03a} Rioboo, Renaud
-``Quelques aspects du calcul exact avec des nombres r\'eels''
-Ph.D. Thesis, Laboratoire d'Informatique Th\'eorique et Programmationg
-%\verb|axiom-developer.org/axiom-website/papers/Riob03a.ps|
+\bibitem[Jenks 74]{Jen74} Jenks, R. D.
+``The SCRATCHPAD language''
+ACM SIGPLAN Notices, 9(4) pp101-111 1974 CODEN SINODQ. ISSN 0362-1340
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Rioboo 03]{Riob03} Rioboo, Renaud
-``Towards Faster Real Algebraic Numbers''
-J. of Symbolic Computation 36 pp 513-533 (2003)
-%\verb|axiom-developer.org/axiom-website/papers/Riob03.pdf|
+\bibitem[Jen76]{Jen76} Jenks, Richard D.
+``A pattern compiler''
+In Richard D. Jenks, editor,
+SYMSAC '76: proceedings of the 1976 ACM Symposium on Symbolic and Algebraic
+Computation, August 10-12, 1976, Yorktown Heights, New York, pp60-65,
+ACM Press, New York, NY 10036, USA, 1976. LCCN QA155.7.EA .A15 1976
+QA9.58.A11 1976
keywords = "axiomref",
- abstract = "
- This paper presents a new encoding scheme for real algebraic number
- manipulations which enhances current Axiom's real closure. Algebraic
- manipulations are performed using different instantiations of
- sub-resultant-like algorithms instead of Euclidean-like algorithms.
- We use these algorithms to compute polynomial gcds and Bezout
- relations, to compute the roots and the signs of algebraic
- numbers. This allows us to work in the ring of real algebraic integers
- instead of the field of read algebraic numbers avoiding many
- denominators."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Robidoux 93]{Rob93} Robidoux, Nicolas
-``Does Axiom Solve Systems of O.D.E's Like Mathematica?''
-July 1993
-%\verb|axiom-developer.org/axiom-website/papers/Rob93.pdf|
+\bibitem[Jenks 79]{Jen79} Jenks, R. D.
+``MODLISP: An Introduction''
+Proc EUROSAM 79, pp466-480, 1979 and IBMRC8073 Jan 1980
keywords = "axiomref",
- abstract = "
- If I were demonstrating Axiom and were asked this question, my reply
- would be ``No, but I am not sure that this is a bad thing''. And I
- would illustrate this with the following example.
-
- Consider the following system of O.D.E.'s
- \[
- \begin{array}{rcl}
- \frac{dx_1}{dt} & = & \left(1+\frac{cos t}{2+sin t}\right)x_1\\
- \frac{dx_2}{dt} & = & x_1 - x_2
- \end{array}
- \]
- This is a very simple system: $x_1$ is actually uncoupled from $x_2$"
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Rioboo 92]{Rio92} Rioboo, R.
-``Real algebraic closure of an ordered field, implementation in Axiom''
-In Wang [Wan92], pp206-215 ISBN 0-89791-489-9 (soft cover)
-0-89791-490-2 (hard cover) LCCN QA76.95.I59 1992
-%\verb|axiom-developer.org/axiom-website/papers/Rio92.pdf|
+\bibitem[Jenks 81]{JT81} Jenks, R.D.; Trager, B.M.
+``A Language for Computational Algebra''
+Proceedings of SYMSAC81, Symposium on Symbolic and Algebraic Manipulation,
+Snowbird, Utah August, 1981
keywords = "axiomref",
- abstract = "
- Real algebraic numbers appear in many Computer Algebra problems. For
- instance the determination of a cylindrical algebraic decomposition
- for an euclidean space requires computing with real algebraic numbers.
- This paper describes an implementation for computations with the real
- roots of a polynomial. This process is designed to be recursively
- used, so the resulting domain of computation is the set of all real
- algebraic numbers. An implementation for the real algebraic closure
- has been done in Axiom (previously called Scratchpad)."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Roesner 95]{Roe95} Roesner, K. G.
-``Verified solutions for parameters of an exact solution for
-non-Newtonian liquids using computer algebra'' Zeitschrift fur Angewandte
-Mathematik und Physik, 75 (suppl. 2):S435-S438, 1995 ISSN 0044-2267
+\bibitem[Jenks 81a]{JT81a} Jenks, R.D.; Trager, B.M.
+``A Language for Computational Algebra''
+SIGPLAN Notices, New York: Association for Computing Machiner, Nov 1981
keywords = "axiomref",
\end{chunk}
-\subsection{S} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Sage 14]{Sage14} Stein, William
-``Sage''
-\verb|www.sagemath.org/doc/reference/interfaces/sage/interfaces/axiom.html|
+\bibitem[Jenks 81b]{JT81b} Jenks, R.D.; Trager, B.M.
+``A Language for Computational Algebra''
+IBM Research Report RC8930 IBM Yorktown Heights, NY
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Salvy 89]{Sal89} Salvy, B.
-``Examples of automatic asymptotic expansions''
-Technical Report 114,
-Inst. Nat. Recherche Inf. Autom., Le Chesnay, France, Dec. 1989 18pp
+\bibitem[Jenks 84a]{Jen84a} Jenks, Richard D.
+``The new SCRATCHPAD language and system for computer algebra''
+In Golden and Hussain [GH84], pp409-??
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Salvy 91]{Sal91} Salvy, B.
-``Examples of automatic asymptotic expansions''
-SIGSAM Bulletin (ACM Special Interest Group on Symbolic and
-Algebraic Manipulation), 25(2) pp4-17
-April 1991 CODEN SIGSBZ ISSN 0163-5824
+\bibitem[Jenks 84b]{Jen84b} Jenks, Richard D.
+``A primer: 11 keys to New Scratchpad''
+In Fitch [Fit84], pp123-147. ISBN 0-387-13350-X LCCN QA155.7.E4 I57 1984
keywords = "axiomref",
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Saun80,
- author = "Saunders, B. David",
- title = "A Survey of Available Systems",
- journal = "SIGSAM Bull.",
- issue_date = "November 1980",
- volume = "14",
- number = "4",
- month = "November",
- year = "1980",
- issn = "0163-5824",
- pages = "12--28",
- numpages = "17",
- url = "http://doi.acm.org/10.1145/1089235.1089237",
- doi = "10.1145/1089235.1089237",
- acmid = "1089237",
- publisher = "ACM",
- address = "New York, NY, USA",
- keywords = "axiomref,survey",
- paper = "Saun80.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Jenks 86]{JWS86} Jenks, Richard D.; Sutor, Robert S.;
+Watt, Stephen M.
+``Scratchpad II: An Abstract Datatype System for Mathematical Computation''
+Research Report RC 12327 (\#55257), Iinternational Business Machines, Inc.,
+Thomas J. Watson Research Center, Yorktown Heights, NY, USA, 1986 23pp
+\verb|www.csd.uwo.ca/~watt/pub/reprints/1987-ima-spadadt.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/JWS86.pdf|
+ keywords = "axiomref",
+ abstract = "
+ Scratchpad II is an abstract datatype language and system that is
+ under development in the Computer Algebra Group, Mathematical Sciences
+ Department, at the IBM Thomas J. Watson Research Center. Some features
+ of APL that made computation particularly elegant have been borrowed.
+ Many different kinds of computational objects and data structures are
+ provided. Facilities for computation include symbolic integration,
+ differentiation, factorization, solution of equations and linear
+ algebra. Code economy and modularity is achieved by having
+ polymorphic packages of functions that may create datatypes. The use
+ of categories makes these facilities as general as possible."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Schu 92]{Sch92} Sch\"u, J.
-``Implementing des Cartan-Kuranishi-Theorems in AXIOM''
-Master's diploma thesis (in german), Institut f\"ur Algorithmen und
-Kognitive Systeme, Universit\"t Karlsruhe 1992
+\bibitem[Jenks 87]{JWS87} Jenks, Richard D.; Sutor, Robert S.;
+Watt, Stephen M.
+``Scratchpad II: an Abstract Datatype System for Mathematical Computation''
+Proceedings Trends in Computer Algebra, Bad Neuenahr, LNCS 296,
+Springer Verlag, (1987)
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Schwarz 88]{Sch88} Schwarz, F.
-``Programming with abstract data types: the symmetry package SPDE
-in Scratchpad''
-In Jan{\ss}en [Jan88], pp167-176, ISBN 3-540-18928-9,
-0-387-18928-9 LCCN QA155.7.E4T74 1988
+\bibitem[Jenks 88]{JSW88} Jenks, R. D.; Sutor, R. S.; Watt, S. M.
+``Scratchpad II: An abstract datatype system for mathematical computation''
+In Jan{\ss}en [Jan88],
+pp12-?? ISBN 3-540-18928-9, 0-387-18928-9 LCCN QA155.7.E4T74 1988
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Schwarz 89]{Sch89} Schwarz, F.
-``A factorization algorithm for linear ordinary differential equations''
-In ACM [ACM89], pp17-25 ISBN 0-89791-325-6 LCCN QA76.95.I59 1989
+\bibitem[Jenks 88a]{Jen88a} Jenks, R. D.
+``A Guide to Programming in BOOT''
+Computer Algebra Group, Mathematical Sciences Department, IBM Research
+Draft September 5, 1988
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Schwarz 91]{Sch91} Schwarz, F.
-``Monomial orderings and Gr{\"o}bner bases''
-SIGSAM Bulletin (ACM Special Interest Group on Symbolic and Algebraic
-Manipulation) 2591) pp10-23 Jan. 1991 CODEN SIGSBZ ISSN 0163-5824
+\bibitem[Jenks 88b]{Jen88b} Jenks, Richard
+``The Scratchpad II Computer Algebra System Interactive Environment Users
+Guide''
+ Spring 1988
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Seiler 94]{Sei94} Seiler, Werner Markus
-``Analysis and Application of the Formal Theory of Partial Differential
-Equations''
-PhD thesis, School of Physics and Materials, Lancaster University (1994)
-\verb|www.mathematik.uni-kassel.de/~seiler/Papers/Diss/diss.ps.gz|
-%\verb|axiom-developer.org/axiom-website/papers/Sei94.pdf|
+\bibitem[Jenks 88c]{JWS88} Jenks, R. D.; Sutor, R. S.; Watt, S. M.
+``Scratchpad II: an abstract datatype system for mathematical computation''
+In Jan{\ss}en
+[Jan88], pp12-37. ISBN 3-540-18928-9, 0-387-18928-9 LCCN QA155.7.E4T74 1988
keywords = "axiomref",
- abstract = "
- An introduction to the formal theory of partial differential equations
- is given emphasizing the properties of involutive symbols and
- equations. An algorithm to complete any differential equation to an
- involutive one is presented. For an involutive equation possible
- values for the number of arbitrary functions in its general solution
- are determined. The existence and uniqueness of solutions for analytic
- equations is proven. Applications of these results include an
- analysis of symmetry and reduction methods and a study of gauge
- systems. It is show that the Dirac algorithm for systems with
- constraints is closely related to the completion of the equation of
- motion to an involutive equation. Specific examples treated comprise
- the Yang-Mills Equations, Einstein Equations, complete and Jacobian
- systems, and some special models in two and three dimensions. To
- facilitate the involved tedious computations an environment for
- geometric approaches to differential equations has been developed in
- the computer algebra system Axiom. The appendices contain among others
- brief introductions into Carten-K{\"a}hler Theory and Janet-Riquier
- Theory."
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@book{Jenk92,
+ author = "Jenks, Richard D. and Sutor, Robert S.",
+ title = "AXIOM: The Scientific Computation System",
+ publisher = "Springer-Verlag, Berlin, Germany",
+ year = "1992",
+ isbn = "0-387-97855-0",
+ keywords = "axiomref"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Seiler 94a]{Sei94a} Seiler, W.M.
-``Completion to involution in AXIOM''
-in Calmet [Cal94] pp103-104
+\bibitem[Jenks 94]{JT94} Jenks, R. D.; Trager, B. M.
+``How to make AXIOM into a Scratchpad''
+In ACM [ACM94], pp32-40 ISBN 0-89791-638-7 LCCN QA76.95.I59 1994
+%\verb|axiom-developer.org/axiom-website/papers/JT94.pdf|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Sieler 94b]{Sei94b} Seiler, W.M.
-``Pseudo differential operators and integrable systems in AXIOM''
-Computer Physics Communications, 79(2) pp329-340 April 1994 CODEN CPHCBZ
-ISSN 0010-4655
-%\verb|axiom-developer.org/axiom-website/papers/Sei94b.pdf|
+\bibitem[Joswig 03]{JT03} Joswig, Michael; Takayama, Nobuki
+``Algebra, geometry, and software systems''
+Springer-Verlag ISBN 3-540-00256-1 p291
keywords = "axiomref",
- abstract = "
- An implementation of the algebra of pseudo differential operators in
- the computer algebra system Axiom is described. In several exmaples
- the application of the package to typical computations in the theory
- of integrable systems is demonstrated."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Seiler 95]{Sei95} Seiler, W.M.
-``Applying AXIOM to partial differential equations''
-Internal Report 95-17, Universit\"at Karlsruhe, Fakult\"at f\"ur Informatik
-1995
-%\verb|axiom-developer.org/axiom-website/papers/Sei95.pdf|
+\bibitem[Joyner 06]{J006} Joyner, David
+``OSCAS - Maxima''
+SIGSAM Communications in Computer Algebra, 157 2006
+\verb|sage.math.washington.edu/home/wdj/sigsam/oscas-cca1.pdf|
keywords = "axiomref",
- abstract = "
- We present an Axiom environment called JET for geometric computations
- with partial differential equations within the framework of the jet
- bundle formalism. This comprises expecially the completion of a given
- differential equation to an involutive one according to the
- Cartan-Kuranishi Theorem and the setting up of the determining system
- for the generators of classical and non-classical Lie
- symmetries. Details of the implementations are described and
- applications are given. An appendix contains tables of all exported
- functions."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Seiler 95b]{SC95} Seiler, W.M.; Calmet, J.
-``JET -- An Axiom Environment for Geometric Computations with Differential
-Equations''
-%\verb|axiom-developer.org/axiom-website/papers/SC95.pdf|
+\bibitem[Joyner 14]{JO14} Joyner, David
+``Links to some open source mathematical programs''
+\verb|www.opensourcemath.org/opensource_math.html|
keywords = "axiomref",
- abstract = "
- JET is an environment within the computer algebra system Axiom to
- perform such computations. The current implementation emphasises the
- two key concepts involution and symmetry. It provides some packages
- for the completion of a given system of differential equations to an
- equivalent involutive one based on the Cartan-Kuranishi theorem and
- for setting up the determining equations for classical and
- non-classical point symmetries."
\end{chunk}
+\subsection{K} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Seiler 97]{Sei97} Seiler, Werner M.
-``Computer Algebra and Differential Equations: An Overview''
-\verb|www.mathematik.uni-kassel.di/~seiler/Papers/Postscript/CADERep.ps.gz|
+\bibitem[Kauers 08]{Kau08} Kauers, Manuel
+``Integration of Algebraic Functions: A Simple Heuristic for Finding
+the Logarithmic Part''
+ISSAC July 2008 ACM 978-1-59593-904 pp133-140
+\verb|www.risc.jku.at/publications/download/risc_3427/Ka01.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Kau08.pdf|
keywords = "axiomref",
abstract = "
- We present an informal overview of a number of approaches to
- differential equations which are popular in computer algebra. This
- includes symmetry and completion theory, local analysis, differential
- ideal and Galois theory, dynamical systems and numerical analysis. A
- large bibliography is provided."
+ A new method is proposed for finding the logarithmic part of an
+ integral over an algebraic function. The method uses Gr{\"o}bner bases
+ and is easy to implement. It does not have the feature of finding a
+ closed form of an integral whenever there is one. But it very often
+ does, as we will show by a comparison with the built-in integrators of
+ some computer algebra systems."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Seiler (a)]{Seixx} Seiler, W.M.
-``DETools: A Library for Differential Equations''
-\verb|iaks-www.ira.uka.de/iaks-calmet/werner/werner.html|
+\bibitem[Keady 94]{KN94} Keady, G.; Nolan, G.
+``Production of Argument SubPrograms in the AXIOM -- NAG
+link: examples involving nonleanr systems''
+Technical Report TR1/94
+ATR/7 (NP2680), Numerical Algorithms Group, Inc., Downer's Grove, IL, USA and
+Oxford, UK, 1994
+\verb|www.nag.co.uk/doc/TechRep/axiomtr.html|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Shannon 88]{SS88} Shannon, D.; Sweedler, M.
-``Using Gr{\"o}bner bases to determine algebra
-membership, split surjective algebra homomorphisms determine birational
-equivalence''
-Journal of Symbolic Computation 6(2-3) pp267-273
-Oct.-Dec. 1988 CODEN JSYCEH ISSN 0747-7171
+\bibitem[Kelsey 99]{Kel99} Kelsey, Tom
+``Formal Methods and Computer Algebra: A Larch Specification of AXIOM
+Categories and Functors''
+Ph.D. Thesis, University of St Andrews, 1999
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Sit 89]{Sit89} Sit, W.Y.
-``On Goldman's algorithm for solving first-order multinomial
-autonomous systems'' In Mora [Mor89], pp386-395 ISBN 3-540-51083-4
-LCCN QA268.A35 1998 Conference held jointly with ISSAC '88
+\bibitem[Kelsey 00a]{Kel00a} Kelsey, Tom
+``Formal specification of computer algebra''
+University of St Andrews, 6th April 2000
+\verb|www.cs.st-andrews.cs.uk/~tom/pub/fscbs.ps|
+%\verb|axiom-developer.org/axiom-website/papers/Kel00a.pdf|
keywords = "axiomref",
+ abstract = "
+ We investigate the use of formal methods languages and tools in the
+ design and development of computer algebra systems (henceforth CAS).
+ We demonstrate that errors in CAS design can be identified and
+ corrected by the use of (i) abstract specifications of types and
+ procedures, (ii) automated proofs of properties of the specifications,
+ and (iii) interface specifications which assist the verification of
+ pre- and post conditions of implemented code."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Sit 92]{Sit92} Sit, W.Y.
-``An algorithm for solving parametric linear systems''
-Journal of Symbolic Computations, 13(4) pp353-394, April 1992 CODEN JSYCEH
-ISSN 0747-7171
-\verb|www.sciencedirect.com/science/article/pii/S0747717108801046/pdf|
-\verb|?md5=00aa65e18e6ea5c4a008c8dfdfcd4b83&|
-\verb|pid=1-s2.0-S0747717108801046-main.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Sit92.pdf|
+\bibitem[Kelsey 00b]{Kel00b} Kelsey, Tom
+``Formal specification of computer algebra''
+(slides) University of St Andrews, Sept 21, 2000
+\verb|www.cs.st-andrews.cs.uk/~tom/pub/fscbstalk.ps|
keywords = "axiomref",
- abstract = "
- We present a theoretical foundation for studying parametric systesm of
- linear equations and prove an efficient algorithm for identifying all
- parametric values (including degnerate cases) for which the system is
- consistent. The algorithm gives a small set of regimes where for each
- regime, the solutions of the specialized systems may be given
- uniformly. For homogeneous linear systems, or for systems were the
- right hand side is arbitrary, this small set is irredunant. We discuss
- in detail practical issues concerning implementations, with particular
- emphasis on simplification of results. Examples are given based on a
- close implementation of the algorithm in SCRATCHPAD II. We also give a
- complexity analysis of the Gaussian elimination method and compare
- that with our algorithm."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Sit 06]{Sit06} Sit, Emil
-``Tools for Repeatable Research''
-\verb|www.emilsit.net/blog/archives/tools-for-repeatable-research|
+\bibitem[Kendall 99a]{Ken99a} Kendall, W.S.
+``Itovsn3 in AXIOM: modules, algebras and stochastic differentials''
+\verb|www2.warwick.ac.uk/fac/sci/statistics/staff/academic-research/|
+\verb|kendall/personal/ppt/328.ps.gz|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Smedley 92]{Sme92} Smedley, Trevor J.
-``Using pictorial and object oriented programming for computer algebra''
-In Hal Berghel et al., editors. Applied computing --
-technologicial challenges of the 199s: proceedings of the 1992 ACM/SIGAPP
-Symposium on Applied Computing, Kansas City Convention Center, March 1-3, 1992
-pp1243-1247. ACM Press, New York, NY 10036, USA, 1992. ISBN 0-89791-502-X
-LCCN QA76.76.A65 S95 1992
+\bibitem[Kendall 99b]{Ken99b} Kendall, W.S.
+``Symbolic It\^o calculus in AXIOM: an ongoing story
+\verb|www2.warwick.ac.uk/fac/sci/statistics/staff/academic-research/|
+\verb|kendall/personal/ppt/327.ps.gz|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Smith 07]{SDJ07} Smith, Jacob; Dos Reis, Gabriel; Jarvi, Jaakko
-``Algorithmic differentiation in Axiom''
-ACM SIGSAM ISSAC Proceedings 2007 Waterloo, Canada 2007 pp347-354
-ISBN 978-1-59593-743-8
-%\verb|axiom-developer.org/axiom-website/papers/SDJ07.pdf|
+\bibitem[Kosleff 91]{Kos91} P.-V. Koseleff
+``Word games in free Lie algebras: several bases and formulas''
+Theoretical Computer Science 79(1) pp241-256 Feb. 1991 CODEN TCSCDI
+ISSN 0304-3975
+ keywords = "axiomref",
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Kusche 89]{KKM89} Kusche, K.; Kutzler, B.; Mayr, H.
+``Implementation of a geometry theorem proving package in SCRATCHPAD II''
+In Davenport [Dav89] pp246-257 ISBN 3-540-51517-8 LCCN QA155.7.E4E86 1987
keywords = "axiomref",
- abstract = "
- This paper describes the design and implementation of an algorithmic
- differentiation framework in the Axiom computer algebra system. Our
- implementation works by transformations on Spad programs at the level
- of the typed abstract syntax tree."
\end{chunk}
+\subsection{L} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[SSC92]{SSC92}.
-``Algorithmic Methods For Lie Pseudogroups''
-In N. Ibragimov, M. Torrisi and A. Valenti, editors, Proc. Modern Group
-Analysis: Advanced Analytical and Computational Methods in Mathematical
-Physics, pp337-344, Acireale (Italy), 1992 Kluwer, Dordrecht 1993
-\verb|iaks-www.ira.uka.de/iaks-calmet/werner/Papers/Acireale92.ps.gz|
+\bibitem[Lahey 08]{Lah08} Lahey, Tim
+``Sage Integration Testing''
+\verb|github.com/tjl/sage_int_testing| Dec. 2008
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[SSV87]{SSV87} Senechaud, P.; Siebert, F.; Villard G.
-``Scratchpad II: Pr{\'e}sentation d'un nouveau langage de calcul formel''
-Technical Report 640-M, TIM 3 (IMAG), Grenoble, France, Feb 1987
+\bibitem[Lambe 89]{Lam89} Lambe, L. A.
+``Scratchpad II as a tool for mathematical research''
+Notices of the AMS, February 1928 pp143-147
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Steele]{Steele} Steele, Guy L.; Gabriel, Richard P.
-``The Evolution of Lisp''
-\verb|www.dreamsongs.com/Files/HOPL2-Uncut.pdf|
+\bibitem[Lambe 91]{Lam91} Lambe, L. A.
+``Resolutions via homological perturbation''
+Journal of Symbolic Computation 12(1) pp71-87 July 1991
+CODEN JSYCEH ISSN 0747-7171
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Sutor 85]{Sut85} Sutor, R.S.
-``The Scratchpad II computer algebra language and system''
-In Buchberger and Caviness [BC85], pp32-33 ISBN 0-387-15983-5 (vol. 1),
-0-387-15984-3 (vol. 2) LCCN QA155.7.E4 E86 1985 Two volumes.
+\bibitem[Lambe 92]{Lam92} Lambe, Larry
+``Next Generation Computer Algebra Systems AXIOM and the Scratchpad
+Concept: Applications to Research in Algebra''
+$21^{st}$ Nordic Congress of Mathematicians 1992
+%\verb|axiom-developer.org/axiom-website/papers/Lam92.pdf|
keywords = "axiomref",
+ abstract = "
+ One way in which mathematicians deal with infinite amounts of data is
+ symbolic representation. A simple example is the quadratic equation
+ \[x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
+ a formula which uses symbolic representation to describe the solutions
+ to an infinite class of equations. Most computer algebra systems can
+ deal with polynomials with symbolic coefficients, but what if symbolic
+ exponents are called for (e.g. $1+t^i$)? What if symbolic limits on
+ summations are also called for, for example
+ \[1+t+\ldots+t^i=\sum_j{t^j}\]
+
+ The ``Scratchpad Concept'' is a theoretical ideal which allows the
+ implementation of objects at this level of abstraction and beyond in a
+ mathematically consistent way. The Axiom computer algebra system is an
+ implementation of a major part of the Scratchpad Concept. Axiom
+ (formerly called Scratchpad) is a language with extensible
+ parameterized types and generic operators which is based on the
+ notions of domains and categories. By examining some aspects of the
+ Axiom system, the Scratchpad Concept will be illustrated. It will be
+ shown how some complex problems in homologicial algebra were solved
+ through the use of this system."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Sutor 87a]{SJ87a} Sutor, R. S.; Jenks, R. D.
-``The type inference and coercion facilities in
-the Scratchpad II interpreter'' In Wexelblat [Wex87], pp56-63
-ISBN 0-89791-235-7 LCCN QA76.7.S54 v22:7 SIGPLAN Notices, v22 n7 (July 1987)
-%\verb|axiom-developer.org/axiom-website/papers/SJ87a.pdf|
+\bibitem[Lambe 93]{Lam93} Lambe, Larry
+``On Using Axiom to Generate Code''
+(preprint) 1993
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Sutor 87b]{Su87} Sutor, Robert S.
-``The Scratchpad II Computer Algebra System. Using and
-Programming the Interpreter''
-IBM Course presentation slide deck Spring 1987
+\bibitem[Lambe 93a]{LL93} Lambe, Larry; Luczak, Richard
+``Object-Oriented Mathematical Programming and Symbolic/Numeric Interface''
+$3^{rd}$ International Conf. on Expert Systems in Numerical Computing 1993
+%\verb|axiom-developer.org/axiom-website/papers/LL93.pdf|
keywords = "axiomref",
+ abstract = "
+ The Axiom language is based on the notions of ``categories'',
+ ``domains'', and ``packages''. These concepts are used to build an
+ interface between symbolic and numeric calculations. In particular, an
+ interface to the NAG Fortran Library and Axiom's algebra and graphics
+ facilities is presented. Some examples of numerical calculations in a
+ symbolic computational environment are also included using the finite
+ element method. While the examples are elementary, we believe that
+ they point to very powerful methods for combining numeric and symbolic
+ computational techniques."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Sutor 87c]{SJ87c} Sutor, Robert S.; Jenks, Richard
-``The type inference and coercion facilities
-in the Scratchpad II interpreter''
-Research report RC 12595 (\#56575),
-IBM Thomas J. Watson Research Center, Yorktown Heights, NY, USA, 1987, 11pp
-%\verb|axiom-developer.org/axiom-website/papers/SJ87c.pdf|
+\bibitem[Lebedev 08]{Leb08} Lebedev, Yuri
+``OpenMath Library for Computing on Riemann Surfaces''
+PhD thesis, Nov 2008 Florida State University
+\verb|www.math.fsu.edu/~ylebedev/research/HyperbolicGeometry.html|
keywords = "axiomref",
- abstract = "
- The Scratchpad II system is an abstract datatype programming language,
- a compiler for the language, a library of packages of polymorphic
- functions and parameterized abstract datatypes, and an interpreter
- that provides sophisticated type inference and coercion facilities.
- Although originally designed for the implementation of symbolic
- mathematical algorithms, Scratchpad II is a general purpose
- programming language. This paper discusses aspects of the
- implementation of the intepreter and how it attempts to provide a user
- friendly and relatively weakly typed front end for the strongly typed
- programming language."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Sutor 88]{Su88} Sutor, Robert S.
-``A guide to programming in the scratchpad 2 interpreter''
-IBM Manual, March 1988
+\bibitem[LeBlanc 91]{LeB91} LeBlanc, S.E.
+``The use of MathCAD and Theorist in the ChE classroom''
+In Anonymous [Ano91], pp287-299 (vol. 1) 2 vols.
keywords = "axiomref",
\end{chunk}
-\subsection{T} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Thompson 00]{Tho00} Thompson, Simon
-``Logic and dependent types in the Aldor Computer Algebra System''
-%\verb|axiom-developer.org/axiom-website/papers/Tho00.pdf|
+\bibitem[Lecerf 96]{Le96} Lecerf, Gr\'egoire
+``Dynamic Evaluation and Real Closure Implementation in Axiom''
+June 29, 1996
+\verb|lecerf.perso.math.cnrs.fr/software/drc/drc.ps|
+%\verb|axiom-developer.org/axiom-website/papers/Le96.ps|
keywords = "axiomref",
- abstract = "
- We show how the Aldor type system can represent propositions of
- first-order logic, by means of the 'propositions as types'
- correspondence. The representation relies on type casts (using
- pretend) but can be viewed as a prototype implementation of a modified
- type system with {\sl type evaluation} reported elsewhere. The logic
- is used to provide an axiomatisation of a number of familiar Aldor
- categories as well as a type of vectors."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Thompson (a)]{TTxx} Thompson, Simon; Timochouk, Leonid
-``The Aldor\-\- language''
-%\verb|axiom-developer.org/axiom-website/papers/TTxx.pdf|
+\bibitem[Lecerf 96a]{Le96a} Lecerf, Gr\'egoire
+``The Dynamic Real Closure implemented in Axiom''
+\verb|lecerf.perso.math.cnrs.fr/software/drc/drc.ps|
keywords = "axiomref",
- abstract = "
- This paper introduces the \verb|Aldor--| language, which is a
- functional programming language with dependent types and a powerful,
- type-based, overloading mechanism. The language is built on a subset
- of Aldor, the 'library compiler' language for the Axiom computer
- algebra system. \verb|Aldor--| is designed with the intention of
- incorporating logical reasoning into computer algebra computations.
-
- The paper contains a formal account of the semantics and type system
- of \verb|Aldor--|; a general discussion of overloading and how the
- overloading in \verb|Aldor--| fits into the general scheme; examples
- of logic within \verb|Aldor--| and notes on the implementation of the
- system."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Touratier 98]{Tou98} Touratier, Emmanuel
-``Etude du typage dans le syst\`eme de calcul scientifique Aldor''
-Universit\'e de Limoges 1998
-%\verb|axiom-developer.org/axiom-website/papers/Tou98.pdf|
+\bibitem[Levelt 95]{Lev95} Levelt, A. H. M. (ed)
+ISSAC '95: Proceedings of the 1995 International
+Symposium on Symbolic and Algebraic Computation: July 10-12, 1995, Montreal,
+Canada ISSAC-PROCEEDINGS-1995. ACM Press, New York, NY 10036, USA, 1995
+ISBN 0-89791-699-9 LCCN QA76.95 I59 1995 ACM order number 505950
keywords = "axiomref",
\end{chunk}
-\subsection{V} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[van der Hoeven 14]{JvdH14} van der Hoeven, Joris
-``Computer algebra systems and TeXmacs''
-\verb|www.texmacs.org/tmweb/plugins/cas.en.html|
+\bibitem[Li 06]{LM06} Li, Xin; Maza, Moreno
+``Efficient Implementation of Polynomial Arithmetic in a Multiple-Level
+Programming Environment''
+Lecture Notes in
+Computer Science Springer Vol 4151/2006 ISBN 978-3-540-38084-9 pp12-23
+Proceedings of International Congress of Mathematical Software ICMS 2006
+\verb|www.csd.uwo.ca/~moreno//Publications/Li-MorenoMaza-ICMS-06.pdf|
keywords = "axiomref",
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Hoei94,
- author = "{van Hoeij}, M.",
- title = "An algorithm for computing an integral basis in an algebraic
- function field",
- journal = "Journal of Symbolic Computation",
- volume = "18",
- number = "4",
- year = "1994",
- pages = "353-363",
- issn = "0747-7171",
+\begin{chunk}{ignore}
+\bibitem[Li 10]{YL10} Li, Yue; Dos Reis, Gabriel
+``A Quantitative Study of Reductions in Algebraic Libraries''
+PASCO 2010
+\verb|www.axiomatics.org/~gdr/concurrency/quant-pasco10.pdf|
keywords = "axiomref",
- paper = "Hoei94.pdf",
- abstract = "
- Algorithms for computing integral bases of an algebraic function field
- are implemented in some computer algebra systems. They are used e.g.
- for the integration of algebraic functions. The method used by Maple
- 5.2 and AXIOM is given by Trager in [Trag84]. He adapted an algorithm
- of Ford and Zassenhaus [Ford, 1978], that computes the ring of
- integers in an algebraic number field, to the case of a function field.
- It turns out that using algebraic geometry one can write a faster
- algorithm. The method we will give is based on Puiseux expansions.
- One cas see this as a variant on the Coates' algorithm as it is
- described in [Davenport, 1981]. Some difficulties in computing with
- Puiseux expansions can be avoided using a sharp bound for the number
- of terms required which will be given in Section 3. In Section 5 we
- derive which denominator is needed in the integral basis. Using this
- result 'intermediate expression swell' can be avoided.
+\end{chunk}
- The Puiseux expansions generally introduce algebraic extensions. These
- extensions will not appear in the resulting integral basis."
-}
+\begin{chunk}{ignore}
+\bibitem[Li 11]{YL11} Li, Yue; Dos Reis, Gabriel
+``An Automatic Parallelization Framework for Algebraic Computation
+Systems''
+ISSAC 2011
+\verb|www.axiomatics.org/~gdr/concurrency/oa-conc-issac11.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/YL11.pdf|
+ keywords = "axiomref",
+ abstract = "
+ This paper proposes a non-intrusive automatic parallelization
+ framework for typeful and property-aware computer algebra systems."
\end{chunk}
-\begin{chunk}{axiom.bib}
-@misc{Hoei08,
- author = "{van Hoeij}, Mark and Novocin, Andrew",
- title = "A Reduction Algorithm for Algebraic Function Fields",
- year = "2008",
- month = "April",
- url = "http://andy.novocin.com/pro/algext.pdf",
- paper = "Hoei08.pdf",
+\begin{chunk}{ignore}
+\bibitem[Ligatsikas 96]{Liga96} Ligatsikas, Zenon; Rioboo, Renaud;
+Roy, Marie Francoise
+``Generic computation of the real closure of an ordered field''
+Math. and Computers in Simulation 42 pp 541-549 (1996)
+%\verb|axiom-developer.org/axiom-website/papers/Liga96.pdf|
+ keywords = "axiomref",
abstract = "
- Computer algebra systesm often produce large expressions involving
- complicated algebraic numbers. In this paper we study variations of
- the {\tt polred} algorithm that can often be used to find better
- representations for algebraic numbers. The main new algorithm
- presented here is an algorithm that treats the same problem for the
- function field case."
-}
+ This paper describes a generalization of the real closure computation
+ of an ordered field (Rioboo, 1991) enabling to use different technques
+ to code a single real algebraic number."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Vasconcelos 99]{Vas99} Vasconcelos, Wolmer
-``Computational Methods in Commutative Algebra and Algebraic Geometry''
-Springer, Algorithms and Computation in Mathematics, Vol 2 1999
-ISBN 3-540-21311-2
+\bibitem[Linton 93]{Lin93} Linton, Steve
+``Vector Enumeration Programs, version 3.04''
+\verb|www.cs.st-andrews.ac.uk/~sal/nme/nme_toc.html#SEC1|
keywords = "axiomref",
\end{chunk}
-\subsection{W} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Wang 89]{Wan89} Wang, D.
-``A program for computing the Liapunov functions and Liapunov
-constants in Scratchpad II''
-SIGSAM Bulletin (ACM Special Interest Group
-on Symbolic and Algebraic Manipulation), 23(4) pp25-31, Oct. 1989,
-CODEN SIGSBZ ISSN 0163-5824
+\bibitem[Liska 97]{LD97} Liska, Richard; Drska, Ladislav; Limpouch, Jiri;
+Sinor, Milan; Wester, Michael; Winkler, Franz
+``Computer Algebra - algorithms, systems and applications''
+June 2, 1997
+\verb|kfe.fjfi.cvut.cz/~liska/ca/all.html|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Wang 91]{Wan91} Wang, Dongming
-``Mechanical manipulation for a class of differential systems''
-Journal of Symbolic Computation, 12(2) pp233-254 Aug. 1991
-CODEN JSYCEH ISSN 0747-7171
+\bibitem[Lucks 86]{Luc86} Lucks, Michael
+``A fast implementation of polynomial factorization''
+In Bruce W. Char, editor, Proceedings of the 1986 Symposium on Symbolic
+and Algebraic Computation: SYMSAC '86, July 21-23, 1986, Waterloo, Ontario,
+pp228-232 ACM Press, New York, NY 10036, USA, 1986. ISBN 0-89791-199-7
+LCCN QA155.7.E4 A281 1986 ACM order number 505860
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Wang 92]{Wan92} Wang, Paul S. (ed)
-International System Symposium on Symbolic and
-Algebraic Computation 92 ACM Press, New York, NY 10036, USA, 1992
-ISBN 0-89791-489-9 (soft cover), 0-89791-490-2 (hard cover),
-LCCN QA76.95.I59 1992
+\bibitem[Lueken 77]{Lue77} Lueken, E.
+``Ueberlegungen zur Implementierung eines Formelmanipulationssystems''
+Master's thesis, Technischen Universit{\"{a}}t Carolo-Wilhelmina zu
+Braunschweig. Braunschweig, Germany, 1977
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Watanabe 90]{WN90} Watanabe, Shunro; Nagata, Morio; (ed)
-ISSAC '90 Proceedings of the
-International Symposium on Symbolic and Algebraic Computation ACM Press,
-New York, NY, 10036, USA. 1990 ISBN 0-89791-401-5 LCCN QA76.95.I57 1990
+\bibitem[Lynch 91]{LM91} Lynch, R.; Mavromatis, H. A.
+``New quantum mechanical perturbation technique
+using an 'electronic scratchpad' on an inexpensive computer''
+American Journal of Pyhsics, 59(3) pp270-273, March 1991.
+CODEN AJPIAS ISSN 0002-9505
keywords = "axiomref",
\end{chunk}
+\subsection{M} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Watt 85]{Wat85} Watt, Stephen
-``Bounded Parallelism in Computer Algebra''
-PhD Thesis, University of Waterloo
-\verb|www.csd.uwo.ca/~watt/pub/reprints/1985-smw-phd.pdf|
+\bibitem[Mahboubi 05]{Mah05} Mahboubi, Assia
+``Programming and certifying the CAD algorithm inside the coq system''
+Mathematics, Algorithms, Proofs, volume 05021 of Dagstuhl
+Seminar Proceedings, Schloss Dagstuhl (2005)
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Watt 86]{Wat86} Watt, S.M.; Della Dora, J.
-``Algebra Snapshot: Linear Ordinary Differential Operators''
-Scratchpad II Newsletter: Vol 1 Num 2 (Jan 1986)
-\verb|www.csd.uwo.ca/~watt/pub/reprints/1986-snews-lodo.pdf|
+\bibitem[Mathews 89]{Mat89} Mathews, J.
+``Symbolic computational algebra applied to Picard iteration''
+Mathematics and computer education, 23(2) pp117-122 Spring 1989 CODEN MCEDDA,
+ISSN 0730-8639
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Watt 87]{Wat87} Watt, Stephen
-``Domains and Subdomains in Scratchpad II''
-in [Wit87], pp3-5
+\bibitem[McJones 11]{McJ11} McJones, Paul
+``Software Presentation Group -- Common Lisp family''
+\verb|www.softwarepreservation.org/projects/LISP/common_lisp_family|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Watt 87a]{WB87} Watt, Stephen M.; Burge, William H.
-``Mapping as First Class Objects''
-in [Wit87], pp13-17
+\bibitem[Melachrinoudis 90]{MR90} Melachrinoudis, E.; Rumpf, D. L.
+``Teaching advantages of transparent computer software -- MathCAD''
+CoED, 10(1) pp71-76, January-March 1990 CODEN CWLJDP ISSN 0736-8607
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Watt 89]{Wat89} Watt, S. M.
-``A fixed point method for power series computation''
-In Gianni [Gia89], pp206-217 ISBN 3-540-51084-2 LCCN QA76.95.I57
-1988 Conference held jointly with AAECC-6
+\bibitem[Miola 90]{Mio90} Miola, A. (ed)
+``Design and Implementation of Symbolic Computation Systems''
+International Symposium DISCO '90, Capri, Italy, April 10-12, 1990, Proceedings
+volume 429 of Lecture Notes in Cmputer Science,
+Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
+1990 ISBN 0-387-52531-9 (New York), 3-540-52531-9 (Berlin) LCCN QA76.9.S88I576
+1990
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Watt 90]{WJST90} Watt, S.M.; Jenks, R.D.; Sutor, R.S.; Trager B.M.
-``The Scratchpad II type system: Domains and subdomains''
-in A.M. Miola, editor Computing Tools
-for Scientific Problem Solving, Academic Press, New York, 1990
+\bibitem[Miola 93]{Mio93} Miola, A. (ed)
+``Design and Implementation of Symbolic Computation Systems''
+International Symposium DISCO '93 Gmunden, Austria, September 15-17, 1993:
+Proceedings.
+Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
+1993 ISBN 3-540-57235-X LCCN QA76.9.S88I576 1993
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Watt 91]{Wat91} Watt, Stephen M. (ed)
-Proceedings of the 1991 International Symposium on
-Symbolic and Algebraic Computation, ISSAC'91, July 15-17, 1991, Bonn, Germany,
-ACM Press, New York, NY 10036, USA, 1991 ISBN 0-89791-437-6
-LCCN QA76.95.I59 1991
+\bibitem[Missura 94]{Miss94} Missura, Stephan A.; Weber, Andreas
+``Using Commutativity Properties for Controlling Coercions''
+\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/|
+\verb|WeberA/MissuraWeber94a.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Miss94.pdf|
keywords = "axiomref",
+ abstract = "
+ This paper investigates some soundness conditions which have to be
+ fulfilled in systems with coercions and generic operators. A result of
+ Reynolds on unrestricted generic operators is extended to generic
+ operators which obey certain constraints. We get natural conditions
+ for such operators, which are expressed within the theoretic framework
+ of category theory. However, in the context of computer algebra, there
+ arise examples of coercions and generic operators which do not fulfil
+ these conditions. We describe a framework -- relaxing the above
+ conditions -- that allows distinguishing between cases of ambiguities
+ which can be resolved in a quite natural sense and those which
+ cannot. An algorithm is presented that detects such unresolvable
+ ambiguities in expressions."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Watt 94a]{Wat94a} Watt, Stephen M.; Dooley, S.S.; Morrison, S.C.;
-Steinback, J.M.; Sutor, R.S.
-``A\# User's Guide''
-Version 1.0.0 O($\epsilon{}^1$) June 8, 1994
+\bibitem[Monagan 87]{Mon87} Monagan, Michael B.
+``Support for Data Structures in Scratchpad II''
+in [Wit87], pp17-18
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Watt 94b]{Wat94} Watt, Stephen M.; Broadbery, Peter A.;
-Dooley, Samuel S.; Iglio, Pietro
-``A First Report on the A\# Compiler (including benchmarks)''
-IBM Research Report RC19529 (85075) May 12, 1994
-%\verb|axiom-developer.org/axiom-website/papers/Wat94.pdf|
+\bibitem[Monagan 93]{Mon93} Monagan, M. B.
+``Gauss: a parameterized domain of computation system with
+support for signature functions''
+In Miola [Mio93], pp81-94 ISBN 3-540-57235-X LCCN QA76.9.S88I576 1993
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Watt 94c]{Wat94c} Watt, Stephen M.
-``A\# Language Reference Version 0.35''
-IBM Research Division Technical Report RC19530 May 1994
+\bibitem[Mora 89]{Mor89} Mora, T. (ed)
+Applied Algebra, Algebraic Algorithms and Error-Correcting
+Codes, 6th International Conference, AAECC-6, Rome, Italy, July 4-8, 1998,
+Proceedings, volume 357 of Lecture Notes in Computer Science
+Springer-Verlag, Berlin, Germany / Heildelberg, Germany / London, UK / etc.,
+1989 ISBN 3-540-51083-4, LCCN QA268.A35 1988 Conference held jointly with
+ISSAC '88
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Watt 95]{Wat95} Watt, S.M.; Broadbery, P.A.; Dooley, S.S.; Iglio, P.
-Steinbach, J.M.; Morrison, S.C.; Sutor, R.S.
-``AXIOM Library Compiler Users Guide''
-The Numerical Algorithms Group (NAG) Ltd, 1994
+\bibitem[Moses 71]{Mos71} Moses, Joel
+``Algebraic Simplification: A Guide for the Perplexed''
+CACM August 1971 Vol 14 No. 8 pp527-537
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Watt 01]{Wat01} Watt, Stephen M.; Broadbery, Peter A.; Iglio, Pietro;
-Morrison, Scott C.; Steinbach, Jonathan M.
-``FOAM: A First Order Abstract Machine Version 0.35''
-IBM T. J. Watson Research Center (2001)
-%\verb|axiom-developer.org/axiom-website/papers/Wat01.pdf|
+\bibitem[Moses 08]{Mos08} Moses, Joel
+``Macsyma: A Personal History''
+Invited Presentation in Milestones in Computer Algebra, May 2008, Tobago
+\verb|esd.mit.edu/Faculty_Pages/moses/Macsyma.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Mos08.pdf|
keywords = "axiomref",
+ abstract = "
+ The Macsyma system arose out of research on mathematical software in
+ the AI group at MIT in the 1960's. Algorithm development in symbolic
+ integration and simplification arose out of the interest of people,
+ such as the author, who were also mathematics students. The later
+ development of algorithms for the GCD of sparse polynomials, for
+ example, arose out of the needs of our user community. During various
+ times in the 1970's the computer on which Macsyma ran was one of the
+ most popular notes on the ARPANET. We discuss the attempts in the late
+ 70's and the 80's to develop Macsyma systems that ran on popular
+ computer architectures. Finally, we discuss the impact of the
+ fundamental ideas in Macsyma on current research on large scale
+ engineering systems."
\end{chunk}
+\subsection{N} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Weber 92]{Webe92} Weber, Andreas
-``Type Systems for Computer Algebra''
-\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber92a.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Webe92.pdf|
+\bibitem[Naylor]{NPxx} Naylor, William; Padget, Julian
+``From Untyped to Polymorphically Typed Objects in Mathematical Web
+Services''
+%\verb|axiom-developer.org/axiom-website/papers/NPxx.pdf|
keywords = "axiomref",
abstract = "
- An important feature of modern computer algebra systems is the support
- of a rich type system with the possibility of type inference. Basic
- features of such a type system are polymorphism and coercion between
- types. Recently the use of order-sorted rewrite systems was proposed
- as a general framework. We will give a quite simple example of a
- family of types arising in computer algebra whose coercion relations
- cannot be captured by a finite set of first-order rewrite rules."
+ OpenMath is a widely recognized approach to the semantic markup of
+ mathematics that is often used for communication between OpenMath
+ compliant systems. The Aldor language has a sophisticated
+ category-based type system that was specifically developed for the
+ purpose of modelling mathematical structures, while the system itself
+ supports the creation of small-footprint applications suitable for
+ deployment as web services. In this paper we present our first results
+ of how one may perform translations from generic OpenMath objects into
+ values in specific Aldor domains, describing how the Aldor interfae
+ domain ExpresstionTree is used to achieve this. We outline our Aldor
+ implementation of an OpenMath translator, and describe an efficient
+ extention of this to the Parser category. In addition, the Aldor
+ service creation and invocation mechanism are explained. Thus we are
+ in a position to develop and deploy mathematical web services whose
+ descriptions may be directly derived from Aldor's rich type language."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Weber 92b]{Webe92b} Weber, Andreas
-``Structuring the Type System of a Computer Algebra System''
-\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber92a.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Webe92b.pdf|
+\bibitem[Naylor 95]{N95} Naylor, Bill
+``Symbolic Interface for an advanced hyperbolic PDE solver''
+\verb|www.sci.csd.uwo.ca/~bill/Papers/symbInterface2.ps|
+%\verb|axiom-developer.org/axiom-website/papers/N95.pdf|
keywords = "axiomref",
abstract = "
- Most existing computer algebra systems are pure symbol manipulating
- systems without language support for the occuring types. This is
- mainly due to the fact taht the occurring types are much more
- complicated than in traditional programming languages. In the last
- decade the study of type systems has become an active area of
- research. We will give a proposal for a type system showing that
- several problems for a type system of a symbolic computation system
- can be solved by using results of this research. We will also provide
- a variety of examples which will show some of the problems that remain
- and that will require further research."
+ An Axiom front end is described, which is used to generate
+ mathematical objects needed by one of the latest NAG routines, to be
+ included in the Mark 17 version of the NAG Numerical library. This
+ routine uses powerful techniques to find the solution to Hyperbolic
+ Partial Differential Equations in conservation form and in one spatial
+ dimension. These mathematical objects are non-trivial, requiring much
+ mathematical knowledge on the part of the user, which is otherwise
+ irrelvant to the physical problem which is to be solved. We discuss
+ the individual mathematical objects, considering the mathematical
+ theory which is relevant, and some of the problems which have been
+ encountered and solved during the FORTRAN generation necessary to
+ realise the object. Finally we display some of our results."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Weber 93b]{Webe93b} Weber, Andreas
-``Type Systems for Computer Algebra''
-\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber93b.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Webe93b.pdf|
+\bibitem[Naylor 00b]{ND00} Naylor, W.A.; Davenport, J.H.
+``A Monte-Carlo Extension to a Category-Based Type System''
+\verb|www.sci.csd.uwo.ca/~bill/Papers/monteCarCat3.ps|
+%\verb|axiom-developer.org/axiom-website/papers/ND00.pdf|
keywords = "axiomref",
abstract = "
- We study type systems for computer algebra systems, which frequently
- correspond to the ``pragmatically developed'' typing constructs used
- in AXIOM. A central concept is that of {\sl type classes} which
- correspond to AXIOM categories. We will show that types can be
- syntactically described as terms of a regular order-sorted signature
- if no type parameters are allowed. Using results obtained for the
- functional programming language Haskell we will show that the problem
- of {\sl type inference} is decidable. This result still holds if
- higher-order functions are present and {\sl parametric polymorphism}
- is used. These additional typing constructs are useful for further
- extensions of existing computer algebra systems: These typing concepts
- can be used to implement category theoretic constructs and there are
- many well known constructive interactions between category theory and
- algebra."
+ The normal claim for mathematics is that all calculations are 100\%
+ accurate and therefore one calculation can rely completely on the
+ results of sub-calculations, hoever there exist {\sl Monte-Carlo}
+ algorithms which are often much faster than the equivalent
+ deterministic ones where the results will have a prescribed
+ probability (presumably small) of being incorrect. However there has
+ been little discussion of how such algorithms can be used as building
+ blocks in Computer Algebra. In this paper we describe how the
+ computational category theory which is the basis of the type structure
+ used in the Axiom computer algebra system may be extended to cover
+ probabilistic algorithms, which use Monte-Carlo techniques. We follow
+ this with a specific example which uses Straight Line Program
+ representation."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Weber 94]{Web94} Weber, Andreas
-``Algorithms for Type Inference with Coercions''
-ISSAC 94 ACM 0-89791-638-7/94/0007
-%\verb|axiom-developer.org/axiom-website/papers/Web94.pdf|
+\bibitem[Norman 75]{Nor75} Norman, A. C.
+``Computing with formal power series''
+ACM Transactions on Mathematical Software, 1(4) pp346-356
+Dec. 1975 CODEN ACMSCU ISSN 0098-3500
keywords = "axiomref",
- abstract = "
- This paper presents algorithms that perform a type inference for a
- type system occurring in the context of computer algebra. The type
- system permits various classes of coercions between types and the
- algorithms are complete for the precisely defined system, which can be
- seen as a formal description of an important subset of the type system
- supported by the computer algebra program Axiom.
-
- Previously only algorithms for much more restricted cases of coercions
- have been described or the frameworks used have been so general that
- the corresponding type inference problems were known to be
- undecidable."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Weber 95]{Webe95} Weber, A.
-``On coherence in computer algebra''
-\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber94e.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Webe95.pdf|
+\bibitem[Norman 75a]{Nor75a} Norman, A.C.
+``The SCRATCHPAD Power Series Package''
+IBM T.J. Watson Research RC4998
keywords = "axiomref",
- abstract = "
- Modern computer algebra systems (e.g. AXIOM) support a rich type
- system including parameterized data types and the possibility of
- implicit coercions between types. In such a type system it will be
- frequently the case that there are different ways of building
- coercions between types. An important requirement is that all
- coercions between two types coincide, a property which is called {\sl
- coherence}. We will prove a coherence theorem for a formal type system
- having several possibilities of coercions covering many important
- examples. Moreover, we will give some informal reasoning why the
- formally defined restrictions can be satisfied by an actual system."
\end{chunk}
+\subsection{O} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Weber 96]{Webe96} Weber, Andreas
-``Computing Radical Expressions for Roots of Unity''
-\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber96a.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Webe96.pdf|
+\bibitem[Ollivier 89]{Oll89} Ollivier, F.
+``Inversibility of rational mappings and structural
+identifiablility in automatics''
+In ACM [ACM89], pp43-54 ISBN 0-89791-325-6 LCCN QA76.95.I59 1989
keywords = "axiomref",
- abstract = "
- We present an improvement of an algorithm given by Gauss to compute a
- radical expression for a $p$-th root of unity. The time complexity of
- the algorithm is $O(p^3m^6log p)$, where $m$ is the largest prime
- factor of $p-1$."
\end{chunk}
-
\begin{chunk}{ignore}
-\bibitem[Weber 99]{Webe99} Weber, Andreas
-``Solving Cyclotomic Polynomials by Radical Expressions''
-\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/|
-\verb|WeberA/WeberKeckeisen99a.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Webe99.pdf|
+\bibitem[Online 72]{Onl72}.
+Online 72: conference proceedings ... international conference on online
+interactive computing, Brunel University, Uxbridge, England, 4-7 September
+1972 ISBN 0-903796-02-3 LCCN QA76.55.O54 1972 Two volumes.
keywords = "axiomref",
- abstract = "
- We describe a Maple package that allows the solution of cyclotomic
- polynomials by radical expressions. We provide a function that is an
- extension of the Maple {\sl solve} command. The major algorithmic
- ingredient of the package is an improvement of a method due to Gauss
- which gives radical expressions for roots of unity. We will give a
- summary for computations up to degree 100, which could be done within
- a few hours of cpu time on a standard workstation."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Wei-Jiang 12]{WJ12} Wei-Jiang
-``Top free algebra System''
-\verb|wei-jiang.com/it/software/top-free-algebra-system-bye-mathematica-bye-maple|
+\bibitem[OpenMath]{OpenMa}.
+``OpenMath Technical Overview''
+\verb|www.openmath.org/overview/technical.html|
keywords = "axiomref",
\end{chunk}
+\subsection{P} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Wester 99]{Wes99} Wester, Michael J.
-``Computer Algebra Systems''
-John Wiley and Sons 1999 ISBN 0-471-98353-5
+\bibitem[Page 07]{Pa07} Page, William S.
+``Axiom - Open Source Computer Algebra System''
+Poster ISSAC 2007 Proceedings Vol 41 No 3 Sept 2007 p114
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Wexelblat 87]{Wex87} Wexelblat, Richard L. (ed)
-Proceedings of the SIGPLAN '87 Symposium on
-Interpreter and Interpretive Techniques, St. Paul, Minnesota, June 24-26, 1987
-ACM Press, New York, NY 10036, USA, 1987 ISBN 0-89791-235-7
-LCCN QA76.7.S54 v22:7 SIGPLAN Notices, vol 22, no 7 (July 1987)
+\bibitem[Petitot 90]{Pet90} Petitot, Michel
+``Types r\'ecursifs en scratchpad, application aux polyn\^omes non
+commutatifs''
+LIFL, 1990
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Wityak 87]{Wit87} Wityak, Sandra
-``Scratchpad II Newsletter''
-Volume 2, Number 1, Nov 1987
+\bibitem[Petitot 93]{Pet93} Petitot, M.
+``Experience with Axiom''
+In Jacob et al. [JOS93], page 240
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[WWW1]{WWW1}.
-Software Preservation Group
-\verb|www.softwarepresentation.org/projects/LISP/common_lisp_family|
+\bibitem[Petric 71]{Pet71} Petric, S. R. (ed)
+Proceedings of the second symposium on Symbolic and
+Algebraic Manipulation, March 23-25, 1971, Los Angeles, California, ACM Press,
+New York, NY 10036, USA, 1971. LCCN QA76.5.S94 1971
keywords = "axiomref",
\end{chunk}
-\subsection{Y} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Yap 00]{Yap00} Yap, Chee Keng
-``Fundamental Problems of Algorithmic Algebra''
-Oxford University Press (2000) ISBN0-19-512516-9
+\bibitem[Pinch 93]{Pin93} Pinch, R.G.E.
+``Some Primality Testing Algorithms''
+Devlin, Keith (ed.)
+Computers and Mathematics November 1993, Vol 40, Number 9 pp1203-1210
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Yapp 07]{Yapp07} Yapp, Clifford; Hebisch, Waldek; Kaminski, Kai
-``Literate Programming Tools Implemented in ANSI Common Lisp''
-\verb|brlcad.org/~starseeker/cl-web-v0.8.lisp.pamphlet|
+\bibitem[Poll (b)]{Polxx} Poll, Erik
+``The type system of Axiom''
+%\verb|axiom-developer.org/axiom-website/papers/Polxx.pdf|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Yun 83]{Yun83} Yun, David Y.Y.
-``Computer Algebra and Complex Analysis''
-Computational Aspects of Complex Analysis pp379-393
-D. Reidel Publishing Company H. Werner et. al. (eds.)
+\bibitem[Purtilo 86]{Pur86} Purtilo, J.
+``Applications of a software interconnection system in mathematical
+problem solving environments'' In Bruce W. Char, editor. Proceedings of the
+1986 Symposium on Symbolic and Algebraic Computation: SYMSAC '86, July 21-23,
+ACM Press, New York, NY 10036, USA, 1986. ISBN 0-89791-199-7 LCCN QA155.7.E4
+A281 1986 ACM order number 505860
keywords = "axiomref",
\end{chunk}
-\subsection{Z} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{R} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
-\bibitem[Zen92]{Zen92} Zenger, Ch.
-``Gr{\"o}bnerbasen f{\"u}r Differentialformen und ihre
-Implementierung in AXIOM''
-Diplomarbeit, Universit{\"a}t Karlsruhe,
-Karlsruhe, Germany, 1992
+\bibitem[Rainer 14]{Rain14} Joswig, Rainer
+``2014: 30+ Years Common Lisp the Language''
+\verb|lispm.de/30ycltl|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Zip92]{Zip92} Zippel, Richard
-``Algebraic Computation''
-(unpublished) Cornell University Ithaca, NY Sept 1992
+\bibitem[Rioboo 03a]{Riob03a} Rioboo, Renaud
+``Quelques aspects du calcul exact avec des nombres r\'eels''
+Ph.D. Thesis, Laboratoire d'Informatique Th\'eorique et Programmationg
+%\verb|axiom-developer.org/axiom-website/papers/Riob03a.ps|
keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Zwi92]{Zwi92} Zwillinger, Daniel
-``Handbook of Integration''
-Jones and Bartlett, 1992, ISBN 0-86720-293-9
+\bibitem[Rioboo 03]{Riob03} Rioboo, Renaud
+``Towards Faster Real Algebraic Numbers''
+J. of Symbolic Computation 36 pp 513-533 (2003)
+%\verb|axiom-developer.org/axiom-website/papers/Riob03.pdf|
keywords = "axiomref",
+ abstract = "
+ This paper presents a new encoding scheme for real algebraic number
+ manipulations which enhances current Axiom's real closure. Algebraic
+ manipulations are performed using different instantiations of
+ sub-resultant-like algorithms instead of Euclidean-like algorithms.
+ We use these algorithms to compute polynomial gcds and Bezout
+ relations, to compute the roots and the signs of algebraic
+ numbers. This allows us to work in the ring of real algebraic integers
+ instead of the field of read algebraic numbers avoiding many
+ denominators."
\end{chunk}
-\section{Axiom Citations of External Sources}
-\subsection{A} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{chunk}{ignore}
+\bibitem[Robidoux 93]{Rob93} Robidoux, Nicolas
+``Does Axiom Solve Systems of O.D.E's Like Mathematica?''
+July 1993
+%\verb|axiom-developer.org/axiom-website/papers/Rob93.pdf|
+ keywords = "axiomref",
+ abstract = "
+ If I were demonstrating Axiom and were asked this question, my reply
+ would be ``No, but I am not sure that this is a bad thing''. And I
+ would illustrate this with the following example.
-\begin{chunk}{axiom.bib}
-@article{Abla98,
- author = "Ablamowicz, Rafal",
- title = "Spinor Representations of Clifford Algebras: A Symbolic Approach",
- journal = "Computer Physics Communications",
- volume = "115",
- number = "2-3",
- month = "December",
- year = "1998",
- pages = "510-535"
-}
+ Consider the following system of O.D.E.'s
+ \[
+ \begin{array}{rcl}
+ \frac{dx_1}{dt} & = & \left(1+\frac{cos t}{2+sin t}\right)x_1\\
+ \frac{dx_2}{dt} & = & x_1 - x_2
+ \end{array}
+ \]
+ This is a very simple system: $x_1$ is actually uncoupled from $x_2$"
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Abra06,
- author = "Abramov, Sergey A.",
- title = "In Memory of Manuel Bronstein",
- journal = "Programming and Computer Software",
- volume = "32",
- number = "1",
- pages = "56-58",
- publisher = "Pleiades Publishing Inc",
- year = "2006",
- paper = "Abra06.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Rioboo 92]{Rio92} Rioboo, R.
+``Real algebraic closure of an ordered field, implementation in Axiom''
+In Wang [Wan92], pp206-215 ISBN 0-89791-489-9 (soft cover)
+0-89791-490-2 (hard cover) LCCN QA76.95.I59 1992
+%\verb|axiom-developer.org/axiom-website/papers/Rio92.pdf|
+ keywords = "axiomref",
+ abstract = "
+ Real algebraic numbers appear in many Computer Algebra problems. For
+ instance the determination of a cylindrical algebraic decomposition
+ for an euclidean space requires computing with real algebraic numbers.
+ This paper describes an implementation for computations with the real
+ roots of a polynomial. This process is designed to be recursively
+ used, so the resulting domain of computation is the set of all real
+ algebraic numbers. An implementation for the real algebraic closure
+ has been done in Axiom (previously called Scratchpad)."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Abramowitz 64]{AS64} Abramowitz, Milton; Stegun, Irene A.
-``Handbook of Mathematical Functions''
-(1964) Dover Publications, NY ISBN 0-486-61272-4
+\bibitem[Roesner 95]{Roe95} Roesner, K. G.
+``Verified solutions for parameters of an exact solution for
+non-Newtonian liquids using computer algebra'' Zeitschrift fur Angewandte
+Mathematik und Physik, 75 (suppl. 2):S435-S438, 1995 ISSN 0044-2267
+ keywords = "axiomref",
\end{chunk}
+\subsection{S} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Abramowitz 68]{AS68} Abramowitz M; Stegun I A
-``Handbook of Mathematical Functions''
-Dover Publications. (1968)
+\bibitem[Sage 14]{Sage14} Stein, William
+``Sage''
+\verb|www.sagemath.org/doc/reference/interfaces/sage/interfaces/axiom.html|
+ keywords = "axiomref",
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Salvy 89]{Sal89} Salvy, B.
+``Examples of automatic asymptotic expansions''
+Technical Report 114,
+Inst. Nat. Recherche Inf. Autom., Le Chesnay, France, Dec. 1989 18pp
+ keywords = "axiomref",
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Salvy 91]{Sal91} Salvy, B.
+``Examples of automatic asymptotic expansions''
+SIGSAM Bulletin (ACM Special Interest Group on Symbolic and
+Algebraic Manipulation), 25(2) pp4-17
+April 1991 CODEN SIGSBZ ISSN 0163-5824
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
-@book{Altm05,
- author = "Altmann, Simon L.",
- title = "Rotations, Quaternions, and Double Groups",
- publisher = "Dover Publications, Inc.",
- year = "2005",
- isbn = "0-486-44518-6"
+@article{Saun80,
+ author = "Saunders, B. David",
+ title = "A Survey of Available Systems",
+ journal = "SIGSAM Bull.",
+ issue_date = "November 1980",
+ volume = "14",
+ number = "4",
+ month = "November",
+ year = "1980",
+ issn = "0163-5824",
+ pages = "12--28",
+ numpages = "17",
+ url = "http://doi.acm.org/10.1145/1089235.1089237",
+ doi = "10.1145/1089235.1089237",
+ acmid = "1089237",
+ publisher = "ACM",
+ address = "New York, NY, USA",
+ keywords = "axiomref,survey",
+ paper = "Saun80.pdf"
}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Ames 77]{Ames77} Ames W F
-``Nonlinear Partial Differential Equations in Engineering''
-Academic Press (2nd Edition). (1977)
+\bibitem[Schu 92]{Sch92} Sch\"u, J.
+``Implementing des Cartan-Kuranishi-Theorems in AXIOM''
+Master's diploma thesis (in german), Institut f\"ur Algorithmen und
+Kognitive Systeme, Universit\"t Karlsruhe 1992
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Amos 86]{Amos86} Amos D E
-``Algorithm 644: A Portable Package for Bessel Functions of a Complex
-Argument and Nonnegative Order''
-ACM Trans. Math. Softw. 12 265--273. (1986)
+\bibitem[Schwarz 88]{Sch88} Schwarz, F.
+``Programming with abstract data types: the symmetry package SPDE
+in Scratchpad''
+In Jan{\ss}en [Jan88], pp167-176, ISBN 3-540-18928-9,
+0-387-18928-9 LCCN QA155.7.E4T74 1988
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Anderson 00]{And00} Anderson, Edward
-``Discontinuous Plane Rotations and the Symmetric Eigenvalue Problem''
-LAPACK Working Note 150, University of Tennessee, UT-CS-00-454,
-December 4, 2000.
+\bibitem[Schwarz 89]{Sch89} Schwarz, F.
+``A factorization algorithm for linear ordinary differential equations''
+In ACM [ACM89], pp17-25 ISBN 0-89791-325-6 LCCN QA76.95.I59 1989
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Anthony 82]{ACH82} Anthony G T; Cox M G; Hayes J G
-``DASL - Data Approximation Subroutine Library''
-National Physical Laboratory. (1982)
+\bibitem[Schwarz 91]{Sch91} Schwarz, F.
+``Monomial orderings and Gr{\"o}bner bases''
+SIGSAM Bulletin (ACM Special Interest Group on Symbolic and Algebraic
+Manipulation) 2591) pp10-23 Jan. 1991 CODEN SIGSBZ ISSN 0163-5824
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Arnon 88]{Arno88} Arno, D.S.; MIgnotte, M.
-``On Mechanical Quantifier Elimination for Elementary Algebra and Geometry''
-J. Symbolic Computation 5, 237-259 (1988)
-\verb|http://www.sciencedirect.com/science/article/pii/S0747717188800142/|
-\verb|pdf?md5=62052077d84e6078cc024bc8e29c23c1&|
-\verb|pid=1-s2.0-S0747717188800142-main.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Arno88.pdf|
+\bibitem[Seiler 94]{Sei94} Seiler, Werner Markus
+``Analysis and Application of the Formal Theory of Partial Differential
+Equations''
+PhD thesis, School of Physics and Materials, Lancaster University (1994)
+\verb|www.mathematik.uni-kassel.de/~seiler/Papers/Diss/diss.ps.gz|
+%\verb|axiom-developer.org/axiom-website/papers/Sei94.pdf|
+ keywords = "axiomref",
abstract = "
- We give solutions to two problems of elementary algebra and geometry:
- (1) find conditions on real numbers $p$, $q$, and $r$ so that the
- polynomial function $f(x)=x^4+px^2+qx+r$ is nonnegative for all real
- $x$ and (2) find conditions on real numbers $a$, $b$, and $c$ so that
- the ellipse $\frac{(x-e)^2}{q^2}+\frac{y^2}{b^2}-1=0$ lies inside the
- unit circle $y^2+x^2-1=0$. Our solutions are obtained by following the
- basic outline of the method of quantifier elimination by cylindrical
- algebraic decomposition (Collins, 1975), but we have developed, and
- have been considerably aided by, modified versions of certain of its
- steps. We have found three equally simple but not obviously equivalent
- solutions for the first problem, illustrating the difficulty of
- obtaining unique ``simplest'' solutions to quantifier elimination
- problems of elementary algebra and geometry."
+ An introduction to the formal theory of partial differential equations
+ is given emphasizing the properties of involutive symbols and
+ equations. An algorithm to complete any differential equation to an
+ involutive one is presented. For an involutive equation possible
+ values for the number of arbitrary functions in its general solution
+ are determined. The existence and uniqueness of solutions for analytic
+ equations is proven. Applications of these results include an
+ analysis of symmetry and reduction methods and a study of gauge
+ systems. It is show that the Dirac algorithm for systems with
+ constraints is closely related to the completion of the equation of
+ motion to an involutive equation. Specific examples treated comprise
+ the Yang-Mills Equations, Einstein Equations, complete and Jacobian
+ systems, and some special models in two and three dimensions. To
+ facilitate the involved tedious computations an environment for
+ geometric approaches to differential equations has been developed in
+ the computer algebra system Axiom. The appendices contain among others
+ brief introductions into Carten-K{\"a}hler Theory and Janet-Riquier
+ Theory."
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Aubr99,
- author = "Aubry, Phillippe and Lazard, Daniel and {Moreno Maza}, Marc",
- title = "On the Theories of Triangular Sets",
- year = "1999",
- pages = "105-124",
- journal = "Journal of Symbolic Computation",
- volume = "28",
- url = "http://www.csd.uwo.ca/~moreno/Publications/Aubry-Lazard-MorenoMaza-1999-JSC.pdf",
- papers = "Aubr99.pdf",
- abstract = "
- Different notions of triangular sets are presented. The relationship
- between these notions are studied. The main result is that four
- different existing notions of {\sl good} triangular sets are
- equivalent."
-}
+\begin{chunk}{ignore}
+\bibitem[Seiler 94a]{Sei94a} Seiler, W.M.
+``Completion to involution in AXIOM''
+in Calmet [Cal94] pp103-104
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Aubry 96]{Aub96} Aubry, Philippe; Maza, Marc Moreno
-``Triangular Sets for Solving Polynomial Systems: a Comparison of Four Methods''
-\verb|www.lip6.fr/lip6/reports/1997/lip6.1997.009.ps.gz|
-%\verb|axiom-developer.org/axiom-website/papers/Aub96.ps|
+\bibitem[Sieler 94b]{Sei94b} Seiler, W.M.
+``Pseudo differential operators and integrable systems in AXIOM''
+Computer Physics Communications, 79(2) pp329-340 April 1994 CODEN CPHCBZ
+ISSN 0010-4655
+%\verb|axiom-developer.org/axiom-website/papers/Sei94b.pdf|
+ keywords = "axiomref",
abstract = "
- Four methods for solving polynomial systems by means of triangular
- sets are presented and implemented in a unified way. These methods are
- those of Wu, Lazard, Kalkbrener, and Wang. They are compared on
- various examples with emphasis on efficiency, conciseness and
- legibility of the outputs."
+ An implementation of the algebra of pseudo differential operators in
+ the computer algebra system Axiom is described. In several exmaples
+ the application of the package to typical computations in the theory
+ of integrable systems is demonstrated."
\end{chunk}
-\subsection{B} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Bailey 66]{Bai66} Bailey P B
-``Sturm-Liouville Eigenvalues via a Phase Function''
-SIAM J. Appl. Math . 14 242--249. (1966)
+\bibitem[Seiler 95]{Sei95} Seiler, W.M.
+``Applying AXIOM to partial differential equations''
+Internal Report 95-17, Universit\"at Karlsruhe, Fakult\"at f\"ur Informatik
+1995
+%\verb|axiom-developer.org/axiom-website/papers/Sei95.pdf|
+ keywords = "axiomref",
+ abstract = "
+ We present an Axiom environment called JET for geometric computations
+ with partial differential equations within the framework of the jet
+ bundle formalism. This comprises expecially the completion of a given
+ differential equation to an involutive one according to the
+ Cartan-Kuranishi Theorem and the setting up of the determining system
+ for the generators of classical and non-classical Lie
+ symmetries. Details of the implementations are described and
+ applications are given. An appendix contains tables of all exported
+ functions."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Baker 96]{BGM96} Baker, George A.; Graves-Morris, Peter
-``Pade Approximants''
-Cambridge University Press, March 1996 ISBN 9870521450072
+\bibitem[Seiler 95b]{SC95} Seiler, W.M.; Calmet, J.
+``JET -- An Axiom Environment for Geometric Computations with Differential
+Equations''
+%\verb|axiom-developer.org/axiom-website/papers/SC95.pdf|
+ keywords = "axiomref",
+ abstract = "
+ JET is an environment within the computer algebra system Axiom to
+ perform such computations. The current implementation emphasises the
+ two key concepts involution and symmetry. It provides some packages
+ for the completion of a given system of differential equations to an
+ equivalent involutive one based on the Cartan-Kuranishi theorem and
+ for setting up the determining equations for classical and
+ non-classical point symmetries."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Baker 10]{Ba10} Baker, Martin
-``3D World Simulation''
-\verb|www.euclideanspace.com|
-
-\end{chunk}
-
-\begin{chunk}{axiom.bib}
-@misc{Bake14,
- author = "Baker, Martin",
- title = "Axiom Architecture",
- year = "2014",
- url = "http://www.euclideanspace.com/prog/scratchpad/internals/ccode"
-}
+\bibitem[Seiler 97]{Sei97} Seiler, Werner M.
+``Computer Algebra and Differential Equations: An Overview''
+\verb|www.mathematik.uni-kassel.di/~seiler/Papers/Postscript/CADERep.ps.gz|
+ keywords = "axiomref",
+ abstract = "
+ We present an informal overview of a number of approaches to
+ differential equations which are popular in computer algebra. This
+ includes symmetry and completion theory, local analysis, differential
+ ideal and Galois theory, dynamical systems and numerical analysis. A
+ large bibliography is provided."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Banks 68]{BK68} Banks D O; Kurowski I
-``Computation of Eigenvalues of Singular Sturm-Liouville Systems''
-Math. Computing. 22 304--310. (1968)
+\bibitem[Seiler (a)]{Seixx} Seiler, W.M.
+``DETools: A Library for Differential Equations''
+\verb|iaks-www.ira.uka.de/iaks-calmet/werner/werner.html|
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Bard 74]{Bard74} Bard Y
-``Nonlinear Parameter Estimation''
-Academic Press. 1974
+\bibitem[Shannon 88]{SS88} Shannon, D.; Sweedler, M.
+``Using Gr{\"o}bner bases to determine algebra
+membership, split surjective algebra homomorphisms determine birational
+equivalence''
+Journal of Symbolic Computation 6(2-3) pp267-273
+Oct.-Dec. 1988 CODEN JSYCEH ISSN 0747-7171
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Barrodale 73]{BR73} Barrodale I; Roberts F D K
-``An Improved Algorithm for Discrete $ll_1$ Linear Approximation''
-SIAM J. Numer. Anal. 10 839--848. (1973)
+\bibitem[Sit 89]{Sit89} Sit, W.Y.
+``On Goldman's algorithm for solving first-order multinomial
+autonomous systems'' In Mora [Mor89], pp386-395 ISBN 3-540-51083-4
+LCCN QA268.A35 1998 Conference held jointly with ISSAC '88
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Barrodale 74]{BR74} Barrodale I; Roberts F D K
-``Solution of an Overdetermined System of Equations in the $ll_1-norm$.''
-Comm. ACM. 17, 6 319--320. (1974)
+\bibitem[Sit 92]{Sit92} Sit, W.Y.
+``An algorithm for solving parametric linear systems''
+Journal of Symbolic Computations, 13(4) pp353-394, April 1992 CODEN JSYCEH
+ISSN 0747-7171
+\verb|www.sciencedirect.com/science/article/pii/S0747717108801046/pdf|
+\verb|?md5=00aa65e18e6ea5c4a008c8dfdfcd4b83&|
+\verb|pid=1-s2.0-S0747717108801046-main.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Sit92.pdf|
+ keywords = "axiomref",
+ abstract = "
+ We present a theoretical foundation for studying parametric systesm of
+ linear equations and prove an efficient algorithm for identifying all
+ parametric values (including degnerate cases) for which the system is
+ consistent. The algorithm gives a small set of regimes where for each
+ regime, the solutions of the specialized systems may be given
+ uniformly. For homogeneous linear systems, or for systems were the
+ right hand side is arbitrary, this small set is irredunant. We discuss
+ in detail practical issues concerning implementations, with particular
+ emphasis on simplification of results. Examples are given based on a
+ close implementation of the algorithm in SCRATCHPAD II. We also give a
+ complexity analysis of the Gaussian elimination method and compare
+ that with our algorithm."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Beauzamy 92]{Bea92} Beauzamy, Bernard
-``Products of polynomials and a priori estimates for
-coefficients in polynomial decompositions: a sharp result''
-J. Symbolic Computation (1992) 13, 463-472
-%\verb|axiom-developer.org/axiom-website/papers/Bea92.pdf|
+\bibitem[Sit 06]{Sit06} Sit, Emil
+``Tools for Repeatable Research''
+\verb|www.emilsit.net/blog/archives/tools-for-repeatable-research|
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Beauzamy 93]{Bea93} Beauzamy, Bernard; Trevisan, Vilmar;
-Wang, Paul S.
-``Polynomial Factorization: Sharp Bounds, Efficient Algorithms''
-J. Symbolic Computation (1993) 15, 393-413
-%\verb|axiom-developer.org/axiom-website/papers/Bea93.pdf|
+\bibitem[Smedley 92]{Sme92} Smedley, Trevor J.
+``Using pictorial and object oriented programming for computer algebra''
+In Hal Berghel et al., editors. Applied computing --
+technologicial challenges of the 199s: proceedings of the 1992 ACM/SIGAPP
+Symposium on Applied Computing, Kansas City Convention Center, March 1-3, 1992
+pp1243-1247. ACM Press, New York, NY 10036, USA, 1992. ISBN 0-89791-502-X
+LCCN QA76.76.A65 S95 1992
+ keywords = "axiomref",
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Bert95,
- author = "Bertrand, Laurent",
- title = "Computing a hyperelliptic integral using arithmetic in the
- jacobian of the curve",
- journal = "Applicable Algebra in Engineering, Communication and Computing",
- volume = "6",
- pages = "275-298",
- year = "1995",
+\begin{chunk}{ignore}
+\bibitem[Smith 07]{SDJ07} Smith, Jacob; Dos Reis, Gabriel; Jarvi, Jaakko
+``Algorithmic differentiation in Axiom''
+ACM SIGSAM ISSAC Proceedings 2007 Waterloo, Canada 2007 pp347-354
+ISBN 978-1-59593-743-8
+%\verb|axiom-developer.org/axiom-website/papers/SDJ07.pdf|
+ keywords = "axiomref",
abstract = "
- In this paper, we describe an efficient algorithm for computing an
- elementary antiderivative of an algebraic function defined on a
- hyperelliptic curve. Our algorithm combines B.M. Trager's integration
- algorithm and a technique for computing in the Jacobian of a
- hyperelliptic curve introduced by D.G. Cantor. Our method has been
- implemented and successfully compared to Trager's general algorithm."
-}
+ This paper describes the design and implementation of an algorithmic
+ differentiation framework in the Axiom computer algebra system. Our
+ implementation works by transformations on Spad programs at the level
+ of the typed abstract syntax tree."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Berzins 87]{BBG87} Berzins M; Brankin R W; Gladwell I.
-``Design of the Stiff Integrators in the NAG Library''
-Technical Report. TR14/87 NAG. (1987)
+\bibitem[SSC92]{SSC92}.
+``Algorithmic Methods For Lie Pseudogroups''
+In N. Ibragimov, M. Torrisi and A. Valenti, editors, Proc. Modern Group
+Analysis: Advanced Analytical and Computational Methods in Mathematical
+Physics, pp337-344, Acireale (Italy), 1992 Kluwer, Dordrecht 1993
+\verb|iaks-www.ira.uka.de/iaks-calmet/werner/Papers/Acireale92.ps.gz|
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Berzins 90]{Ber90} Berzins M
-``Developments in the NAG Library Software for Parabolic Equations''
-Scientific Software Systems. (ed J C Mason and M G Cox)
-Chapman and Hall. 59--72. (1990)
+\bibitem[SSV87]{SSV87} Senechaud, P.; Siebert, F.; Villard G.
+``Scratchpad II: Pr{\'e}sentation d'un nouveau langage de calcul formel''
+Technical Report 640-M, TIM 3 (IMAG), Grenoble, France, Feb 1987
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Birkhoff 62]{BR62} Birkhoff, G; Rota, G C
-``Ordinary Differential Equations''
-Ginn \& Co., Boston and New York. (1962)
+\bibitem[Steele]{Steele} Steele, Guy L.; Gabriel, Richard P.
+``The Evolution of Lisp''
+\verb|www.dreamsongs.com/Files/HOPL2-Uncut.pdf|
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Boyd9 3a]{Boyd93a} Boyd, David W.
-``Bounds for the Height of a Factor of a Polynomial in
-Terms of Bombieri's Norms: I. The Largest Factor''
-J. Symbolic Computation (1993) 16, 115-130
-%\verb|axiom-developer.org/axiom-website/Boyd93a.pdf|
+\bibitem[Sutor 85]{Sut85} Sutor, R.S.
+``The Scratchpad II computer algebra language and system''
+In Buchberger and Caviness [BC85], pp32-33 ISBN 0-387-15983-5 (vol. 1),
+0-387-15984-3 (vol. 2) LCCN QA155.7.E4 E86 1985 Two volumes.
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Boyd 93b]{Boyd93b} Boyd, David W.
-``Bounds for the Height of a Factor of a Polynomial in
-Terms of Bombieri's Norms: II. The Smallest Factor''
-J. Symbolic Computation (1993) 16, 131-145
-%\verb|axiom-developer.org/axiom-website/Boyd93b.pdf|
+\bibitem[Sutor 87a]{SJ87a} Sutor, R. S.; Jenks, R. D.
+``The type inference and coercion facilities in
+the Scratchpad II interpreter'' In Wexelblat [Wex87], pp56-63
+ISBN 0-89791-235-7 LCCN QA76.7.S54 v22:7 SIGPLAN Notices, v22 n7 (July 1987)
+%\verb|axiom-developer.org/axiom-website/papers/SJ87a.pdf|
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Braman 02a]{BBM02a} Braman, K.; Byers, R.; Mathias, R.
-``The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts,
-and Level 3 Performance''
-SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002.
+\bibitem[Sutor 87b]{Su87} Sutor, Robert S.
+``The Scratchpad II Computer Algebra System. Using and
+Programming the Interpreter''
+IBM Course presentation slide deck Spring 1987
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Braman 02b]{BBM02b} Braman, K.; Byers, R.; Mathias, R.
-``The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation''
-SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.
+\bibitem[Sutor 87c]{SJ87c} Sutor, Robert S.; Jenks, Richard
+``The type inference and coercion facilities
+in the Scratchpad II interpreter''
+Research report RC 12595 (\#56575),
+IBM Thomas J. Watson Research Center, Yorktown Heights, NY, USA, 1987, 11pp
+%\verb|axiom-developer.org/axiom-website/papers/SJ87c.pdf|
+ keywords = "axiomref",
+ abstract = "
+ The Scratchpad II system is an abstract datatype programming language,
+ a compiler for the language, a library of packages of polymorphic
+ functions and parameterized abstract datatypes, and an interpreter
+ that provides sophisticated type inference and coercion facilities.
+ Although originally designed for the implementation of symbolic
+ mathematical algorithms, Scratchpad II is a general purpose
+ programming language. This paper discusses aspects of the
+ implementation of the intepreter and how it attempts to provide a user
+ friendly and relatively weakly typed front end for the strongly typed
+ programming language."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Brent 75]{Bre75} Brent, R. P.
-``Multiple-Precision Zero-Finding Methods and the Complexity
-of Elementary Function Evaluation, Analytic Computational Complexity''
-J. F. Traub, Ed., Academic Press, New York 1975, 151-176
+\bibitem[Sutor 88]{Su88} Sutor, Robert S.
+``A guide to programming in the scratchpad 2 interpreter''
+IBM Manual, March 1988
+ keywords = "axiomref",
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Brent 78]{BK78} Brent, R. P.; Kung, H. T.
-``Fast Algorithms for Manipulating Formal Power Series''
-Journal of the Association for Computing Machinery,
-Vol. 25, No. 4, October 1978, 581-595
-
-\end{chunk}
+\subsection{T} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
-\bibitem[Brigham 73]{Bri73} Brigham E O
-``The Fast Fourier Transform''
-Prentice-Hall. (1973)
+\bibitem[Thompson 00]{Tho00} Thompson, Simon
+``Logic and dependent types in the Aldor Computer Algebra System''
+%\verb|axiom-developer.org/axiom-website/papers/Tho00.pdf|
+ keywords = "axiomref",
+ abstract = "
+ We show how the Aldor type system can represent propositions of
+ first-order logic, by means of the 'propositions as types'
+ correspondence. The representation relies on type casts (using
+ pretend) but can be viewed as a prototype implementation of a modified
+ type system with {\sl type evaluation} reported elsewhere. The logic
+ is used to provide an axiomatisation of a number of familiar Aldor
+ categories as well as a type of vectors."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Brillhart 69]{Bri69} Brillhart, John
-``On the Euler and Bernoulli polynomials''
-J. Reine Angew. Math., v. 234, (1969), pp. 45-64
+\bibitem[Thompson (a)]{TTxx} Thompson, Simon; Timochouk, Leonid
+``The Aldor\-\- language''
+%\verb|axiom-developer.org/axiom-website/papers/TTxx.pdf|
+ keywords = "axiomref",
+ abstract = "
+ This paper introduces the \verb|Aldor--| language, which is a
+ functional programming language with dependent types and a powerful,
+ type-based, overloading mechanism. The language is built on a subset
+ of Aldor, the 'library compiler' language for the Axiom computer
+ algebra system. \verb|Aldor--| is designed with the intention of
+ incorporating logical reasoning into computer algebra computations.
+
+ The paper contains a formal account of the semantics and type system
+ of \verb|Aldor--|; a general discussion of overloading and how the
+ overloading in \verb|Aldor--| fits into the general scheme; examples
+ of logic within \verb|Aldor--| and notes on the implementation of the
+ system."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Brillhart 90]{Bri90} Brillhart, John
-``Note on Irreducibility Testing''
-Mathematics of Computation, vol. 35, num. 35, Oct. 1980, 1379-1381
+\bibitem[Touratier 98]{Tou98} Touratier, Emmanuel
+``Etude du typage dans le syst\`eme de calcul scientifique Aldor''
+Universit\'e de Limoges 1998
+%\verb|axiom-developer.org/axiom-website/papers/Tou98.pdf|
+ keywords = "axiomref",
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Bronstein 98a]{Bro98a} Bronstein, M.; Grabmeier, J.; Weispfenning, V. (eds)
-``Symbolic Rewriting Techniques''
-Progress in Computer Science and Applied Logic 15, Birkhauser-Verlag, Basel
-ISBN 3-7643-5901-3 (1998)
-
-\end{chunk}
+\subsection{V} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
-\bibitem[Bronstein 88]{Bro88} Bronstein, Manual
-``The Transcendental Risch Differential Equation''
-J. Symbolic Computation (1990) 9, pp49-60 Feb 1988
-IBM Research Report RC13460 IBM Corp. Yorktown Heights, NY
-\verb|www.sciencedirect.com/science/article/pii/S0747717108800065|
-%\verb|axiom-developer.org/axiom-website/papers/Bro88.pdf|
- abstract = "
- We present a new rational algorithm for solving Risch differential
- equations in towers of transcendental elementary extensions. In
- contrast to a recent algorithm by Davenport we do not require a
- progressive reduction of the denominators involved, but use weak
- normality to obtain a formula for the denominator of a possible
- solution. Implementation timings show this approach to be faster than
- a Hermite-like reduction."
+\bibitem[van der Hoeven 14]{JvdH14} van der Hoeven, Joris
+``Computer algebra systems and TeXmacs''
+\verb|www.texmacs.org/tmweb/plugins/cas.en.html|
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{axiom.bib}
-@techreport{Bron98,
- author = "Bronstein, Manuel",
- title = "The lazy hermite reduction",
- type = "Rapport de Recherche",
- number = "RR-3562",
- year = "1998",
- institution = "French Institute for Research in Computer Science",
- paper = "Bron98.pdf",
+@article{Hoei94,
+ author = "{van Hoeij}, M.",
+ title = "An algorithm for computing an integral basis in an algebraic
+ function field",
+ journal = "Journal of Symbolic Computation",
+ volume = "18",
+ number = "4",
+ year = "1994",
+ pages = "353-363",
+ issn = "0747-7171",
+ keywords = "axiomref",
+ paper = "Hoei94.pdf",
abstract = "
- The Hermite reduction is a symbolic integration technique that reduces
- algebraic functions to integrands having only simple affine
- poles. While it is very effective in the case of simple radical
- extensions, its use in more general algebraic extensions requires the
- precomputation of an integral basis, which makes the reduction
- impractical for either multiple algebraic extensions or complicated
- ground fields. In this paper, we show that the Hermite reduction can
- be performed without {\sl a priori} computation of either a primitive
- element or integral basis, computing the smallest order necessary for
- a particular integrand along the way."
+ Algorithms for computing integral bases of an algebraic function field
+ are implemented in some computer algebra systems. They are used e.g.
+ for the integration of algebraic functions. The method used by Maple
+ 5.2 and AXIOM is given by Trager in [Trag84]. He adapted an algorithm
+ of Ford and Zassenhaus [Ford, 1978], that computes the ring of
+ integers in an algebraic number field, to the case of a function field.
+
+ It turns out that using algebraic geometry one can write a faster
+ algorithm. The method we will give is based on Puiseux expansions.
+ One cas see this as a variant on the Coates' algorithm as it is
+ described in [Davenport, 1981]. Some difficulties in computing with
+ Puiseux expansions can be avoided using a sharp bound for the number
+ of terms required which will be given in Section 3. In Section 5 we
+ derive which denominator is needed in the integral basis. Using this
+ result 'intermediate expression swell' can be avoided.
+
+ The Puiseux expansions generally introduce algebraic extensions. These
+ extensions will not appear in the resulting integral basis."
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@misc{Bro98b,
- author = "Bronstein, Manuel",
- title = "Symbolic Integration Tutorial",
- series = "ISSAC'98",
- year = "1998",
- address = "INRIA Sophia Antipolis",
- url =
- "http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac98.pdf",
- paper = "Bro98b.pdf"
+@misc{Hoei08,
+ author = "{van Hoeij}, Mark and Novocin, Andrew",
+ title = "A Reduction Algorithm for Algebraic Function Fields",
+ year = "2008",
+ month = "April",
+ url = "http://andy.novocin.com/pro/algext.pdf",
+ paper = "Hoei08.pdf",
+ abstract = "
+ Computer algebra systesm often produce large expressions involving
+ complicated algebraic numbers. In this paper we study variations of
+ the {\tt polred} algorithm that can often be used to find better
+ representations for algebraic numbers. The main new algorithm
+ presented here is an algorithm that treats the same problem for the
+ function field case."
}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Brown 99]{Brow99} Brown, Christopher W.
-``Solution Formula Construction for Truth Invariant CADs''
-Ph.D Thesis, Univ. Delaware (1999)
-\verb|www.usna.edu/Users/cs/wcbrown/research/thesis.ps.gz|
-%\verb|axiom-developer.org/axiom-website/papers/Brow99.pdf|
- abstract = "
- The CAD-based quantifier elimination algorithm takes a formula from
- the elementary theory of real closed fields as input, and constructs a
- CAD of the space of the formula's unquantified variables. This
- decomposition is truth invariant with respect to the input formula,
- meaning that the formula is either identically true or identically
- false in each cell of the decomposition. The method determines the
- truth of the input formula for each cell of the CAD, and then uses the
- CAD to construct a solution formula -- a quantifier free formula that
- is equivalent to the input formula. This final phase of the algorithm,
- the solution formula construction phase, is the focus of this thesis.
+\bibitem[Vasconcelos 99]{Vas99} Vasconcelos, Wolmer
+``Computational Methods in Commutative Algebra and Algebraic Geometry''
+Springer, Algorithms and Computation in Mathematics, Vol 2 1999
+ISBN 3-540-21311-2
+ keywords = "axiomref",
- An optimal solution formula construction algorithm would be {\sl
- complete} -- i.e. applicable to any truth-invariant CAD, would be {\sl
- efficient}, and would produce {\sl simple} solution formulas. Prior to
- this thesis, no method was available with even two of these three
- properties. Several algorithms are presented, all addressing problems
- related to solution formula construction. In combination, these
- provide an efficient and complete method for constructing solution
- formulas that are simple in a variety of ways.
+\end{chunk}
- Algorithms presented in this thesis have been implemented using the
- SACLIB library, and integrated into QEPCAD, a SACLIB-based
- implementation of quantifier elimination by CAD. Example computations
- based on these implementations are discussed."
+\subsection{W} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{ignore}
+\bibitem[Wang 89]{Wan89} Wang, D.
+``A program for computing the Liapunov functions and Liapunov
+constants in Scratchpad II''
+SIGSAM Bulletin (ACM Special Interest Group
+on Symbolic and Algebraic Manipulation), 23(4) pp25-31, Oct. 1989,
+CODEN SIGSBZ ISSN 0163-5824
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Brown 02]{Brow02} Brown, Christopher W.
-``QEPCAD B -- A program for computing with semi-algebraic sets using CADs''
-%\verb|axiom-developer.org/axiom-website/papers/Brow02.pdf|
- abstract = "
- This report introduces QEPCAD B, a program for computing with real
- algebraic sets using cylindrical algebraic decomposition (CAD). QEPCAD
- B both extends and improves upon the QEPCAD system for quantifier
- elimination by partial cylindrical algebraic decomposition written by
- Hoon Hong in the early 1990s. This paper briefly discusses some of the
- improvements in the implementation of CAD and quantifier elimination
- vis CAD, and provides somewhat more detail on extensions to the system
- that go beyond quantifier elimination. The author is responsible for
- most of the extended features of QEPCAD B, but improvements to the
- basic CAD implementation and to the SACLIB library on which QEPCAD is
- based are the results of many people's work."
+\bibitem[Wang 91]{Wan91} Wang, Dongming
+``Mechanical manipulation for a class of differential systems''
+Journal of Symbolic Computation, 12(2) pp233-254 Aug. 1991
+CODEN JSYCEH ISSN 0747-7171
+ keywords = "axiomref",
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Burg74,
- author = "William H. Burge",
- title = "Stream Processing Functions",
- year = "1974",
- month = "January",
- journal = "IBM Journal of Research and Development",
- volume = "19",
- issue = "1",
- pages = "12-25",
- papers = "Burg74.pdf",
- abstract = "
- One principle of structured programming is that a program should be
- separated into meaningful independent subprograms, which are then
- combined so that the relation of the parts to the whole can be clearly
- established. This paper describes several alternative ways to compose
- programs. The main method used is to permit the programmer to denote
- by an expression the sequence of values taken on by a variable. The
- sequence is represented by a function called a stream, which is a
- functional analog of a coroutine. The conventional while and for loops
- of structured programming may be composed by a technique of stream
- processing (analogous to list processing), which results in more
- structured programs than the orignals. This technique makes it
- possible to structure a program in a natural way into its logically
- separate parts, which can then be considered independently."
-}
+\begin{chunk}{ignore}
+\bibitem[Wang 92]{Wan92} Wang, Paul S. (ed)
+International System Symposium on Symbolic and
+Algebraic Computation 92 ACM Press, New York, NY 10036, USA, 1992
+ISBN 0-89791-489-9 (soft cover), 0-89791-490-2 (hard cover),
+LCCN QA76.95.I59 1992
+ keywords = "axiomref",
\end{chunk}
-\subsection{C} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{chunk}{ignore}
+\bibitem[Watanabe 90]{WN90} Watanabe, Shunro; Nagata, Morio; (ed)
+ISSAC '90 Proceedings of the
+International Symposium on Symbolic and Algebraic Computation ACM Press,
+New York, NY, 10036, USA. 1990 ISBN 0-89791-401-5 LCCN QA76.95.I57 1990
+ keywords = "axiomref",
+
+\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Carlson 65]{Car65} Carlson B C
-``On Computing Elliptic Integrals and Functions''
-J Math Phys. 44 36--51. (1965)
+\bibitem[Watt 85]{Wat85} Watt, Stephen
+``Bounded Parallelism in Computer Algebra''
+PhD Thesis, University of Waterloo
+\verb|www.csd.uwo.ca/~watt/pub/reprints/1985-smw-phd.pdf|
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Carlson 77a]{Car77a} Carlson B C
-``Elliptic Integrals of the First Kind''
-SIAM J Math Anal. 8 231--242. (1977)
+\bibitem[Watt 86]{Wat86} Watt, S.M.; Della Dora, J.
+``Algebra Snapshot: Linear Ordinary Differential Operators''
+Scratchpad II Newsletter: Vol 1 Num 2 (Jan 1986)
+\verb|www.csd.uwo.ca/~watt/pub/reprints/1986-snews-lodo.pdf|
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Carlson 77b]{Car77b} Carlson B C
-``Special Functions of Applied Mathematics''
-Academic Press. (1977)
+\bibitem[Watt 87]{Wat87} Watt, Stephen
+``Domains and Subdomains in Scratchpad II''
+in [Wit87], pp3-5
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Carlson 78]{Car78} Carlson B C,
-``Computing Elliptic Integrals by Duplication''
-(Preprint) Department of Physics, Iowa State University. (1978)
+\bibitem[Watt 87a]{WB87} Watt, Stephen M.; Burge, William H.
+``Mapping as First Class Objects''
+in [Wit87], pp13-17
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Carlson 88]{Car88} Carlson B C,
-``A Table of Elliptic Integrals of the Third Kind''
-Math. Comput. 51 267--280. (1988)
+\bibitem[Watt 89]{Wat89} Watt, S. M.
+``A fixed point method for power series computation''
+In Gianni [Gia89], pp206-217 ISBN 3-540-51084-2 LCCN QA76.95.I57
+1988 Conference held jointly with AAECC-6
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Cauchy 1829]{Cau1829} Augustin-Lux Cauchy
-``Exercices de Math\'ematiques Quatri\`eme Ann\'ee. De Bure Fr\`eres''
-Paris 1829 (reprinted Oeuvres, II S\'erie, Tome IX,
-Gauthier-Villars, Paris, 1891).
+\bibitem[Watt 90]{WJST90} Watt, S.M.; Jenks, R.D.; Sutor, R.S.; Trager B.M.
+``The Scratchpad II type system: Domains and subdomains''
+in A.M. Miola, editor Computing Tools
+for Scientific Problem Solving, Academic Press, New York, 1990
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Ch\`eze 07]{Chez07} Ch\'eze, Guillaume; Lecerf, Gr\'egoire
-``Lifting and recombination techniques for absolute factorization''
-Journal of Complexity, VOl 23 Issue 3 June 2007 pp 380-420
-\verb|www.sciencedirect.com/science/article/pii/S0885064X07000465|
-%\verb|axiom-developer.org/axiom-website/papers/Chez07.pdf|
- abstract = "
- In the vein of recent algorithmic advances in polynomial factorization
- based on lifting and recombination techniques, we present new faster
- algorithms for computing the absolute factorization of a bivariate
- polynomial. The running time of our probabilistic algorithm is less
- than quadratic in the dense size of the polynomial to be factored."
+\bibitem[Watt 91]{Wat91} Watt, Stephen M. (ed)
+Proceedings of the 1991 International Symposium on
+Symbolic and Algebraic Computation, ISSAC'91, July 15-17, 1991, Bonn, Germany,
+ACM Press, New York, NY 10036, USA, 1991 ISBN 0-89791-437-6
+LCCN QA76.95.I59 1991
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Childs 79]{CSDDN79} Childs B; Scott M; Daniel J W; Denman E;
-Nelson P (eds)
-``Codes for Boundary-value Problems in Ordinary Differential Equations''
-Lecture Notes in Computer Science. 76 (1979) Springer-Verlag
+\bibitem[Watt 94a]{Wat94a} Watt, Stephen M.; Dooley, S.S.; Morrison, S.C.;
+Steinback, J.M.; Sutor, R.S.
+``A\# User's Guide''
+Version 1.0.0 O($\epsilon{}^1$) June 8, 1994
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Clausen 89]{Cla89} Clausen, M.; Fortenbacher, A.
-``Efficient Solution of Linear Diophantine Equations''
-JSC (1989) 8, 201-216
+\bibitem[Watt 94b]{Wat94} Watt, Stephen M.; Broadbery, Peter A.;
+Dooley, Samuel S.; Iglio, Pietro
+``A First Report on the A\# Compiler (including benchmarks)''
+IBM Research Report RC19529 (85075) May 12, 1994
+%\verb|axiom-developer.org/axiom-website/papers/Wat94.pdf|
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Clenshaw 55]{Cle55} Clenshaw C W,
-``A Note on the Summation of Chebyshev Series''
-Math. Tables Aids Comput. 9 118--120. (1955)
+\bibitem[Watt 94c]{Wat94c} Watt, Stephen M.
+``A\# Language Reference Version 0.35''
+IBM Research Division Technical Report RC19530 May 1994
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Clenshaw 60]{Cle60} Clenshaw C W
-``Curve Fitting with a Digital Computer''
-Comput. J. 2 170--173. (1960)
+\bibitem[Watt 95]{Wat95} Watt, S.M.; Broadbery, P.A.; Dooley, S.S.; Iglio, P.
+Steinbach, J.M.; Morrison, S.C.; Sutor, R.S.
+``AXIOM Library Compiler Users Guide''
+The Numerical Algorithms Group (NAG) Ltd, 1994
+ keywords = "axiomref",
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Watt 01]{Wat01} Watt, Stephen M.; Broadbery, Peter A.; Iglio, Pietro;
+Morrison, Scott C.; Steinbach, Jonathan M.
+``FOAM: A First Order Abstract Machine Version 0.35''
+IBM T. J. Watson Research Center (2001)
+%\verb|axiom-developer.org/axiom-website/papers/Wat01.pdf|
+ keywords = "axiomref",
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Weber 92]{Webe92} Weber, Andreas
+``Type Systems for Computer Algebra''
+\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber92a.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Webe92.pdf|
+ keywords = "axiomref",
+ abstract = "
+ An important feature of modern computer algebra systems is the support
+ of a rich type system with the possibility of type inference. Basic
+ features of such a type system are polymorphism and coercion between
+ types. Recently the use of order-sorted rewrite systems was proposed
+ as a general framework. We will give a quite simple example of a
+ family of types arising in computer algebra whose coercion relations
+ cannot be captured by a finite set of first-order rewrite rules."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Clenshaw 62]{Cle62} Clenshaw C W
-``Mathematical Tables. Chebyshev Series for Mathematical Functions''
-HMSO. (1962)
+\bibitem[Weber 92b]{Webe92b} Weber, Andreas
+``Structuring the Type System of a Computer Algebra System''
+\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber92a.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Webe92b.pdf|
+ keywords = "axiomref",
+ abstract = "
+ Most existing computer algebra systems are pure symbol manipulating
+ systems without language support for the occuring types. This is
+ mainly due to the fact taht the occurring types are much more
+ complicated than in traditional programming languages. In the last
+ decade the study of type systems has become an active area of
+ research. We will give a proposal for a type system showing that
+ several problems for a type system of a symbolic computation system
+ can be solved by using results of this research. We will also provide
+ a variety of examples which will show some of the problems that remain
+ and that will require further research."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Cline 84]{CR84} Cline A K; Renka R L,
-``A Storage-efficient Method for Construction of a Thiessen Triangulation''
-Rocky Mountain J. Math. 14 119--139. (1984)
+\bibitem[Weber 93b]{Webe93b} Weber, Andreas
+``Type Systems for Computer Algebra''
+\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber93b.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Webe93b.pdf|
+ keywords = "axiomref",
+ abstract = "
+ We study type systems for computer algebra systems, which frequently
+ correspond to the ``pragmatically developed'' typing constructs used
+ in AXIOM. A central concept is that of {\sl type classes} which
+ correspond to AXIOM categories. We will show that types can be
+ syntactically described as terms of a regular order-sorted signature
+ if no type parameters are allowed. Using results obtained for the
+ functional programming language Haskell we will show that the problem
+ of {\sl type inference} is decidable. This result still holds if
+ higher-order functions are present and {\sl parametric polymorphism}
+ is used. These additional typing constructs are useful for further
+ extensions of existing computer algebra systems: These typing concepts
+ can be used to implement category theoretic constructs and there are
+ many well known constructive interactions between category theory and
+ algebra."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Conway 87]{CCNPW87} Conway, J.; Curtis, R.; Norton, S.; Parker, R.;
-Wilson, R.
-``Atlas of Finite Groups''
-Oxford, Clarendon Press, 1987
-
-\end{chunk}
+\bibitem[Weber 94]{Web94} Weber, Andreas
+``Algorithms for Type Inference with Coercions''
+ISSAC 94 ACM 0-89791-638-7/94/0007
+%\verb|axiom-developer.org/axiom-website/papers/Web94.pdf|
+ keywords = "axiomref",
+ abstract = "
+ This paper presents algorithms that perform a type inference for a
+ type system occurring in the context of computer algebra. The type
+ system permits various classes of coercions between types and the
+ algorithms are complete for the precisely defined system, which can be
+ seen as a formal description of an important subset of the type system
+ supported by the computer algebra program Axiom.
-\begin{chunk}{ignore}
-\bibitem[Conway 03]{CS03} Conway, John H.; Smith, Derek, A.
-``On Quaternions and Octonions''
-A.K Peters, Natick, MA. (2003) ISBN 1-56881-134-9
+ Previously only algorithms for much more restricted cases of coercions
+ have been described or the frameworks used have been so general that
+ the corresponding type inference problems were known to be
+ undecidable."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Cox 72]{Cox72} Cox M G
-``The Numerical Evaluation of B-splines''
-J. Inst. Math. Appl. 10 134--149. (1972)
+\bibitem[Weber 95]{Webe95} Weber, A.
+``On coherence in computer algebra''
+\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber94e.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Webe95.pdf|
+ keywords = "axiomref",
+ abstract = "
+ Modern computer algebra systems (e.g. AXIOM) support a rich type
+ system including parameterized data types and the possibility of
+ implicit coercions between types. In such a type system it will be
+ frequently the case that there are different ways of building
+ coercions between types. An important requirement is that all
+ coercions between two types coincide, a property which is called {\sl
+ coherence}. We will prove a coherence theorem for a formal type system
+ having several possibilities of coercions covering many important
+ examples. Moreover, we will give some informal reasoning why the
+ formally defined restrictions can be satisfied by an actual system."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[CH 73]{CH73} Cox M G; Hayes J G
-``Curve fitting: a guide and suite of algorithms for the
-non-specialist user''
-Report NAC26. National Physical Laboratory. (1973)
+\bibitem[Weber 96]{Webe96} Weber, Andreas
+``Computing Radical Expressions for Roots of Unity''
+\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/WeberA/Weber96a.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Webe96.pdf|
+ keywords = "axiomref",
+ abstract = "
+ We present an improvement of an algorithm given by Gauss to compute a
+ radical expression for a $p$-th root of unity. The time complexity of
+ the algorithm is $O(p^3m^6log p)$, where $m$ is the largest prime
+ factor of $p-1$."
\end{chunk}
+
\begin{chunk}{ignore}
-\bibitem[Cox 74a]{Cox74a} Cox M G
-``A Data-fitting Package for the Non-specialist User''
-Software for Numerical Mathematics. (ed D J Evans) Academic Press. (1974)
+\bibitem[Weber 99]{Webe99} Weber, Andreas
+``Solving Cyclotomic Polynomials by Radical Expressions''
+\verb|cg.cs.uni-bonn.de/personal-pages/weber/publications/pdf/|
+\verb|WeberA/WeberKeckeisen99a.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Webe99.pdf|
+ keywords = "axiomref",
+ abstract = "
+ We describe a Maple package that allows the solution of cyclotomic
+ polynomials by radical expressions. We provide a function that is an
+ extension of the Maple {\sl solve} command. The major algorithmic
+ ingredient of the package is an improvement of a method due to Gauss
+ which gives radical expressions for roots of unity. We will give a
+ summary for computations up to degree 100, which could be done within
+ a few hours of cpu time on a standard workstation."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Cox 74b]{Cox74b} Cox M G
-``Numerical methods for the interpolation and approximation of data
-by spline functions''
-PhD Thesis. City University, London. (1975)
+\bibitem[Wei-Jiang 12]{WJ12} Wei-Jiang
+``Top free algebra System''
+\verb|wei-jiang.com/it/software/top-free-algebra-system-bye-mathematica-bye-maple|
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Cox 75]{Cox75} Cox M G
-``An Algorithm for Spline Interpolation''
-J. Inst. Math. Appl. 15 95--108. (1975)
+\bibitem[Wester 99]{Wes99} Wester, Michael J.
+``Computer Algebra Systems''
+John Wiley and Sons 1999 ISBN 0-471-98353-5
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Cox 77]{Cox77} Cox M G
-``A Survey of Numerical Methods for Data and Function Approximation''
-The State of the Art in Numerical Analysis. (ed D A H Jacobs)
-Academic Press. 627--668. (1977)
- keywords = "survey",
+\bibitem[Wexelblat 87]{Wex87} Wexelblat, Richard L. (ed)
+Proceedings of the SIGPLAN '87 Symposium on
+Interpreter and Interpretive Techniques, St. Paul, Minnesota, June 24-26, 1987
+ACM Press, New York, NY 10036, USA, 1987 ISBN 0-89791-235-7
+LCCN QA76.7.S54 v22:7 SIGPLAN Notices, vol 22, no 7 (July 1987)
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Cox 78]{Cox78} Cox M G
-``The Numerical Evaluation of a Spline from its B-spline Representation''
-J. Inst. Math. Appl. 21 135--143. (1978)
+\bibitem[Wityak 87]{Wit87} Wityak, Sandra
+``Scratchpad II Newsletter''
+Volume 2, Number 1, Nov 1987
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Curtis 74]{CPR74} Curtis A R; Powell M J D; Reid J K
-``On the Estimation of Sparse Jacobian Matrices''
-J. Inst. Maths Applics. 13 117--119. (1974)
+\bibitem[WWW1]{WWW1}.
+Software Preservation Group
+\verb|www.softwarepresentation.org/projects/LISP/common_lisp_family|
+ keywords = "axiomref",
\end{chunk}
-\subsection{D} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Y} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
-\bibitem[Dahlquist 74]{DB74} Dahlquist G; Bjork A
-``Numerical Methods''
-Prentice- Hall. (1974)
+\bibitem[Yap 00]{Yap00} Yap, Chee Keng
+``Fundamental Problems of Algorithmic Algebra''
+Oxford University Press (2000) ISBN0-19-512516-9
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Dalmas 98]{DA98} Dalmas, Stephane; Arsac, Olivier
-``The INRIA OpenMath Library''
-Projet SAFIR, INRIA Sophia Antipolis Nov 25, 1998
+\bibitem[Yapp 07]{Yapp07} Yapp, Clifford; Hebisch, Waldek; Kaminski, Kai
+``Literate Programming Tools Implemented in ANSI Common Lisp''
+\verb|brlcad.org/~starseeker/cl-web-v0.8.lisp.pamphlet|
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Dantzig 63]{Dan63} Dantzig G B
-``Linear Programming and Extensions''
-Princeton University Press. (1963)
+\bibitem[Yun 83]{Yun83} Yun, David Y.Y.
+``Computer Algebra and Complex Analysis''
+Computational Aspects of Complex Analysis pp379-393
+D. Reidel Publishing Company H. Werner et. al. (eds.)
+ keywords = "axiomref",
\end{chunk}
+\subsection{Z} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Davenport]{Dav} Davenport, James
-``On Brillhart Irreducibility.''
-To appear.
+\bibitem[Zen92]{Zen92} Zenger, Ch.
+``Gr{\"o}bnerbasen f{\"u}r Differentialformen und ihre
+Implementierung in AXIOM''
+Diplomarbeit, Universit{\"a}t Karlsruhe,
+Karlsruhe, Germany, 1992
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Davenport 93]{Ref-Dav93} Davenport, J.H.
-``Primality testing revisited''
-Technical Report TR2/93
-(ATR/6)(NP2556) Numerical Algorithms Group, Inc., Downer's Grove, IL, USA
-and Oxford, UK, August 1993
-\verb|www.nag.co.uk/doc/TechRep/axiomtr.html|
+\bibitem[Zip92]{Zip92} Zippel, Richard
+``Algebraic Computation''
+(unpublished) Cornell University Ithaca, NY Sept 1992
+ keywords = "axiomref",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Davis 67]{DR67} Davis P J; Rabinowitz P
-``Numerical Integration''
-Blaisdell Publishing Company. 33--52. (1967)
+\bibitem[Zwi92]{Zwi92} Zwillinger, Daniel
+``Handbook of Integration''
+Jones and Bartlett, 1992, ISBN 0-86720-293-9
+ keywords = "axiomref",
\end{chunk}
+\section{Axiom Citations of External Sources}
-\begin{chunk}{ignore}
-\bibitem[Davis 75]{DR75} Davis P J; Rabinowitz P
-``Methods of Numerical Integration''
-Academic Press. (1975)
+\subsection{A} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@article{Abla98,
+ author = "Ablamowicz, Rafal",
+ title = "Spinor Representations of Clifford Algebras: A Symbolic Approach",
+ journal = "Computer Physics Communications",
+ volume = "115",
+ number = "2-3",
+ month = "December",
+ year = "1998",
+ pages = "510-535"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[DeBoor 72]{DeB72} De Boor C
-``On Calculating with B-splines''
-J. Approx. Theory. 6 50--62. (1972)
+\begin{chunk}{axiom.bib}
+@article{Abra06,
+ author = "Abramov, Sergey A.",
+ title = "In Memory of Manuel Bronstein",
+ journal = "Programming and Computer Software",
+ volume = "32",
+ number = "1",
+ pages = "56-58",
+ publisher = "Pleiades Publishing Inc",
+ year = "2006",
+ paper = "Abra06.pdf"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[De Doncker 78]{DeD78} De Doncker E,
-``An Adaptive Extrapolation Algorithm for Automatic Integration''
-Signum Newsletter. 13 (2) 12--18. (1978)
+\bibitem[Abramowitz 64]{AS64} Abramowitz, Milton; Stegun, Irene A.
+``Handbook of Mathematical Functions''
+(1964) Dover Publications, NY ISBN 0-486-61272-4
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Demmel 89]{Dem89} Demmel J W
-``On Floating-point Errors in Cholesky''
-LAPACK Working Note No. 14. University of Tennessee, Knoxville. 1989
+\bibitem[Abramowitz 68]{AS68} Abramowitz M; Stegun I A
+``Handbook of Mathematical Functions''
+Dover Publications. (1968)
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Dennis 77]{DM77} Dennis J E Jr; More J J
-``Quasi-Newton Methods, Motivation and Theory''
-SIAM Review. 19 46--89. 1977
+\begin{chunk}{axiom.bib}
+@book{Altm05,
+ author = "Altmann, Simon L.",
+ title = "Rotations, Quaternions, and Double Groups",
+ publisher = "Dover Publications, Inc.",
+ year = "2005",
+ isbn = "0-486-44518-6"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Dennis 81]{DS81} Dennis J E Jr; Schnabel R B
-``A New Derivation of Symmetric Positive-Definite Secant Updates''
-Nonlinear Programming 4. (ed O L Mangasarian, R R Meyer and S M. Robinson)
-Academic Press. 167--199. (1981)
+\bibitem[Ames 77]{Ames77} Ames W F
+``Nonlinear Partial Differential Equations in Engineering''
+Academic Press (2nd Edition). (1977)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Dennis 83]{DS83} Dennis J E Jr; Schnabel R B
-``Numerical Methods for Unconstrained Optimixation and Nonlinear Equations''
-Prentice-Hall.(1983)
+\bibitem[Amos 86]{Amos86} Amos D E
+``Algorithm 644: A Portable Package for Bessel Functions of a Complex
+Argument and Nonnegative Order''
+ACM Trans. Math. Softw. 12 265--273. (1986)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Dierckx 75]{Die75} Dierckx P
-``An Algorithm for Smoothing, Differentiating and Integration of
-Experimental Data Using Spline Functions''
-J. Comput. Appl. Math. 1 165--184. (1975)
+\bibitem[Anderson 00]{And00} Anderson, Edward
+``Discontinuous Plane Rotations and the Symmetric Eigenvalue Problem''
+LAPACK Working Note 150, University of Tennessee, UT-CS-00-454,
+December 4, 2000.
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Dierckx 81]{Die81} Dierckx P
-``An Improved Algorithm for Curve Fitting with Spline Functions''
-Report TW54. Dept. of Computer Science, Katholieke Universiteit Leuven. 1981
+\bibitem[Anthony 82]{ACH82} Anthony G T; Cox M G; Hayes J G
+``DASL - Data Approximation Subroutine Library''
+National Physical Laboratory. (1982)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Dierckx 82]{Die82} Dierckx P
-``A Fast Algorithm for Smoothing Data on a Rectangular Grid while using
-Spline Functions''
-SIAM J. Numer. Anal. 19 1286--1304. (1982)
+\bibitem[Arnon 88]{Arno88} Arno, D.S.; MIgnotte, M.
+``On Mechanical Quantifier Elimination for Elementary Algebra and Geometry''
+J. Symbolic Computation 5, 237-259 (1988)
+\verb|http://www.sciencedirect.com/science/article/pii/S0747717188800142/|
+\verb|pdf?md5=62052077d84e6078cc024bc8e29c23c1&|
+\verb|pid=1-s2.0-S0747717188800142-main.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Arno88.pdf|
+ abstract = "
+ We give solutions to two problems of elementary algebra and geometry:
+ (1) find conditions on real numbers $p$, $q$, and $r$ so that the
+ polynomial function $f(x)=x^4+px^2+qx+r$ is nonnegative for all real
+ $x$ and (2) find conditions on real numbers $a$, $b$, and $c$ so that
+ the ellipse $\frac{(x-e)^2}{q^2}+\frac{y^2}{b^2}-1=0$ lies inside the
+ unit circle $y^2+x^2-1=0$. Our solutions are obtained by following the
+ basic outline of the method of quantifier elimination by cylindrical
+ algebraic decomposition (Collins, 1975), but we have developed, and
+ have been considerably aided by, modified versions of certain of its
+ steps. We have found three equally simple but not obviously equivalent
+ solutions for the first problem, illustrating the difficulty of
+ obtaining unique ``simplest'' solutions to quantifier elimination
+ problems of elementary algebra and geometry."
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Dongarra 79]{DMBS79} Dongarra J J; Moler C B; Bunch J R;
-Stewart G W
-``LINPACK Users' Guide''
-SIAM, Philadelphia. (1979)
+\begin{chunk}{axiom.bib}
+@article{Aubr99,
+ author = "Aubry, Phillippe and Lazard, Daniel and {Moreno Maza}, Marc",
+ title = "On the Theories of Triangular Sets",
+ year = "1999",
+ pages = "105-124",
+ journal = "Journal of Symbolic Computation",
+ volume = "28",
+ url = "http://www.csd.uwo.ca/~moreno/Publications/Aubry-Lazard-MorenoMaza-1999-JSC.pdf",
+ papers = "Aubr99.pdf",
+ abstract = "
+ Different notions of triangular sets are presented. The relationship
+ between these notions are studied. The main result is that four
+ different existing notions of {\sl good} triangular sets are
+ equivalent."
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Dongarra 85]{DCHH85} Dongarra J J; Du Croz J J; Hammarling S;
-Hanson R J
-``A Proposal for an Extended set of Fortran Basic Linear
-Algebra Subprograms''
-SIGNUM Newsletter. 20 (1) 2--18. (1985)
+\bibitem[Aubry 96]{Aub96} Aubry, Philippe; Maza, Marc Moreno
+``Triangular Sets for Solving Polynomial Systems: a Comparison of Four Methods''
+\verb|www.lip6.fr/lip6/reports/1997/lip6.1997.009.ps.gz|
+%\verb|axiom-developer.org/axiom-website/papers/Aub96.ps|
+ abstract = "
+ Four methods for solving polynomial systems by means of triangular
+ sets are presented and implemented in a unified way. These methods are
+ those of Wu, Lazard, Kalkbrener, and Wang. They are compared on
+ various examples with emphasis on efficiency, conciseness and
+ legibility of the outputs."
\end{chunk}
+\subsection{B} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Dongarra 88]{REF-DON88} Dongarra, Jack J.; Du Croz, Jeremy;
-Hammarling, Sven; Hanson, Richard J.
-``An Extended Set of FORTRAN Basic Linear Algebra Subroutines''
-ACM Transactions on Mathematical Software, Vol 14, No 1, March 1988,
-pp 1-17
+\bibitem[Bailey 66]{Bai66} Bailey P B
+``Sturm-Liouville Eigenvalues via a Phase Function''
+SIAM J. Appl. Math . 14 242--249. (1966)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Dongarra 88a]{REF-DON88a} Dongarra, Jack J.; Du Croz, Jeremy;
-Hammarling, Sven; Hanson, Richard J.
-``ALGORITHM 656: An Extended Set of Basic Linear Algebra Subprograms:
-Model Implementation and Test Programs''
-ACM Transactions on Mathematical Software, Vol 14, No 1, March 1988,
-pp 18-32
+\bibitem[Baker 96]{BGM96} Baker, George A.; Graves-Morris, Peter
+``Pade Approximants''
+Cambridge University Press, March 1996 ISBN 9870521450072
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Dongarra 90]{REF-DON90} Dongarra, Jack J.; Du Croz, Jeremy;
-Hammarling, Sven; Duff, Iain S.
-``A Set of Level 3 Basic Linear Algebra Subprograms''
-ACM Transactions on Mathematical Software, Vol 16, No 1, March 1990,
-pp 1-17
+\bibitem[Baker 10]{Ba10} Baker, Martin
+``3D World Simulation''
+\verb|www.euclideanspace.com|
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Dongarra 90a]{REF-DON90a} Dongarra, Jack J.; Du Croz, Jeremy;
-Hammarling, Sven; Duff, Iain S.
-``ALGORITHM 679: A Set of Level 3 Basic Linear Algebra Subprograms:
-Model Implementation and Test Programs''
-ACM Transactions on Mathematical Software, Vol 16, No 1, March 1990,
-pp 18-28
+\begin{chunk}{axiom.bib}
+@misc{Bake14,
+ author = "Baker, Martin",
+ title = "Axiom Architecture",
+ year = "2014",
+ url = "http://www.euclideanspace.com/prog/scratchpad/internals/ccode"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Ducos 00]{Duc00} Ducos, Lionel
-``Optimizations of the subresultant algorithm''
-Journal of Pure and Applied Algebra V145 No 2 Jan 2000 pp149-163
+\bibitem[Banks 68]{BK68} Banks D O; Kurowski I
+``Computation of Eigenvalues of Singular Sturm-Liouville Systems''
+Math. Computing. 22 304--310. (1968)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Duff 77]{Duff77} Duff I S,
-``MA28 -- a set of Fortran subroutines for sparse unsymmetric linear
-equations''
-A.E.R.E. Report R.8730. HMSO. (1977)
+\bibitem[Bard 74]{Bard74} Bard Y
+``Nonlinear Parameter Estimation''
+Academic Press. 1974
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Duval 96a]{Duva96a} Duval, D.; Gonz\'alez-Vega, L.
-``Dynamic Evaluation and Real Closure''
-Mathematics and Computers in Simulation 42 pp 551-560 (1996)
-%\verb|axiom-developer.org/axiom-website/papers/Duva96a.pdf|
- abstract = "
- The aim of this paper is to present how the dynamic evaluation method
- can be used to deal with the real closure of an ordered field. Two
- kinds of questions, or tests, may be asked in an ordered field:
- equality tests $(a=b?)$ and sign tests $(a > b?)$. Equality tests are
- handled through splittings, exactly as in the algebraic closure of a
- field. Sign tests are handled throug a structure called ``Tarski data
- type''."
+\bibitem[Barrodale 73]{BR73} Barrodale I; Roberts F D K
+``An Improved Algorithm for Discrete $ll_1$ Linear Approximation''
+SIAM J. Numer. Anal. 10 839--848. (1973)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Duval 96]{Duva96} Duval, D.; Reynaud, J.C.
-``Sketches and Computations over Fields''
-Mathematics and Computers in Simulation 42 pp 363-373 (1996)
-%\verb|axiom-developer.org/axiom-website/papers/Duva96.pdf|
- abstract = "
- The goal of this short paper is to describe one possible use of
- sketches in computer algebra. We show that sketches are a powerful
- tool for the description of mathematical structures and for the
- description of computations."
+\bibitem[Barrodale 74]{BR74} Barrodale I; Roberts F D K
+``Solution of an Overdetermined System of Equations in the $ll_1-norm$.''
+Comm. ACM. 17, 6 319--320. (1974)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Duval 94a]{Duva94a} Duval, D.; Reynaud, J.C.
-``Sketches and Computation (Part I): Basic Definitions and Static Evaluation''
-Mathematical Structures in Computer Science, 4, p 185-238 Cambridge University Press (1994)
-\verb|journals.cambridge.org/abstract_S0960129500000438|
-%\verb|axiom-developer.org/axiom-website/papers/Duva94a.pdf|
- abstract = "
- We define a categorical framework, based on the notion of {\sl
- sketch}, for specification and evaluation in the sense of algebraic
- specifications and algebraic programming. This framework goes far
- beyond our initial motivations, which was to specify computation with
- algebraic numbers. We begin by redefining sketches in order to deal
- explicitly with programs. Expressions and terms are carefully defined
- and studied, then {\sl quasi-projective sketches} are introduced. We
- describe {\sl static evaluation} in these sketches: we propose a
- rigorous basis for evaluation in the corresponding structures. These
- structures admit an initial model, but are not necessarily
- equational. In Part II (Duval and Reynaud 1994), we study a more
- general process, called {\sl dynamic evaluation}, for structures that
- may have no initial model."
+\bibitem[Beauzamy 92]{Bea92} Beauzamy, Bernard
+``Products of polynomials and a priori estimates for
+coefficients in polynomial decompositions: a sharp result''
+J. Symbolic Computation (1992) 13, 463-472
+%\verb|axiom-developer.org/axiom-website/papers/Bea92.pdf|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Duval 94b]{Duva94b} Duval, D.; Reynaud, J.C.
-``Sketches and Computation (Part II): Dynamic Evaluation and Applications''
-Mathematical Structures in Computer Science, 4, p 239-271. Cambridge University Press (1994)
-\verb|journals.cambridge.org/abstract_S096012950000044X|
-%\verb|axiom-developer.org/axiom-website/papers/Duva94b.pdf|
- abstract = "
- In the first part of this paper (Duval and Reynaud 1994), we defined a
- categorical framework, based on the notion of {\sl sketch}, for
- specification and evaluation in the senses of algebraic specification
- and algebraic programming. {\sl Static evaluation} in {\sl
- quasi-projective sketches} was defined in Part I; in this paper, {\sl
- dynamic evaluation} is introduced. It deals with more general
- structures, which may have no initial model. Until now, this process
- has not been used in algebraic specification systems, but computer
- algebra systems are beginning to use it as a basic tool. Finally, we
- give some applications of dynamic evaluation to computation in field
- extensions."
+\bibitem[Beauzamy 93]{Bea93} Beauzamy, Bernard; Trevisan, Vilmar;
+Wang, Paul S.
+``Polynomial Factorization: Sharp Bounds, Efficient Algorithms''
+J. Symbolic Computation (1993) 15, 393-413
+%\verb|axiom-developer.org/axiom-website/papers/Bea93.pdf|
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Duval 94c]{Duva94c} Duval, Dominique
-``Algebraic Numbers: An Example of Dynamic Evaluation''
-J. Symbolic Computation 18, 429-445 (1994)
-\verb|www.sciencedirect.com/science/article/pii/S0747717106000551|
-%\verb|axiom-developer.org/axiom-website/papers/Duva94c.pdf|
+\begin{chunk}{axiom.bib}
+@article{Bert95,
+ author = "Bertrand, Laurent",
+ title = "Computing a hyperelliptic integral using arithmetic in the
+ jacobian of the curve",
+ journal = "Applicable Algebra in Engineering, Communication and Computing",
+ volume = "6",
+ pages = "275-298",
+ year = "1995",
abstract = "
- Dynamic evaluation is presented through examples: computations
- involving algebraic numbers, automatic case discussion according to
- the characteristic of a field. Implementation questions are addressed
- too. Finally, branches are presented as ``dual'' to binary functions,
- according to the approach of sketch theory."
+ In this paper, we describe an efficient algorithm for computing an
+ elementary antiderivative of an algebraic function defined on a
+ hyperelliptic curve. Our algorithm combines B.M. Trager's integration
+ algorithm and a technique for computing in the Jacobian of a
+ hyperelliptic curve introduced by D.G. Cantor. Our method has been
+ implemented and successfully compared to Trager's general algorithm."
+}
\end{chunk}
-\subsection{F} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Fateman 08]{Fat08} Fateman, Richard
-``Revisiting numeric/symbolic indefinite integration of rational functions, and extensions''
-\verb|www.eecs.berkeley.edu/~fateman/papers/integ.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Fat08.pdf|
- abstract = "
- We know we can solve this problem: Given any rational function
- $f(x)=p(x)/q(x)$, where $p$ and $q$ are univariate polynomials over
- the rationals, compute its {\sl indefinite} integral, using if
- necessary, algebraic numbers. But in many circumstances an approximate
- result is more likely to be of use. Furthermore, it is plausible that
- it would be more useful to solve the problem to allow definite
- integration, or introduce additional parameters so that we can solve
- multiple definite integrations. How can a computer algebra system
- best answer the more useful questions? Finally, what if the integrand
- is not a ratio of polynomials, but something more challenging?"
+\bibitem[Berzins 87]{BBG87} Berzins M; Brankin R W; Gladwell I.
+``Design of the Stiff Integrators in the NAG Library''
+Technical Report. TR14/87 NAG. (1987)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@misc{Flet01,
- author = "Fletcher, John P.",
- title = "Symbolic processing of Clifford Numbers in C++",
- year = "2001",
- journal = "Paper 25, AGACSE 2001."
-}
+\begin{chunk}{ignore}
+\bibitem[Berzins 90]{Ber90} Berzins M
+``Developments in the NAG Library Software for Parabolic Equations''
+Scientific Software Systems. (ed J C Mason and M G Cox)
+Chapman and Hall. 59--72. (1990)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@misc{Flet09,
- author = "Fletcher, John P.",
- title = "Clifford Numbers and their inverses calculated using the matrix
- representation",
- publisher = "Chemical Engineering and Applied Chemistry, School of
- Engineering and Applied Science, Aston University, Aston Triangle,
- Birmingham B4 7 ET, U. K.",
- url =
- "http://www.ceac.aston.ac.uk/research/staff/jpf/papers/paper24/index.php"
-}
+\begin{chunk}{ignore}
+\bibitem[Birkhoff 62]{BR62} Birkhoff, G; Rota, G C
+``Ordinary Differential Equations''
+Ginn \& Co., Boston and New York. (1962)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Fletcher 81]{Fle81} Fletcher R
-``Practical Methods of Optimization''
-Vol 2. Constrained Optimization. Wiley. (1981)
+\bibitem[Boyd9 3a]{Boyd93a} Boyd, David W.
+``Bounds for the Height of a Factor of a Polynomial in
+Terms of Bombieri's Norms: I. The Largest Factor''
+J. Symbolic Computation (1993) 16, 115-130
+%\verb|axiom-developer.org/axiom-website/Boyd93a.pdf|
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Floy63,
- author = "Floyd, R. W.",
- title = "Semantic Analysis and Operator Precedence",
- journal = "JACM",
- volume = "10",
- number = "3",
- pages = "316-333",
- year = "1963"
-}
+\begin{chunk}{ignore}
+\bibitem[Boyd 93b]{Boyd93b} Boyd, David W.
+``Bounds for the Height of a Factor of a Polynomial in
+Terms of Bombieri's Norms: II. The Smallest Factor''
+J. Symbolic Computation (1993) 16, 131-145
+%\verb|axiom-developer.org/axiom-website/Boyd93b.pdf|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Forsythe 57]{For57} Forsythe G E,
-``Generation and use of orthogonal polynomials for data fitting
-with a digital computer''
-J. Soc. Indust. Appl. Math. 5 74--88. (1957)
+\bibitem[Braman 02a]{BBM02a} Braman, K.; Byers, R.; Mathias, R.
+``The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts,
+and Level 3 Performance''
+SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002.
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Fortenbacher 90]{REF-For90} Fortenbacher, A.
-``Efficient type inference and coercion in computer algebra''
-Design and Implementation of Symbolic Computation Systems (DISCO 90)
-A. Miola, (ed) vol 429 of Lecture Notes in Computer Science
-Springer-Verlag, pp56-60
- abstract = "
- Computer algebra systems of the new generation, like Scratchpad, are
- characterized by a very rich type concept, which models the
- relationship between mathematical domains of computation. To use these
- systems interactively, however, the user should be freed of type
- information. A type inference mechanism determines the appropriate
- function to call. All known models which allow to define a semantics
- for type inference cannot express the rich ``mathematical'' type
- structure, so presently type inference is done heuristically. The
- following paper defines a semantics for a subproblem thereof, namely
- coercion, which is based on rewrite rules. From this definition, and
- efficient coercion algorith for Scratchpad is constructed using graph
- techniques."
+\bibitem[Braman 02b]{BBM02b} Braman, K.; Byers, R.; Mathias, R.
+``The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation''
+SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Fox 68]{Fox68} Fox L.; Parker I B.
-``Chebyshev Polynomials in Numerical Analysis''
-Oxford University Press. (1968)
+\bibitem[Brent 75]{Bre75} Brent, R. P.
+``Multiple-Precision Zero-Finding Methods and the Complexity
+of Elementary Function Evaluation, Analytic Computational Complexity''
+J. F. Traub, Ed., Academic Press, New York 1975, 151-176
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Franke 80]{FN80} Franke R.; Nielson G
-``Smooth Interpolation of Large Sets of Scattered Data''
-Internat. J. Num. Methods Engrg. 15 1691--1704. (1980)
+\bibitem[Brent 78]{BK78} Brent, R. P.; Kung, H. T.
+``Fast Algorithms for Manipulating Formal Power Series''
+Journal of the Association for Computing Machinery,
+Vol. 25, No. 4, October 1978, 581-595
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Fritsch 82]{Fri82} Fritsch F N
-``PCHIP Final Specifications''
-Report UCID-30194. Lawrence Livermore National Laboratory. (1982)
+\bibitem[Brigham 73]{Bri73} Brigham E O
+``The Fast Fourier Transform''
+Prentice-Hall. (1973)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Fritsch 84]{FB84} Fritsch F N.; Butland J.
-``A Method for Constructing Local Monotone Piecewise Cubic Interpolants''
-SIAM J. Sci. Statist. Comput. 5 300--304. (1984)
+\bibitem[Brillhart 69]{Bri69} Brillhart, John
+``On the Euler and Bernoulli polynomials''
+J. Reine Angew. Math., v. 234, (1969), pp. 45-64
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Froberg 65]{Fro65} Froberg C E.
-``Introduction to Numerical Analysis''
-Addison-Wesley. 181--187. (1965)
+\bibitem[Brillhart 90]{Bri90} Brillhart, John
+``Note on Irreducibility Testing''
+Mathematics of Computation, vol. 35, num. 35, Oct. 1980, 1379-1381
\end{chunk}
-\subsection{G} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Garcia 95]{Ga95} Garcia, A.; Stichtenoth, H.
-``A tower of Artin-Schreier extensions of function fields attaining the
-Drinfeld-Vladut bound''
-Invent. Math., vol. 121, 1995, pp. 211--222.
+\bibitem[Bronstein 98a]{Bro98a} Bronstein, M.; Grabmeier, J.; Weispfenning, V. (eds)
+``Symbolic Rewriting Techniques''
+Progress in Computer Science and Applied Logic 15, Birkhauser-Verlag, Basel
+ISBN 3-7643-5901-3 (1998)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gathen 90a]{Gat90a} Gathen, Joachim von zur; Giesbrecht, Mark
-``Constructing Normal Bases in Finite Fields''
-J. Symbolic Computation pp 547-570 (1990)
-%\verb|axiom-developer.org/axiom-website/papers/Gat90a.pdf|
+\bibitem[Bronstein 88]{Bro88} Bronstein, Manual
+``The Transcendental Risch Differential Equation''
+J. Symbolic Computation (1990) 9, pp49-60 Feb 1988
+IBM Research Report RC13460 IBM Corp. Yorktown Heights, NY
+\verb|www.sciencedirect.com/science/article/pii/S0747717108800065|
+%\verb|axiom-developer.org/axiom-website/papers/Bro88.pdf|
abstract = "
- An efficient probabilistic algorithm to find a normal basis in a
- finite field is presented. It can, in fact, find an element of
- arbitrary prescribed additive order. It is based on a density estimate
- for normal elements. A similar estimate yields a probabilistic
- polynomial-time reduction from finding primitive normal elements to
- finding primitive elements."
+ We present a new rational algorithm for solving Risch differential
+ equations in towers of transcendental elementary extensions. In
+ contrast to a recent algorithm by Davenport we do not require a
+ progressive reduction of the denominators involved, but use weak
+ normality to obtain a formula for the denominator of a possible
+ solution. Implementation timings show this approach to be faster than
+ a Hermite-like reduction."
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Gathen 90]{Gat90} Gathen, Joachim von zur
-``Functional Decomposition Polynomials: the Tame Case''
-Journal of Symbolic Computation (1990) 9, 281-299
+\begin{chunk}{axiom.bib}
+@techreport{Bron98,
+ author = "Bronstein, Manuel",
+ title = "The lazy hermite reduction",
+ type = "Rapport de Recherche",
+ number = "RR-3562",
+ year = "1998",
+ institution = "French Institute for Research in Computer Science",
+ paper = "Bron98.pdf",
+ abstract = "
+ The Hermite reduction is a symbolic integration technique that reduces
+ algebraic functions to integrands having only simple affine
+ poles. While it is very effective in the case of simple radical
+ extensions, its use in more general algebraic extensions requires the
+ precomputation of an integral basis, which makes the reduction
+ impractical for either multiple algebraic extensions or complicated
+ ground fields. In this paper, we show that the Hermite reduction can
+ be performed without {\sl a priori} computation of either a primitive
+ element or integral basis, computing the smallest order necessary for
+ a particular integrand along the way."
+}
\end{chunk}
\begin{chunk}{axiom.bib}
-@book{Gath99,
- author = {{von zur Gathen}, Joachim and Gerhard, J\"urgen},
- title = "Modern Computer Algebra",
- publisher = "Cambridge University Press",
- year = "1999",
- isbn = "0-521-64176-4"
+@misc{Bro98b,
+ author = "Bronstein, Manuel",
+ title = "Symbolic Integration Tutorial",
+ series = "ISSAC'98",
+ year = "1998",
+ address = "INRIA Sophia Antipolis",
+ url =
+ "http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac98.pdf",
+ paper = "Bro98b.pdf"
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Gautschi 79a]{Gau79a} Gautschi W.
-``A Computational Procedure for Incomplete Gamma Functions''
-ACM Trans. Math. Softw. 5 466--481. (1979)
+\begin{chunk}{ignore}
+\bibitem[Brown 99]{Brow99} Brown, Christopher W.
+``Solution Formula Construction for Truth Invariant CADs''
+Ph.D Thesis, Univ. Delaware (1999)
+\verb|www.usna.edu/Users/cs/wcbrown/research/thesis.ps.gz|
+%\verb|axiom-developer.org/axiom-website/papers/Brow99.pdf|
+ abstract = "
+ The CAD-based quantifier elimination algorithm takes a formula from
+ the elementary theory of real closed fields as input, and constructs a
+ CAD of the space of the formula's unquantified variables. This
+ decomposition is truth invariant with respect to the input formula,
+ meaning that the formula is either identically true or identically
+ false in each cell of the decomposition. The method determines the
+ truth of the input formula for each cell of the CAD, and then uses the
+ CAD to construct a solution formula -- a quantifier free formula that
+ is equivalent to the input formula. This final phase of the algorithm,
+ the solution formula construction phase, is the focus of this thesis.
+
+ An optimal solution formula construction algorithm would be {\sl
+ complete} -- i.e. applicable to any truth-invariant CAD, would be {\sl
+ efficient}, and would produce {\sl simple} solution formulas. Prior to
+ this thesis, no method was available with even two of these three
+ properties. Several algorithms are presented, all addressing problems
+ related to solution formula construction. In combination, these
+ provide an efficient and complete method for constructing solution
+ formulas that are simple in a variety of ways.
+
+ Algorithms presented in this thesis have been implemented using the
+ SACLIB library, and integrated into QEPCAD, a SACLIB-based
+ implementation of quantifier elimination by CAD. Example computations
+ based on these implementations are discussed."
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Brown 02]{Brow02} Brown, Christopher W.
+``QEPCAD B -- A program for computing with semi-algebraic sets using CADs''
+%\verb|axiom-developer.org/axiom-website/papers/Brow02.pdf|
+ abstract = "
+ This report introduces QEPCAD B, a program for computing with real
+ algebraic sets using cylindrical algebraic decomposition (CAD). QEPCAD
+ B both extends and improves upon the QEPCAD system for quantifier
+ elimination by partial cylindrical algebraic decomposition written by
+ Hoon Hong in the early 1990s. This paper briefly discusses some of the
+ improvements in the implementation of CAD and quantifier elimination
+ vis CAD, and provides somewhat more detail on extensions to the system
+ that go beyond quantifier elimination. The author is responsible for
+ most of the extended features of QEPCAD B, but improvements to the
+ basic CAD implementation and to the SACLIB library on which QEPCAD is
+ based are the results of many people's work."
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@article{Burg74,
+ author = "William H. Burge",
+ title = "Stream Processing Functions",
+ year = "1974",
+ month = "January",
+ journal = "IBM Journal of Research and Development",
+ volume = "19",
+ issue = "1",
+ pages = "12-25",
+ papers = "Burg74.pdf",
+ abstract = "
+ One principle of structured programming is that a program should be
+ separated into meaningful independent subprograms, which are then
+ combined so that the relation of the parts to the whole can be clearly
+ established. This paper describes several alternative ways to compose
+ programs. The main method used is to permit the programmer to denote
+ by an expression the sequence of values taken on by a variable. The
+ sequence is represented by a function called a stream, which is a
+ functional analog of a coroutine. The conventional while and for loops
+ of structured programming may be composed by a technique of stream
+ processing (analogous to list processing), which results in more
+ structured programs than the orignals. This technique makes it
+ possible to structure a program in a natural way into its logically
+ separate parts, which can then be considered independently."
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Gautschi 79b]{Gau79b} Gautschi W.
-``Algorithm 542: Incomplete Gamma Functions''
-ACM Trans. Math. Softw. 5 482--489. (1979)
-
-\end{chunk}
+\subsection{C} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
-\bibitem[Gentlemen 69]{Gen69} Gentlemen W M
-``An Error Analysis of Goertzel's (Watt's) Method for Computing
-Fourier Coefficients''
-Comput. J. 12 160--165. (1969)
+\bibitem[Carlson 65]{Car65} Carlson B C
+``On Computing Elliptic Integrals and Functions''
+J Math Phys. 44 36--51. (1965)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gentleman 73]{Gen73} Gentleman W M.
-``Least-squares Computations by Givens Transformations without Square Roots''
-J. Inst. Math. Applic. 12 329--336. (1973)
+\bibitem[Carlson 77a]{Car77a} Carlson B C
+``Elliptic Integrals of the First Kind''
+SIAM J Math Anal. 8 231--242. (1977)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gentleman 74]{Gen74} Gentleman W M.
-``Algorithm AS 75. Basic Procedures for Large Sparse or
-Weighted Linear Least-squares Problems''
-Appl. Statist. 23 448--454. (1974)
+\bibitem[Carlson 77b]{Car77b} Carlson B C
+``Special Functions of Applied Mathematics''
+Academic Press. (1977)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gentlemen 74a]{GM74a} Gentleman W. M.; Marovich S. B.
-``More on algorithms that reveal properties of floating point
-arithmetic units''
-Comms. of the ACM, 17, 276-277. (1974)
+\bibitem[Carlson 78]{Car78} Carlson B C,
+``Computing Elliptic Integrals by Duplication''
+(Preprint) Department of Physics, Iowa State University. (1978)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Genz 80]{GM80} Genz A C.; Malik A A.
-``An Adaptive Algorithm for Numerical Integration over an N-dimensional
-Rectangular Region''
-J. Comput. Appl. Math. 6 295--302. (1980)
+\bibitem[Carlson 88]{Car88} Carlson B C,
+``A Table of Elliptic Integrals of the Third Kind''
+Math. Comput. 51 267--280. (1988)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gill 72]{GM72} Gill P E.; Miller G F.
-``An Algorithm for the Integration of Unequally Spaced Data''
-Comput. J. 15 80--83. (1972)
+\bibitem[Cauchy 1829]{Cau1829} Augustin-Lux Cauchy
+``Exercices de Math\'ematiques Quatri\`eme Ann\'ee. De Bure Fr\`eres''
+Paris 1829 (reprinted Oeuvres, II S\'erie, Tome IX,
+Gauthier-Villars, Paris, 1891).
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gill 74b]{GM74b} Gill P E.; Murray W. (eds)
-``Numerical Methods for Constrained Optimization''
-Academic Press. (1974)
+\bibitem[Ch\`eze 07]{Chez07} Ch\'eze, Guillaume; Lecerf, Gr\'egoire
+``Lifting and recombination techniques for absolute factorization''
+Journal of Complexity, VOl 23 Issue 3 June 2007 pp 380-420
+\verb|www.sciencedirect.com/science/article/pii/S0885064X07000465|
+%\verb|axiom-developer.org/axiom-website/papers/Chez07.pdf|
+ abstract = "
+ In the vein of recent algorithmic advances in polynomial factorization
+ based on lifting and recombination techniques, we present new faster
+ algorithms for computing the absolute factorization of a bivariate
+ polynomial. The running time of our probabilistic algorithm is less
+ than quadratic in the dense size of the polynomial to be factored."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gill 76a]{GM76a} Gill P E.; Murray W.
-``Minimization subject to bounds on the variables''
-Report NAC 72. National Physical Laboratory. (1976)
+\bibitem[Childs 79]{CSDDN79} Childs B; Scott M; Daniel J W; Denman E;
+Nelson P (eds)
+``Codes for Boundary-value Problems in Ordinary Differential Equations''
+Lecture Notes in Computer Science. 76 (1979) Springer-Verlag
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gill 76b]{GM76b} Gill P E.; Murray W.
-``Algorithms for the Solution of the Nonlinear Least-squares Problem''
-NAC 71 National Physical Laboratory. (1976)
+\bibitem[Clausen 89]{Cla89} Clausen, M.; Fortenbacher, A.
+``Efficient Solution of Linear Diophantine Equations''
+JSC (1989) 8, 201-216
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gill 78]{GM78} Gill P E.; Murray W.
-``Algorithms for the Solution of the Nonlinear Least-squares Problem''
-SIAM J. Numer. Anal. 15 977--992. (1978)
+\bibitem[Clenshaw 55]{Cle55} Clenshaw C W,
+``A Note on the Summation of Chebyshev Series''
+Math. Tables Aids Comput. 9 118--120. (1955)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gill 79]{GM79} Gill P E.; Murray W;
-``Conjugate-gradient Methods for Large-scale Nonlinear Optimization''
-Technical Report SOL 79-15. Department of Operations Research,
-Stanford University. (1979)
+\bibitem[Clenshaw 60]{Cle60} Clenshaw C W
+``Curve Fitting with a Digital Computer''
+Comput. J. 2 170--173. (1960)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gill 81]{GMW81} Gill P E.; Murray W.; Wright M H.
-``Practical Optimization''
-Academic Press. 1981
+\bibitem[Clenshaw 62]{Cle62} Clenshaw C W
+``Mathematical Tables. Chebyshev Series for Mathematical Functions''
+HMSO. (1962)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gill 82]{GMW82} Gill P E.; Murray W.; Saunders M A.; Wright M H.
-``The design and implementation of a quadratic programming algorithm''
-Report SOL 82-7. Department of Operations Research,
-Stanford University. (1982)
+\bibitem[Cline 84]{CR84} Cline A K; Renka R L,
+``A Storage-efficient Method for Construction of a Thiessen Triangulation''
+Rocky Mountain J. Math. 14 119--139. (1984)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gill 84a]{GMSW84a} Gill P E.; Murray W.; Saunders M A.; Wright M H
-``User's Guide for SOL/QPSOL Version 3.2''
-Report SOL 84-5. Department of Operations Research, Stanford University. 1984
+\bibitem[Conway 87]{CCNPW87} Conway, J.; Curtis, R.; Norton, S.; Parker, R.;
+Wilson, R.
+``Atlas of Finite Groups''
+Oxford, Clarendon Press, 1987
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gill 84b]{GMSW84b} Gill P E.; Murray W.; Saunders M A.; Wright M H
-``Procedures for Optimization Problems with a Mixture of
-Bounds and General Linear Constraints''
-ACM Trans. Math. Softw. 10 282--298. 1984
+\bibitem[Conway 03]{CS03} Conway, John H.; Smith, Derek, A.
+``On Quaternions and Octonions''
+A.K Peters, Natick, MA. (2003) ISBN 1-56881-134-9
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gill 86a]{GMSW86a} Gill P E.; Hammarling S.; Murray W.;
-Saunders M A.; Wright M H.
-``User's Guide for LSSOL (Version 1.0)''
-Report SOL 86-1. Department of Operations Research, Stanford University. 1986
+\bibitem[Cox 72]{Cox72} Cox M G
+``The Numerical Evaluation of B-splines''
+J. Inst. Math. Appl. 10 134--149. (1972)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gill 86b]{GMSW86b} Gill P E.; Murray W.; Saunders M A.; Wright M H.
-``Some Theoretical Properties of an Augmented Lagrangian Merit Function''
-Report SOL 86-6R. Department of Operations Research, Stanford University. 1986
+\bibitem[CH 73]{CH73} Cox M G; Hayes J G
+``Curve fitting: a guide and suite of algorithms for the
+non-specialist user''
+Report NAC26. National Physical Laboratory. (1973)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gladwell 79]{Gla79} Gladwell I
-``Initial Value Routines in the NAG Library''
-ACM Trans Math Softw. 5 386--400. (1979)
+\bibitem[Cox 74a]{Cox74a} Cox M G
+``A Data-fitting Package for the Non-specialist User''
+Software for Numerical Mathematics. (ed D J Evans) Academic Press. (1974)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gladwell 80]{GS80} Gladwell I.; Sayers D K
-``Computational Techniques for Ordinary Differential Equations''
-Academic Press. 1980
+\bibitem[Cox 74b]{Cox74b} Cox M G
+``Numerical methods for the interpolation and approximation of data
+by spline functions''
+PhD Thesis. City University, London. (1975)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gladwell 86]{Gla86} Gladwell I
-``Vectorisation of one dimensional quadrature codes''
-Techincal Report. TR7/86 NAG. (1986)
+\bibitem[Cox 75]{Cox75} Cox M G
+``An Algorithm for Spline Interpolation''
+J. Inst. Math. Appl. 15 95--108. (1975)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gladwell 87]{Gla87} Gladwell I
-``The NAG Library Boundary Value Codes''
-Numerical Analysis Report. 134 Manchester University. (1987)
+\bibitem[Cox 77]{Cox77} Cox M G
+``A Survey of Numerical Methods for Data and Function Approximation''
+The State of the Art in Numerical Analysis. (ed D A H Jacobs)
+Academic Press. 627--668. (1977)
+ keywords = "survey",
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Goedel 40]{God40} Goedel
-``The consistency of the continuum hypothesis''
-Ann. Math. Studies, Princeton Univ. Press, 1940
+\bibitem[Cox 78]{Cox78} Cox M G
+``The Numerical Evaluation of a Spline from its B-spline Representation''
+J. Inst. Math. Appl. 21 135--143. (1978)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Goldman 87]{Gold87} Goldman, L.
-``Integrals of multinomial systems of ordinary differential equations''
-J. of Pure and Applied Algebra, 45, 225-240 (1987)
-\verb|www.sciencedirect.com/science/article/pii/0022404987900727/pdf|
-\verb|?md5=5a0c70643eab514ccf47d80e4fc6ec5a&|
-\verb|pid=1-s2.0-0022404987900727-main.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Gold87.pdf|
+\bibitem[Curtis 74]{CPR74} Curtis A R; Powell M J D; Reid J K
+``On the Estimation of Sparse Jacobian Matrices''
+J. Inst. Maths Applics. 13 117--119. (1974)
\end{chunk}
+\subsection{D} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Gollan 90]{GG90} H. Gollan; J. Grabmeier
-``Algorithms in Representation Theory and
-their Realization in the Computer Algebra System Scratchpad''
-Bayreuther Mathematische Schriften, Heft 33, 1990, 1-23
+\bibitem[Dahlquist 74]{DB74} Dahlquist G; Bjork A
+``Numerical Methods''
+Prentice- Hall. (1974)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Golub 89]{GL89} Golub, Gene H.; Van Loan, Charles F.
-``Matrix Computations''
-Johns Hopkins University Press ISBN 0-8018-3772-3 (1989)
+\bibitem[Dalmas 98]{DA98} Dalmas, Stephane; Arsac, Olivier
+``The INRIA OpenMath Library''
+Projet SAFIR, INRIA Sophia Antipolis Nov 25, 1998
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Golub 96]{GL96} Golub, Gene H.; Van Loan, Charles F.
-``Matrix Computations''
-Johns Hopkins University Press ISBN 978-0-8018-5414-9 (1996)
+\bibitem[Dantzig 63]{Dan63} Dantzig G B
+``Linear Programming and Extensions''
+Princeton University Press. (1963)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Grabmeier]{Grab} Grabmeier, J.
-``On Plesken's root finding algorithm''
-in preparation
+\bibitem[Davenport]{Dav} Davenport, James
+``On Brillhart Irreducibility.''
+To appear.
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Grebmeier 87]{GK87} Grabmeier, J.; Kerber, A.;
-``The Evaluation of Irreducible Polynomial Representations of the General
-Linear Groups and of the Unitary Groups over Fields of Characteristic 0''
-Acta Appl. Math. 8 (1987), 271-291
+\bibitem[Davenport 93]{Ref-Dav93} Davenport, J.H.
+``Primality testing revisited''
+Technical Report TR2/93
+(ATR/6)(NP2556) Numerical Algorithms Group, Inc., Downer's Grove, IL, USA
+and Oxford, UK, August 1993
+\verb|www.nag.co.uk/doc/TechRep/axiomtr.html|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Grabmeier 92]{REF-GS92} Grabmeier, J.; Scheerhorn, A.
-``Finite fields in Axiom''
-AXIOM Technical Report TR7/92 (ATR/5)(NP2522),
-Numerical Algorithms Group, Inc., Downer's
-Grove, IL, USA and Oxford, UK, 1992
-\verb|www.nag.co.uk/doc/TechRep/axiomtr.html|
+\bibitem[Davis 67]{DR67} Davis P J; Rabinowitz P
+``Numerical Integration''
+Blaisdell Publishing Company. 33--52. (1967)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Granville 1911]{Gran1911} Granville, William Anthony
-``Elements of the Differential and Integral Calculus''
-\verb|djm.cc/library/Elements_Differential_Integral_Calculus_Granville_edited_2.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Gran1911.pdf|
+\bibitem[Davis 75]{DR75} Davis P J; Rabinowitz P
+``Methods of Numerical Integration''
+Academic Press. (1975)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Gruntz 93]{Gru93} Gruntz, Dominik
-``Limit computation in computer algebra''
-\verb|algo.inria.fr/seminars/sem92-93/gruntz.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Gru93.pdf|
- abstract = "
- The automatic computation of limits can be reduced to two main
- sub-problems. The first one is asymptotic comparison where one must
- decide automatically which one of two functions in a specified class
- dominates the other one asymptotically. The second one is asymptotic
- cancellation and is usually exemplified by
- \[e^x[exp(1/x+e^{-x})-exp(1/x)],\quad{}x \leftarrow \infty\]
+\bibitem[DeBoor 72]{DeB72} De Boor C
+``On Calculating with B-splines''
+J. Approx. Theory. 6 50--62. (1972)
- In this example, if the sum is expanded in powers of $1/x$, the
- expansion always yields $O(x^{-k})$, and this is not enough to
- conclude.
+\end{chunk}
- In 1990, J.Shackell found an algorithm that solved both these problems
- for the case of $exp-log$ functions, i.e. functions build by recursive
- application of exponential, logarithm, algebraic extension and field
- operations to one variable and the rational numbers. D. Gruntz and
- G. Gonnet propose a slightly different algorithm for exp-log
- functions. Extensions to larger classes of functions are also
- discussed."
+\begin{chunk}{ignore}
+\bibitem[De Doncker 78]{DeD78} De Doncker E,
+``An Adaptive Extrapolation Algorithm for Automatic Integration''
+Signum Newsletter. 13 (2) 12--18. (1978)
\end{chunk}
-\subsection{H} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{chunk}{axiom.bib}
-@article{Hach95,
- author = "Hach\'e, G. and Le Brigand, D.",
- title = "Effective construction of algebraic geometry codes",
- journal = "IEEE Transaction on Information Theory",
- volume = "41",
- month = "November",
- year = "1995",
- pages = "1615--1628"
-}
+\begin{chunk}{ignore}
+\bibitem[Demmel 89]{Dem89} Demmel J W
+``On Floating-point Errors in Cholesky''
+LAPACK Working Note No. 14. University of Tennessee, Knoxville. 1989
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Hach95a,
- author = "Hach\'e, G.",
- title = "Computation in algebraic function fields for effective
- construction of algebraic-geometric codes",
- journal = "Lecture Notes in Computer Science",
- volume = "948",
- year = "1995",
- pages = "262--278"
-}
+\begin{chunk}{ignore}
+\bibitem[Dennis 77]{DM77} Dennis J E Jr; More J J
+``Quasi-Newton Methods, Motivation and Theory''
+SIAM Review. 19 46--89. 1977
\end{chunk}
-\begin{chunk}{axiom.bib}
-@phdthesis{Hach96,
- author = "Hach\'e, G.",
- title = "Construction effective des codes g\'eom\'etriques",
- school = "l'Universit\'e Pierre et Marie Curie (Paris 6)",
- year = "1996",
- month = "Septembre"
-}
+\begin{chunk}{ignore}
+\bibitem[Dennis 81]{DS81} Dennis J E Jr; Schnabel R B
+``A New Derivation of Symmetric Positive-Definite Secant Updates''
+Nonlinear Programming 4. (ed O L Mangasarian, R R Meyer and S M. Robinson)
+Academic Press. 167--199. (1981)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Hall 76]{HW76} Hall G.; Watt J M. (eds),
-``Modern Numerical Methods for Ordinary Differential Equations''
-Clarendon Press. (1976)
+\bibitem[Dennis 83]{DS83} Dennis J E Jr; Schnabel R B
+``Numerical Methods for Unconstrained Optimixation and Nonlinear Equations''
+Prentice-Hall.(1983)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Hamdy 04]{Ham04} Hamdy, S.
-``LiDIA A library for computational number theory''
-Reference manual Edition 2.1.1 May 2004
-\verb|www.cdc.informatik.tu-darmstadt.de/TI/LiDIA|
+\bibitem[Dierckx 75]{Die75} Dierckx P
+``An Algorithm for Smoothing, Differentiating and Integration of
+Experimental Data Using Spline Functions''
+J. Comput. Appl. Math. 1 165--184. (1975)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Hammarling 85]{Ham85} Hammarling S.
-`` The Singular Value Decomposition in Multivariate Statistics''
-ACM Signum Newsletter. 20, 3 2--25. (1985)
+\bibitem[Dierckx 81]{Die81} Dierckx P
+``An Improved Algorithm for Curve Fitting with Spline Functions''
+Report TW54. Dept. of Computer Science, Katholieke Universiteit Leuven. 1981
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Hammersley 67]{HH67} Hammersley J M; Handscomb D C.
-``Monte-Carlo Methods''
-Methuen. (1967)
+\bibitem[Dierckx 82]{Die82} Dierckx P
+``A Fast Algorithm for Smoothing Data on a Rectangular Grid while using
+Spline Functions''
+SIAM J. Numer. Anal. 19 1286--1304. (1982)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@misc{Hath1896,
- author = "Hathway, Arthur S.",
- title = "A Primer Of Quaternions",
- year = "1896"
-}
+\begin{chunk}{ignore}
+\bibitem[Dongarra 79]{DMBS79} Dongarra J J; Moler C B; Bunch J R;
+Stewart G W
+``LINPACK Users' Guide''
+SIAM, Philadelphia. (1979)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@book{Haya05,
- author = "Hayashi, K. and Kangkook, J. and Lascu, O. and Pienaar, H. and
- Schreitmueller, S. and Tarquinio, T. and Thompson, J.",
- title = "AIX 5L Practical Performance Tools and Tuning Guide",
- publisher = "IBM",
- year = "2005",
- url = "http://www.redbooks.ibm.com/redbooks/pdfs/sg246478.pdf",
- paper = "Haya05.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Dongarra 85]{DCHH85} Dongarra J J; Du Croz J J; Hammarling S;
+Hanson R J
+``A Proposal for an Extended set of Fortran Basic Linear
+Algebra Subprograms''
+SIGNUM Newsletter. 20 (1) 2--18. (1985)
\end{chunk}
+
\begin{chunk}{ignore}
-\bibitem[Hayes 70]{Hay70} Hayes J G.
-``Curve Fitting by Polynomials in One Variable''
-Numerical Approximation to Functions and Data.
-(ed J G Hayes) Athlone Press, London. (1970)
+\bibitem[Dongarra 88]{REF-DON88} Dongarra, Jack J.; Du Croz, Jeremy;
+Hammarling, Sven; Hanson, Richard J.
+``An Extended Set of FORTRAN Basic Linear Algebra Subroutines''
+ACM Transactions on Mathematical Software, Vol 14, No 1, March 1988,
+pp 1-17
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Hayes 74]{Hay74} Hayes J G.
-``Numerical Methods for Curve and Surface Fitting''
-Bull Inst Math Appl. 10 144--152. (1974)
+\bibitem[Dongarra 88a]{REF-DON88a} Dongarra, Jack J.; Du Croz, Jeremy;
+Hammarling, Sven; Hanson, Richard J.
+``ALGORITHM 656: An Extended Set of Basic Linear Algebra Subprograms:
+Model Implementation and Test Programs''
+ACM Transactions on Mathematical Software, Vol 14, No 1, March 1988,
+pp 18-32
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Hayes 74a]{HH74} Hayes J G.; Halliday J,
-``The Least-squares Fitting of Cubic Spline Surfaces to General Data Sets''
-J. Inst. Math. Appl. 14 89--103. (1974)
+\bibitem[Dongarra 90]{REF-DON90} Dongarra, Jack J.; Du Croz, Jeremy;
+Hammarling, Sven; Duff, Iain S.
+``A Set of Level 3 Basic Linear Algebra Subprograms''
+ACM Transactions on Mathematical Software, Vol 16, No 1, March 1990,
+pp 1-17
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Henrici 56]{Hen56} Henrici, Peter
-``Automatic Computations with Power Series''
-Journal of the Association for Computing Machinery, Volume 3, No. 1,
-January 1956, 10-15
+\bibitem[Dongarra 90a]{REF-DON90a} Dongarra, Jack J.; Du Croz, Jeremy;
+Hammarling, Sven; Duff, Iain S.
+``ALGORITHM 679: A Set of Level 3 Basic Linear Algebra Subprograms:
+Model Implementation and Test Programs''
+ACM Transactions on Mathematical Software, Vol 16, No 1, March 1990,
+pp 18-28
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Higham 88]{Hig88} Higham, N.J.
-``FORTRAN codes for estimating the one-norm of a
-real or complex matrix, with applications to condition estimation''
-ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.
+\bibitem[Ducos 00]{Duc00} Ducos, Lionel
+``Optimizations of the subresultant algorithm''
+Journal of Pure and Applied Algebra V145 No 2 Jan 2000 pp149-163
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Higham 02]{Hig02} Higham, Nicholas J.
-``Accuracy and stability of numerical algorithms''
-SIAM Philadelphia, PA ISBN 0-89871-521-0 (2002)
+\bibitem[Duff 77]{Duff77} Duff I S,
+``MA28 -- a set of Fortran subroutines for sparse unsymmetric linear
+equations''
+A.E.R.E. Report R.8730. HMSO. (1977)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Hock 81]{HS81} Hock W.; Schittkowski K.
-``Test Examples for Nonlinear Programming Codes''
-Lecture Notes in Economics and Mathematical Systems. 187 Springer-Verlag. 1981
+\bibitem[Duval 96a]{Duva96a} Duval, D.; Gonz\'alez-Vega, L.
+``Dynamic Evaluation and Real Closure''
+Mathematics and Computers in Simulation 42 pp 551-560 (1996)
+%\verb|axiom-developer.org/axiom-website/papers/Duva96a.pdf|
+ abstract = "
+ The aim of this paper is to present how the dynamic evaluation method
+ can be used to deal with the real closure of an ordered field. Two
+ kinds of questions, or tests, may be asked in an ordered field:
+ equality tests $(a=b?)$ and sign tests $(a > b?)$. Equality tests are
+ handled through splittings, exactly as in the algebraic closure of a
+ field. Sign tests are handled throug a structure called ``Tarski data
+ type''."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Householder 70]{Hou70} Householder A S.
-``The Numerical Treatment of a Single Nonlinear Equation''
-McGraw-Hill. (1970)
+\bibitem[Duval 96]{Duva96} Duval, D.; Reynaud, J.C.
+``Sketches and Computations over Fields''
+Mathematics and Computers in Simulation 42 pp 363-373 (1996)
+%\verb|axiom-developer.org/axiom-website/papers/Duva96.pdf|
+ abstract = "
+ The goal of this short paper is to describe one possible use of
+ sketches in computer algebra. We show that sketches are a powerful
+ tool for the description of mathematical structures and for the
+ description of computations."
\end{chunk}
-\begin{chunk}{axiom.bib}
-@book{Hous81,
- author = "Householder, Alston S.",
- title = "Principles of Numerical Analysis",
- publisher = "Dover Publications, Mineola, NY",
- year = "1981",
- isbn = "0-486-45312-X"
-}
+\begin{chunk}{ignore}
+\bibitem[Duval 94a]{Duva94a} Duval, D.; Reynaud, J.C.
+``Sketches and Computation (Part I): Basic Definitions and Static Evaluation''
+Mathematical Structures in Computer Science, 4, p 185-238 Cambridge University Press (1994)
+\verb|journals.cambridge.org/abstract_S0960129500000438|
+%\verb|axiom-developer.org/axiom-website/papers/Duva94a.pdf|
+ abstract = "
+ We define a categorical framework, based on the notion of {\sl
+ sketch}, for specification and evaluation in the sense of algebraic
+ specifications and algebraic programming. This framework goes far
+ beyond our initial motivations, which was to specify computation with
+ algebraic numbers. We begin by redefining sketches in order to deal
+ explicitly with programs. Expressions and terms are carefully defined
+ and studied, then {\sl quasi-projective sketches} are introduced. We
+ describe {\sl static evaluation} in these sketches: we propose a
+ rigorous basis for evaluation in the corresponding structures. These
+ structures admit an initial model, but are not necessarily
+ equational. In Part II (Duval and Reynaud 1994), we study a more
+ general process, called {\sl dynamic evaluation}, for structures that
+ may have no initial model."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Huang 96]{HI96} Huang, M.D.; Ierardi, D.
-``Efficient algorithms for Riemann-Roch problem and for addition in the
-jacobian of a curve''
-Proceedings 32nd Annual Symposium on Foundations of Computer Sciences.
-IEEE Comput. Soc. Press, pp. 678--687.
+\bibitem[Duval 94b]{Duva94b} Duval, D.; Reynaud, J.C.
+``Sketches and Computation (Part II): Dynamic Evaluation and Applications''
+Mathematical Structures in Computer Science, 4, p 239-271. Cambridge University Press (1994)
+\verb|journals.cambridge.org/abstract_S096012950000044X|
+%\verb|axiom-developer.org/axiom-website/papers/Duva94b.pdf|
+ abstract = "
+ In the first part of this paper (Duval and Reynaud 1994), we defined a
+ categorical framework, based on the notion of {\sl sketch}, for
+ specification and evaluation in the senses of algebraic specification
+ and algebraic programming. {\sl Static evaluation} in {\sl
+ quasi-projective sketches} was defined in Part I; in this paper, {\sl
+ dynamic evaluation} is introduced. It deals with more general
+ structures, which may have no initial model. Until now, this process
+ has not been used in algebraic specification systems, but computer
+ algebra systems are beginning to use it as a basic tool. Finally, we
+ give some applications of dynamic evaluation to computation in field
+ extensions."
\end{chunk}
-\subsection{I} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[IBM]{IBM}.
-SCRIPT Mathematical Formula Formatter User's Guide, SH20-6453,
-IBM Corporation, Publishing Systems Information Development,
-Dept. G68, P.O. Box 1900, Boulder, Colorado, USA 80301-9191.
+\bibitem[Duval 94c]{Duva94c} Duval, Dominique
+``Algebraic Numbers: An Example of Dynamic Evaluation''
+J. Symbolic Computation 18, 429-445 (1994)
+\verb|www.sciencedirect.com/science/article/pii/S0747717106000551|
+%\verb|axiom-developer.org/axiom-website/papers/Duva94c.pdf|
+ abstract = "
+ Dynamic evaluation is presented through examples: computations
+ involving algebraic numbers, automatic case discussion according to
+ the characteristic of a field. Implementation questions are addressed
+ too. Finally, branches are presented as ``dual'' to binary functions,
+ according to the approach of sketch theory."
\end{chunk}
+\subsection{F} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Itoh 88]{Itoh88} Itoh, T.;, Tsujii, S.
-``A fast algorithm for computing multiplicative inverses
-in $GF(2^m)$ using normal bases''
-Inf. and Comp. 78, pp.171-177, 1988
-%\verb|axiom-developer.org/axiom-website/Itoh88.pdf|
+\bibitem[Fateman 08]{Fat08} Fateman, Richard
+``Revisiting numeric/symbolic indefinite integration of rational functions, and extensions''
+\verb|www.eecs.berkeley.edu/~fateman/papers/integ.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Fat08.pdf|
abstract = "
- This paper proposes a fast algorithm for computing multiplicative
- inverses in $GF(2^m)$ using normal bases. Normal bases have the
- following useful property: In the case that an element $x$ in
- $GF(2^m)$ is represented by normal bases, $2^k$ power operation of an
- element $x$ in $GF(2^m)$ can be carried out by $k$ times cyclic shift
- of its vector representation. C.C. Wang et al. proposed an algorithm
- for computing multiplicative inverses using normal bases, which
- requires $(m-2)$ multiplications in $GF(2^m)$ and $(m-1)$ cyclic
- shifts. The fast algorithm proposed in this paper also uses normal
- bases, and computes multiplicative inverses iterating multiplications
- in $GF(2^m)$. It requires at most $2[log_2(m-1)]$ multiplications in
- $GF(2^m)$ and $(m-1)$ cyclic shifts, which are much less than those
- required in Wang's method. The same idea of the proposed fast
- algorithm is applicable to the general power operation in $GF(2^m)$
- and the computation of multiplicative inverses in $GF(q^m)$
- $(q=2^n)$."
+ We know we can solve this problem: Given any rational function
+ $f(x)=p(x)/q(x)$, where $p$ and $q$ are univariate polynomials over
+ the rationals, compute its {\sl indefinite} integral, using if
+ necessary, algebraic numbers. But in many circumstances an approximate
+ result is more likely to be of use. Furthermore, it is plausible that
+ it would be more useful to solve the problem to allow definite
+ integration, or introduce additional parameters so that we can solve
+ multiple definite integrations. How can a computer algebra system
+ best answer the more useful questions? Finally, what if the integrand
+ is not a ratio of polynomials, but something more challenging?"
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Iyanaga 77]{Iya77} Iyanaga, Shokichi; Iyanaga, Yukiyosi Kawada
-``Encyclopedic Dictionary of Mathematics''
-1977
+\begin{chunk}{axiom.bib}
+@misc{Flet01,
+ author = "Fletcher, John P.",
+ title = "Symbolic processing of Clifford Numbers in C++",
+ year = "2001",
+ journal = "Paper 25, AGACSE 2001."
+}
\end{chunk}
-\subsection{J} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{chunk}{axiom.bib}
+@misc{Flet09,
+ author = "Fletcher, John P.",
+ title = "Clifford Numbers and their inverses calculated using the matrix
+ representation",
+ publisher = "Chemical Engineering and Applied Chemistry, School of
+ Engineering and Applied Science, Aston University, Aston Triangle,
+ Birmingham B4 7 ET, U. K.",
+ url =
+ "http://www.ceac.aston.ac.uk/research/staff/jpf/papers/paper24/index.php"
+}
+
+\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jacobson 68]{Jac68} Jacobson, N.
-``Structure and Representations of Jordan Algebras''
-AMS, Colloquium Publications Volume 39
+\bibitem[Fletcher 81]{Fle81} Fletcher R
+``Practical Methods of Optimization''
+Vol 2. Constrained Optimization. Wiley. (1981)
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[James 81]{JK81} James, Gordon; Kerber, Adalbert
-``The Representation Theory of the Symmetric Group''
-Encyclopedia of Mathematics and its Applications Vol. 16
-Addison-Wesley, 1981
+\begin{chunk}{axiom.bib}
+@article{Floy63,
+ author = "Floyd, R. W.",
+ title = "Semantic Analysis and Operator Precedence",
+ journal = "JACM",
+ volume = "10",
+ number = "3",
+ pages = "316-333",
+ year = "1963"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jaswon 77]{JS77} Jaswon, M A.; Symm G T.
-``Integral Equation Methods in Potential Theory and Elastostatics''
-Academic Press. (1977)
+\bibitem[Forsythe 57]{For57} Forsythe G E,
+``Generation and use of orthogonal polynomials for data fitting
+with a digital computer''
+J. Soc. Indust. Appl. Math. 5 74--88. (1957)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jeffrey 04]{Je04} Jeffrey, Alan
-``Handbook of Mathematical Formulas and Integrals''
-Third Edition, Elsevier Academic Press ISBN 0-12-382256-4
+\bibitem[Fortenbacher 90]{REF-For90} Fortenbacher, A.
+``Efficient type inference and coercion in computer algebra''
+Design and Implementation of Symbolic Computation Systems (DISCO 90)
+A. Miola, (ed) vol 429 of Lecture Notes in Computer Science
+Springer-Verlag, pp56-60
+ abstract = "
+ Computer algebra systems of the new generation, like Scratchpad, are
+ characterized by a very rich type concept, which models the
+ relationship between mathematical domains of computation. To use these
+ systems interactively, however, the user should be freed of type
+ information. A type inference mechanism determines the appropriate
+ function to call. All known models which allow to define a semantics
+ for type inference cannot express the rich ``mathematical'' type
+ structure, so presently type inference is done heuristically. The
+ following paper defines a semantics for a subproblem thereof, namely
+ coercion, which is based on rewrite rules. From this definition, and
+ efficient coercion algorith for Scratchpad is constructed using graph
+ techniques."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Jenning 66]{Jen66} Jennings A
-``A Compact Storage Scheme for the Solution of Symmetric Linear
-Simultaneous Equations''
-Comput. J. 9 281--285. (1966)
+\bibitem[Fox 68]{Fox68} Fox L.; Parker I B.
+``Chebyshev Polynomials in Numerical Analysis''
+Oxford University Press. (1968)
\end{chunk}
-\subsection{K} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{chunk}{ignore}
+\bibitem[Franke 80]{FN80} Franke R.; Nielson G
+``Smooth Interpolation of Large Sets of Scattered Data''
+Internat. J. Num. Methods Engrg. 15 1691--1704. (1980)
+
+\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Kalkbrener 91]{Kal91} Kalkbrener, M.
-``Three contributions to elimination theory''
-Ph. D. Thesis, University of Linz, Austria, 1991
+\bibitem[Fritsch 82]{Fri82} Fritsch F N
+``PCHIP Final Specifications''
+Report UCID-30194. Lawrence Livermore National Laboratory. (1982)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Kalkbrener 98]{Kal98} Kalkbrener, M.
-``Algorithmic properties of polynomial rings''
-Journal of Symbolic Computation 1998
+\bibitem[Fritsch 84]{FB84} Fritsch F N.; Butland J.
+``A Method for Constructing Local Monotone Piecewise Cubic Interpolants''
+SIAM J. Sci. Statist. Comput. 5 300--304. (1984)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Kantor 89]{Kan89} Kantor,I.L.; Solodovnikov, A.S.
-``Hypercomplex Numbers''
-Springer Verlag Heidelberg, 1989, ISBN 0-387-96980-2
+\bibitem[Froberg 65]{Fro65} Froberg C E.
+``Introduction to Numerical Analysis''
+Addison-Wesley. 181--187. (1965)
\end{chunk}
+\subsection{G} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Kaufmann 00]{KMJ00} Kaufmann, Matt; Manolios, Panagiotis;
-Moore J Strother
-``Computer-Aided Reasoning: An Approach''
-Springer, July 31. 2000 ISBN 0792377443
+\bibitem[Garcia 95]{Ga95} Garcia, A.; Stichtenoth, H.
+``A tower of Artin-Schreier extensions of function fields attaining the
+Drinfeld-Vladut bound''
+Invent. Math., vol. 121, 1995, pp. 211--222.
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Knuth 71]{Knu71} Knuth, Donald
-``The Art of Computer Programming''
-2nd edition Vol. 2 (Seminumerical Algorithms) 1st edition, 2nd printing,
-Addison-Wesley 1971, p. 397-398
+\bibitem[Gathen 90a]{Gat90a} Gathen, Joachim von zur; Giesbrecht, Mark
+``Constructing Normal Bases in Finite Fields''
+J. Symbolic Computation pp 547-570 (1990)
+%\verb|axiom-developer.org/axiom-website/papers/Gat90a.pdf|
+ abstract = "
+ An efficient probabilistic algorithm to find a normal basis in a
+ finite field is presented. It can, in fact, find an element of
+ arbitrary prescribed additive order. It is based on a density estimate
+ for normal elements. A similar estimate yields a probabilistic
+ polynomial-time reduction from finding primitive normal elements to
+ finding primitive elements."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Knuth 84]{Knu84} Knuth, Donald
-{\it The \TeX{}book}.
-Reading, Massachusetts, Addison-Wesley Publishing Company, Inc.,
-1984. ISBN 0-201-13448-9
+\bibitem[Gathen 90]{Gat90} Gathen, Joachim von zur
+``Functional Decomposition Polynomials: the Tame Case''
+Journal of Symbolic Computation (1990) 9, 281-299
\end{chunk}
\begin{chunk}{axiom.bib}
-@book{Knut92,
- author = "Knuth, Donald E.",
- title = "Literate Programming",
- publisher = "Center for the Study of Language and Information, Stanford CA",
- year = "1992",
- isbn = "0-937073-81-4"
-}
+@book{Gath99,
+ author = {{von zur Gathen}, Joachim and Gerhard, J\"urgen},
+ title = "Modern Computer Algebra",
+ publisher = "Cambridge University Press",
+ year = "1999",
+ isbn = "0-521-64176-4"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Knu98]{Knu98} Donald Knuth
-``The Art of Computer Programming'' Vol. 3
-(Sorting and Searching)
-Addison-Wesley 1998
+\bibitem[Gautschi 79a]{Gau79a} Gautschi W.
+``A Computational Procedure for Incomplete Gamma Functions''
+ACM Trans. Math. Softw. 5 466--481. (1979)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Kobayashi 89]{Koba89} Kobayashi, H.; Moritsugu, S.; Hogan, R.W.
-``On Radical Zero-Dimensional Ideals''
-J. Symbolic Computations 8, 545-552 (1989)
-\verb|www.sciencedirect.com/science/article/pii/S0747717189800604/pdf|
-\verb|?md5=f06dc6269514c90dcae57f0184bcbe65&|
-\verb|pid=1-s2.0-S0747717189800604-main.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Koba88.pdf|
+\bibitem[Gautschi 79b]{Gau79b} Gautschi W.
+``Algorithm 542: Incomplete Gamma Functions''
+ACM Trans. Math. Softw. 5 482--489. (1979)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Kolchin 73]{Kol73} Kolchin, E.R.
-``Differential Algebra and Algebraic Groups''
-(Academic Press, 1973).
+\bibitem[Gentlemen 69]{Gen69} Gentlemen W M
+``An Error Analysis of Goertzel's (Watt's) Method for Computing
+Fourier Coefficients''
+Comput. J. 12 160--165. (1969)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Koutschan 10]{Kou10} Koutschan, Christoph
-``Axiom / FriCAS''
-\verb|www.risc.jku.at/education/courses/ws2010/cas/axiom.pdf|
+\bibitem[Gentleman 73]{Gen73} Gentleman W M.
+``Least-squares Computations by Givens Transformations without Square Roots''
+J. Inst. Math. Applic. 12 329--336. (1973)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Kozen 86]{KL86} Kozen, Dexter; Landau, Susan
-``Polynomial Decomposition Algorithms''
-Journal of Symbolic Computation (1989) 7, 445-456
+\bibitem[Gentleman 74]{Gen74} Gentleman W M.
+``Algorithm AS 75. Basic Procedures for Large Sparse or
+Weighted Linear Least-squares Problems''
+Appl. Statist. 23 448--454. (1974)
\end{chunk}
-\subsection{L} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-
-\begin{chunk}{axiom.bib}
-@book{Lamp86,
- author = "Lamport, Leslie",
- title = "LaTeX: A Document Preparation System",
- publisher = "Addison-Wesley Publishing Company, Reading, Massachusetts",
- year = "1986",
- isbn = "0-201-15790-X"
-}
+\begin{chunk}{ignore}
+\bibitem[Gentlemen 74a]{GM74a} Gentleman W. M.; Marovich S. B.
+``More on algorithms that reveal properties of floating point
+arithmetic units''
+Comms. of the ACM, 17, 276-277. (1974)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lautrup 71]{Lau71} Lautrup B.
-``An Adaptive Multi-dimensional Integration Procedure''
-Proc. 2nd Coll. on Advanced Methods in Theoretical Physics, Marseille. (1971)
+\bibitem[Genz 80]{GM80} Genz A C.; Malik A A.
+``An Adaptive Algorithm for Numerical Integration over an N-dimensional
+Rectangular Region''
+J. Comput. Appl. Math. 6 295--302. (1980)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lawson 77]{Law77} Lawson C L.
-``Software for C Surface Interpolation''
-Mathematical Software III. (ed J R Rice) Academic Press. 161--194. (1977)
+\bibitem[Gill 72]{GM72} Gill P E.; Miller G F.
+``An Algorithm for the Integration of Unequally Spaced Data''
+Comput. J. 15 80--83. (1972)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lawson 74]{LH74} Lawson C L.; Hanson R J.
-``Solving Least-squares Problems''
-Prentice-Hall. (1974)
+\bibitem[Gill 74b]{GM74b} Gill P E.; Murray W. (eds)
+``Numerical Methods for Constrained Optimization''
+Academic Press. (1974)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Laws79,
- author = "Lawson, C.L. and Hanson R.J. and Kincaid, D.R. and Krogh, F.T.",
- title = "Algorithm 539: Basic linear algebra subprograms for FORTRAN usage",
- journal = "ACM Transactions on Mathematical Software",
- volume = "5",
- number = "3",
- month = "September",
- year = "1979",
- pages = "308-323"
-}
+\begin{chunk}{ignore}
+\bibitem[Gill 76a]{GM76a} Gill P E.; Murray W.
+``Minimization subject to bounds on the variables''
+Report NAC 72. National Physical Laboratory. (1976)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lawson 79]{LHKK79} Lawson C L; Hanson R J; Kincaid D R;
- Krogh F T
-``Basic Linear Algebra Subprograms for Fortran Usage''
-ACM Trans. Math. Softw. 5 308--325. (1979)
+\bibitem[Gill 76b]{GM76b} Gill P E.; Murray W.
+``Algorithms for the Solution of the Nonlinear Least-squares Problem''
+NAC 71 National Physical Laboratory. (1976)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lazard 91]{Laz91} Lazard, D.
-``A new method for solving algebraic systems of positive dimension''
-Discr. App. Math. 33:147-160,1991
+\bibitem[Gill 78]{GM78} Gill P E.; Murray W.
+``Algorithms for the Solution of the Nonlinear Least-squares Problem''
+SIAM J. Numer. Anal. 15 977--992. (1978)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lazard92]{Laz92} Lazard, D.
-``Solving Zero-dimensional Algebraic Systems''
-Journal of Symbolic Computation, 1992, 13, 117-131
+\bibitem[Gill 79]{GM79} Gill P E.; Murray W;
+``Conjugate-gradient Methods for Large-scale Nonlinear Optimization''
+Technical Report SOL 79-15. Department of Operations Research,
+Stanford University. (1979)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Laza90,
- author = "Lazard, Daniel and Rioboo, Renaud",
- title = "Integration of rational functions: Rational computation of the
- logarithmic part",
- journal = "Journal of Symbolic Computation",
- volume = "9",
- number = "2",
- year = "1990",
- month = "February",
- pages = "113-115",
- keywords = "axiomref",
- paper = "Laza90.pdf",
- abstract = "
- A new formula is given for the logarithmic part of the integral of a
- rational function, one that strongly improves previous algorithms and
- does not need any computation in an algebraic extension of the field
- of constants, nor any factorisation since only polynomial arithmetic
- and GCD computations are used. This formula was independently found
- and implemented in SCRATCHPAD by B.M. Trager."
-}
+\begin{chunk}{ignore}
+\bibitem[Gill 81]{GMW81} Gill P E.; Murray W.; Wright M H.
+``Practical Optimization''
+Academic Press. 1981
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{LeBr88,
- author = "Le Brigand, D.; Risler, J.J.",
- title = "Algorithme de Brill-Noether et codes de Goppa",
- journal = "Bull. Soc. Math. France",
- volume = "116",
- year = "1988",
- pages = "231--253"
-}
+\begin{chunk}{ignore}
+\bibitem[Gill 82]{GMW82} Gill P E.; Murray W.; Saunders M A.; Wright M H.
+``The design and implementation of a quadratic programming algorithm''
+Report SOL 82-7. Department of Operations Research,
+Stanford University. (1982)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@book{Lege11,
- author = "Legendre, George L. and Grazini, Stefano",
- title = "Pasta by Design",
- publisher = "Thames and Hudson",
- isbn = "978-0-500-51580-8",
- year = "2011"
-}
+\begin{chunk}{ignore}
+\bibitem[Gill 84a]{GMSW84a} Gill P E.; Murray W.; Saunders M A.; Wright M H
+``User's Guide for SOL/QPSOL Version 3.2''
+Report SOL 84-5. Department of Operations Research, Stanford University. 1984
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lenstra 87]{LS87} Lenstra, H. W.; Schoof, R. J.
-``Primitivive Normal Bases for Finite Fields''
-Math. Comp. 48, 1987, pp. 217-231
+\bibitem[Gill 84b]{GMSW84b} Gill P E.; Murray W.; Saunders M A.; Wright M H
+``Procedures for Optimization Problems with a Mixture of
+Bounds and General Linear Constraints''
+ACM Trans. Math. Softw. 10 282--298. 1984
\end{chunk}
-\begin{chunk}{axiom.bib}
-@misc{Leop03,
- author = "Leopardi, Paul",
- title = "A quick introduction to Clifford Algebras",
- publisher = "School of Mathematics, University of New South Wales",
- year = "2003",
- paper = "Leop03.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Gill 86a]{GMSW86a} Gill P E.; Hammarling S.; Murray W.;
+Saunders M A.; Wright M H.
+``User's Guide for LSSOL (Version 1.0)''
+Report SOL 86-1. Department of Operations Research, Stanford University. 1986
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lewis 77]{Lew77} Lewis J G,
-``Algorithms for sparse matrix eigenvalue problems''
-Technical Report STAN-CS-77-595. Computer Science Department,
-Stanford University. (1977)
+\bibitem[Gill 86b]{GMSW86b} Gill P E.; Murray W.; Saunders M A.; Wright M H.
+``Some Theoretical Properties of an Augmented Lagrangian Merit Function''
+Report SOL 86-6R. Department of Operations Research, Stanford University. 1986
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lidl 83]{LN83} Lidl, R.; Niederreiter, H.
-``Finite Field, Encycoldia of Mathematics and Its Applications''
-Vol. 20, Cambridge Univ. Press, 1983 ISBN 0-521-30240-4
+\bibitem[Gladwell 79]{Gla79} Gladwell I
+``Initial Value Routines in the NAG Library''
+ACM Trans Math Softw. 5 386--400. (1979)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Linger 79]{LMW79} Linger, Richard C.; Mills, Harlan D.;
-Witt, Bernard I.
-``Structured Programming: Theory and Practice''
-Addison-Wesley (March 1979) ISBN 0201144611
+\bibitem[Gladwell 80]{GS80} Gladwell I.; Sayers D K
+``Computational Techniques for Ordinary Differential Equations''
+Academic Press. 1980
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lipson 81]{Lip81} Lipson, D.
-``Elements of Algebra and Algebraic Computing''
-The Benjamin/Cummings Publishing Company, Inc.-Menlo Park, California, 1981.
+\bibitem[Gladwell 86]{Gla86} Gladwell I
+``Vectorisation of one dimensional quadrature codes''
+Techincal Report. TR7/86 NAG. (1986)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@misc{Loet09,
- author = "Loetzsch, Martin and Bleys, Joris and Wellens, Pieter",
- title = "Understanding the Dynamics of Complex Lisp Programs",
- year = "2009",
- url = "http://www.martin-loetzsch.de/papers/loetzsch09understanding.pdf",
- paper = "Loet09.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Gladwell 87]{Gla87} Gladwell I
+``The NAG Library Boundary Value Codes''
+Numerical Analysis Report. 134 Manchester University. (1987)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@misc{Loet00,
- author = "Loetzsch, M.",
- title = "GTFL - A graphical terminal for Lisp",
- year = "2000",
- url = "http://martin-loetzsch.de/gtfl"
-}
+\begin{chunk}{ignore}
+\bibitem[Goedel 40]{God40} Goedel
+``The consistency of the continuum hypothesis''
+Ann. Math. Studies, Princeton Univ. Press, 1940
\end{chunk}
-\begin{chunk}{axiom.bib}
-@book{Losc60,
- author = {L\"osch, Friedrich},
- title = "Tables of Higher Functions",
- publisher = "McGraw-Hill Book Company",
- year = "1960"
-}
+\begin{chunk}{ignore}
+\bibitem[Goldman 87]{Gold87} Goldman, L.
+``Integrals of multinomial systems of ordinary differential equations''
+J. of Pure and Applied Algebra, 45, 225-240 (1987)
+\verb|www.sciencedirect.com/science/article/pii/0022404987900727/pdf|
+\verb|?md5=5a0c70643eab514ccf47d80e4fc6ec5a&|
+\verb|pid=1-s2.0-0022404987900727-main.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Gold87.pdf|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[LTU10]{LTU10}.
-``Lambda the Ultimate''
-\verb|lambda-the-ultimate.org/node/3663#comment-62440|
+\bibitem[Gollan 90]{GG90} H. Gollan; J. Grabmeier
+``Algorithms in Representation Theory and
+their Realization in the Computer Algebra System Scratchpad''
+Bayreuther Mathematische Schriften, Heft 33, 1990, 1-23
\end{chunk}
-\begin{chunk}{axiom.bib}
-@book{Luke69a,
- author = "Luke, Yudell L.",
- title = "The Special Functions and their Approximations",
- volume = "1",
- publisher = "Academic Press",
- year = "1969",
- booktitle = "Mathematics in Science and Engineering Volume 53-I"
-}
+\begin{chunk}{ignore}
+\bibitem[Golub 89]{GL89} Golub, Gene H.; Van Loan, Charles F.
+``Matrix Computations''
+Johns Hopkins University Press ISBN 0-8018-3772-3 (1989)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@book{Luke69b,
- author = "Luke, Yudell L.",
- title = "The Special Functions and their Approximations",
- volume = "2",
- publisher = "Academic Press",
- year = "1969",
- booktitle = "Mathematics in Science and Engineering Volume 53-I"
-}
+\begin{chunk}{ignore}
+\bibitem[Golub 96]{GL96} Golub, Gene H.; Van Loan, Charles F.
+``Matrix Computations''
+Johns Hopkins University Press ISBN 978-0-8018-5414-9 (1996)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Lyness 83]{Lyn83} Lyness J N.
-``When not to use an automatic quadrature routine''
-SIAM Review. 25 63--87. (1983)
+\bibitem[Grabmeier]{Grab} Grabmeier, J.
+``On Plesken's root finding algorithm''
+in preparation
\end{chunk}
-\subsection{M} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Mac Lane 79]{MB79} Mac Lane, Saunders; Birkhoff, Garret
-``Algebra''
-AMS Chelsea Publishing ISBN 0821816462
+\bibitem[Grebmeier 87]{GK87} Grabmeier, J.; Kerber, A.;
+``The Evaluation of Irreducible Polynomial Representations of the General
+Linear Groups and of the Unitary Groups over Fields of Characteristic 0''
+Acta Appl. Math. 8 (1987), 271-291
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Malcolm 72]{Mal72} Malcolm M. A.
-``Algorithms to reveal properties of floating-point arithmetic''
-Comms. of the ACM, 15, 949-951. (1972)
+\bibitem[Grabmeier 92]{REF-GS92} Grabmeier, J.; Scheerhorn, A.
+``Finite fields in Axiom''
+AXIOM Technical Report TR7/92 (ATR/5)(NP2522),
+Numerical Algorithms Group, Inc., Downer's
+Grove, IL, USA and Oxford, UK, 1992
+\verb|www.nag.co.uk/doc/TechRep/axiomtr.html|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Malcolm 76]{MS76} Malcolm M A.; Simpson R B.
-``Local Versus Global Strategies for Adaptive Quadrature''
-ACM Trans. Math. Softw. 1 129--146. (1976)
+\bibitem[Granville 1911]{Gran1911} Granville, William Anthony
+``Elements of the Differential and Integral Calculus''
+\verb|djm.cc/library/Elements_Differential_Integral_Calculus_Granville_edited_2.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Gran1911.pdf|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Marden 66]{Mar66} Marden M.
-``Geometry of Polynomials''
-Mathematical Surveys. 3 Am. Math. Soc., Providence, RI. (1966)
+\bibitem[Gruntz 93]{Gru93} Gruntz, Dominik
+``Limit computation in computer algebra''
+\verb|algo.inria.fr/seminars/sem92-93/gruntz.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Gru93.pdf|
+ abstract = "
+ The automatic computation of limits can be reduced to two main
+ sub-problems. The first one is asymptotic comparison where one must
+ decide automatically which one of two functions in a specified class
+ dominates the other one asymptotically. The second one is asymptotic
+ cancellation and is usually exemplified by
+ \[e^x[exp(1/x+e^{-x})-exp(1/x)],\quad{}x \leftarrow \infty\]
+
+ In this example, if the sum is expanded in powers of $1/x$, the
+ expansion always yields $O(x^{-k})$, and this is not enough to
+ conclude.
+
+ In 1990, J.Shackell found an algorithm that solved both these problems
+ for the case of $exp-log$ functions, i.e. functions build by recursive
+ application of exponential, logarithm, algebraic extension and field
+ operations to one variable and the rational numbers. D. Gruntz and
+ G. Gonnet propose a slightly different algorithm for exp-log
+ functions. Extensions to larger classes of functions are also
+ discussed."
\end{chunk}
+\subsection{H} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{axiom.bib}
-@misc{Mars07,
- author = "Marshak, U.",
- title = "HT-AJAX - AJAX framework for Hunchentoot",
- year = "2007",
- url = "http://common-lisp.net/project/ht-ajax/ht-ajax.html"
+@article{Hach95,
+ author = "Hach\'e, G. and Le Brigand, D.",
+ title = "Effective construction of algebraic geometry codes",
+ journal = "IEEE Transaction on Information Theory",
+ volume = "41",
+ month = "November",
+ year = "1995",
+ pages = "1615--1628"
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Maza 95]{MR95} Maza, M. Moreno; Rioboo, R.
-``Computations of gcd over algebraic towers of simple extensions''
-In proceedings of AAECC11 Paris, 1995.
+\begin{chunk}{axiom.bib}
+@article{Hach95a,
+ author = "Hach\'e, G.",
+ title = "Computation in algebraic function fields for effective
+ construction of algebraic-geometric codes",
+ journal = "Lecture Notes in Computer Science",
+ volume = "948",
+ year = "1995",
+ pages = "262--278"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Maza 97]{Maz97} Maza, M. Moreno
-``Calculs de pgcd au-dessus des tours
-d'extensions simples et resolution des systemes d'equations algebriques''
-These, Universite P.etM. Curie, Paris, 1997.
+\begin{chunk}{axiom.bib}
+@phdthesis{Hach96,
+ author = "Hach\'e, G.",
+ title = "Construction effective des codes g\'eom\'etriques",
+ school = "l'Universit\'e Pierre et Marie Curie (Paris 6)",
+ year = "1996",
+ month = "Septembre"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Maza 98]{Maz98} Maza, M. Moreno
-``A new algorithm for computing triangular
-decomposition of algebraic varieties''
- NAG Tech. Rep. 4/98.
+\bibitem[Hall 76]{HW76} Hall G.; Watt J M. (eds),
+``Modern Numerical Methods for Ordinary Differential Equations''
+Clarendon Press. (1976)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Mignotte 82]{Mig82} Mignotte, Maurice
-``Some Useful Bounds''
-Computing, Suppl. 4, 259-263 (1982), Springer-Verlag
+\bibitem[Hamdy 04]{Ham04} Hamdy, S.
+``LiDIA A library for computational number theory''
+Reference manual Edition 2.1.1 May 2004
+\verb|www.cdc.informatik.tu-darmstadt.de/TI/LiDIA|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[McCarthy 83]{McC83} McCarthy G J.
-``Investigation into the Multigrid Code MGD1''
-Report AERE-R 10889. Harwell. (1983)
+\bibitem[Hammarling 85]{Ham85} Hammarling S.
+`` The Singular Value Decomposition in Multivariate Statistics''
+ACM Signum Newsletter. 20, 3 2--25. (1985)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Mie97]{Mie97} Mielenz, Klaus D.
-``Computation of Fresnel Integrals''
-J. Res. Natl. Inst. Stand. Technol. (NIST) V102 No3 May-June 1997 pp363-365
+\bibitem[Hammersley 67]{HH67} Hammersley J M; Handscomb D C.
+``Monte-Carlo Methods''
+Methuen. (1967)
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Mie00]{Mie00} Mielenz, Klaus D.
-``Computation of Fresnel Integrals II''
-J. Res. Natl. Inst. Stand. Technol. (NIST) V105 No4 July-Aug 2000 pp589-590
+\begin{chunk}{axiom.bib}
+@misc{Hath1896,
+ author = "Hathway, Arthur S.",
+ title = "A Primer Of Quaternions",
+ year = "1896"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Millen 68]{Mil68} Millen, J. K.
-``CHARYBDIS: A LISP program to display mathematical expressions on
-typewriter-like devices''
-Interactive Systems for Experimental and Applied Mathematics
-M. Klerer and J. Reinfelds, eds., Academic Press, New York 1968, pp79-90
-%\verb|axiom-developer.org/axiom-website/papers/Mil68.pdf|
+\begin{chunk}{axiom.bib}
+@book{Haya05,
+ author = "Hayashi, K. and Kangkook, J. and Lascu, O. and Pienaar, H. and
+ Schreitmueller, S. and Tarquinio, T. and Thompson, J.",
+ title = "AIX 5L Practical Performance Tools and Tuning Guide",
+ publisher = "IBM",
+ year = "2005",
+ url = "http://www.redbooks.ibm.com/redbooks/pdfs/sg246478.pdf",
+ paper = "Haya05.pdf"
+}
\end{chunk}
-
\begin{chunk}{ignore}
-\bibitem[Minc 79]{Min79} Henryk Minc
-``Evaluation of Permanents''
-Proc. of the Edinburgh Math. Soc.(1979), 22/1 pp 27-32.
+\bibitem[Hayes 70]{Hay70} Hayes J G.
+``Curve Fitting by Polynomials in One Variable''
+Numerical Approximation to Functions and Data.
+(ed J G Hayes) Athlone Press, London. (1970)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[More 74]{MGH74} More J J.; Garbow B S.; Hillstrom K E.
-``User Guide for Minpack-1''
-ANL-80-74 Argonne National Laboratory. (1974)
+\bibitem[Hayes 74]{Hay74} Hayes J G.
+``Numerical Methods for Curve and Surface Fitting''
+Bull Inst Math Appl. 10 144--152. (1974)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Mikhlin 67]{MS67} Mikhlin S G.; Smolitsky K L.
-``Approximate Methods for the Solution of Differential and
-Integral Equations''
-Elsevier. (1967)
+\bibitem[Hayes 74a]{HH74} Hayes J G.; Halliday J,
+``The Least-squares Fitting of Cubic Spline Surfaces to General Data Sets''
+J. Inst. Math. Appl. 14 89--103. (1974)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Mitchell 80]{MG80} Mitchell A R.; Griffiths D F.
-``The Finite Difference Method in Partial Differential Equations''
-Wiley. (1980)
+\bibitem[Henrici 56]{Hen56} Henrici, Peter
+``Automatic Computations with Power Series''
+Journal of the Association for Computing Machinery, Volume 3, No. 1,
+January 1956, 10-15
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Moler 73]{MS73} Moler C B.; Stewart G W.
-``An Algorithm for Generalized Matrix Eigenproblems''
-SIAM J. Numer. Anal. 10 241--256. 1973
+\bibitem[Higham 88]{Hig88} Higham, N.J.
+``FORTRAN codes for estimating the one-norm of a
+real or complex matrix, with applications to condition estimation''
+ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.
\end{chunk}
-\begin{chunk}{axiom.bib}
-@article{Muld97,
- author = "Mulders, Thom",
- title = "A Note on Subresultants and the Lazard/Rioboo/Trager Formula in
- Rational Function Integration",
- journal = "Journal of Symbolic Computation",
- year = "1997",
- volume = "24",
- number = "1",
- month = "July",
- pages = "45-50",
- paper = "Muld97.pdf",
- abstract = "
- An ambiguity in a formula of Lazard, Rioboo and Trager, connecting
- subresultants and rational function integration, is indicated and
- examples of incorrect interpretations are given."
-}
+\begin{chunk}{ignore}
+\bibitem[Higham 02]{Hig02} Higham, Nicholas J.
+``Accuracy and stability of numerical algorithms''
+SIAM Philadelphia, PA ISBN 0-89871-521-0 (2002)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Munksgaard 80]{Mun80} Munksgaard N.
-``Solving Sparse Symmetric Sets of Linear Equations by Pre-conditioned
-Conjugate Gradients''
-ACM Trans. Math. Softw. 6 206--219. (1980)
+\bibitem[Hock 81]{HS81} Hock W.; Schittkowski K.
+``Test Examples for Nonlinear Programming Codes''
+Lecture Notes in Economics and Mathematical Systems. 187 Springer-Verlag. 1981
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Murray 72]{Mur72} Murray W, (ed)
-``Numerical Methods for Unconstrained Optimization''
-Academic Press. (1972)
+\bibitem[Householder 70]{Hou70} Householder A S.
+``The Numerical Treatment of a Single Nonlinear Equation''
+McGraw-Hill. (1970)
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Murtagh 83]{MS83} Murtagh B A.; Saunders M A
-``MINOS 5.0 User's Guide''
-Report SOL 83-20. Department of Operations Research, Stanford University 1983
+\begin{chunk}{axiom.bib}
+@book{Hous81,
+ author = "Householder, Alston S.",
+ title = "Principles of Numerical Analysis",
+ publisher = "Dover Publications, Mineola, NY",
+ year = "1981",
+ isbn = "0-486-45312-X"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Musser 78]{Mus78} Musser, David R.
-``On the Efficiency of a Polynomial Irreducibility Test''
-Journal of the ACM, Vol. 25, No. 2, April 1978, pp. 271-282
+\bibitem[Huang 96]{HI96} Huang, M.D.; Ierardi, D.
+``Efficient algorithms for Riemann-Roch problem and for addition in the
+jacobian of a curve''
+Proceedings 32nd Annual Symposium on Foundations of Computer Sciences.
+IEEE Comput. Soc. Press, pp. 678--687.
\end{chunk}
-\subsection{N} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{I} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
-\bibitem[Nijenhuis 78]{NW78} Nijenhuis and Wilf
-``Combinatorical Algorithms''
-Academic Press, New York 1978.
+\bibitem[IBM]{IBM}.
+SCRIPT Mathematical Formula Formatter User's Guide, SH20-6453,
+IBM Corporation, Publishing Systems Information Development,
+Dept. G68, P.O. Box 1900, Boulder, Colorado, USA 80301-9191.
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Nikolai 79]{Nik79} Nikolai P J.
-``Algorithm 538: Eigenvectors and eigenvalues of real generalized
-symmetric matrices by simultaneous iteration''
-ACM Trans. Math. Softw. 5 118--125. (1979)
-
-\end{chunk}
-
-\subsection{O} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{chunk}{axiom.bib}
-@misc{OCAM14,
- author = "unknown",
- title = "The OCAML website",
- url = "http://ocaml.org"
-}
+\bibitem[Itoh 88]{Itoh88} Itoh, T.;, Tsujii, S.
+``A fast algorithm for computing multiplicative inverses
+in $GF(2^m)$ using normal bases''
+Inf. and Comp. 78, pp.171-177, 1988
+%\verb|axiom-developer.org/axiom-website/Itoh88.pdf|
+ abstract = "
+ This paper proposes a fast algorithm for computing multiplicative
+ inverses in $GF(2^m)$ using normal bases. Normal bases have the
+ following useful property: In the case that an element $x$ in
+ $GF(2^m)$ is represented by normal bases, $2^k$ power operation of an
+ element $x$ in $GF(2^m)$ can be carried out by $k$ times cyclic shift
+ of its vector representation. C.C. Wang et al. proposed an algorithm
+ for computing multiplicative inverses using normal bases, which
+ requires $(m-2)$ multiplications in $GF(2^m)$ and $(m-1)$ cyclic
+ shifts. The fast algorithm proposed in this paper also uses normal
+ bases, and computes multiplicative inverses iterating multiplications
+ in $GF(2^m)$. It requires at most $2[log_2(m-1)]$ multiplications in
+ $GF(2^m)$ and $(m-1)$ cyclic shifts, which are much less than those
+ required in Wang's method. The same idea of the proposed fast
+ algorithm is applicable to the general power operation in $GF(2^m)$
+ and the computation of multiplicative inverses in $GF(q^m)$
+ $(q=2^n)$."
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Ollagnier 94]{Olla94} Ollagnier, Jean Moulin
-``Algorithms and methods in differential algebra''
-\verb|www.lix.polytechnique.fr/~moulin/papiers/atelier.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Olla94.pdf|
+\bibitem[Iyanaga 77]{Iya77} Iyanaga, Shokichi; Iyanaga, Yukiyosi Kawada
+``Encyclopedic Dictionary of Mathematics''
+1977
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Olver 10]{NIST10} Olver, Frank W.; Lozier, Daniel W.;
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-(2010) Cambridge University Press ISBN 978-0-521-19225-5
-
-\end{chunk}
+\subsection{J} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
-\bibitem[OpenM]{OpenM}.
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-\verb|www.openmath.org/overview/technical.html|
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+\bibitem[James 81]{JK81} James, Gordon; Kerber, Adalbert
+``The Representation Theory of the Symmetric Group''
+Encyclopedia of Mathematics and its Applications Vol. 16
+Addison-Wesley, 1981
\end{chunk}
-\begin{chunk}{axiom.bib}
-@misc{Ostr1845,
- author = "Ostrogradsky. M.W.",
- title = "De l'int\'{e}gration des fractions rationelles.",
- journal = "Bulletin de la Classe Physico-Math\'{e}matiques de
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- volume = "IV",
- pages = "145-167,286-300",
- year = "1845"
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+``Integral Equation Methods in Potential Theory and Elastostatics''
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\end{chunk}
-\subsection{P} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
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+\bibitem[Jeffrey 04]{Je04} Jeffrey, Alan
+``Handbook of Mathematical Formulas and Integrals''
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\end{chunk}
+\subsection{K} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
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+\bibitem[Kalkbrener 91]{Kal91} Kalkbrener, M.
+``Three contributions to elimination theory''
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-\bibitem[Parnas 10]{PJ10} Parnas, David Lorge; Jin, Ying
-``Defining the meaning of tabular mathematical expressions''
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+\bibitem[Kaufmann 00]{KMJ00} Kaufmann, Matt; Manolios, Panagiotis;
+Moore J Strother
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+Springer, July 31. 2000 ISBN 0792377443
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-\bibitem[Parnas 95]{PM95} Parnas, David Lorge; Madey, Jan
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+``The Art of Computer Programming''
+2nd edition Vol. 2 (Seminumerical Algorithms) 1st edition, 2nd printing,
+Addison-Wesley 1971, p. 397-398
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\end{chunk}
\begin{chunk}{axiom.bib}
-@book{Pear56,
- author = "Pearcey, T.",
- title = "Table of the Fresnel Integral",
- publisher = "Cambridge University Press",
- year = "1956"
-}
+@book{Knut92,
+ author = "Knuth, Donald E.",
+ title = "Literate Programming",
+ publisher = "Center for the Study of Language and Information, Stanford CA",
+ year = "1992",
+ isbn = "0-937073-81-4"
+}
\end{chunk}
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-(ed B Childs, M Scott, J W Daniel, E Denman and P Nelson) 76
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+\bibitem[Knu98]{Knu98} Donald Knuth
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+Addison-Wesley 1998
\end{chunk}
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+``On Radical Zero-Dimensional Ideals''
+J. Symbolic Computations 8, 545-552 (1989)
+\verb|www.sciencedirect.com/science/article/pii/S0747717189800604/pdf|
+\verb|?md5=f06dc6269514c90dcae57f0184bcbe65&|
+\verb|pid=1-s2.0-S0747717189800604-main.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Koba88.pdf|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Peters 67b]{Pet67b} Peters G.
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-\bibitem[Pierce 82]{Pie82} R.S. Pierce
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-Graduate Texts in Mathematics 88
-Springer-Verlag, Heidelberg, 1982, ISBN 0-387-90693-2
+\subsection{L} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\begin{chunk}{axiom.bib}
+@book{Lamp86,
+ author = "Lamport, Leslie",
+ title = "LaTeX: A Document Preparation System",
+ publisher = "Addison-Wesley Publishing Company, Reading, Massachusetts",
+ year = "1986",
+ isbn = "0-201-15790-X"
+}
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-``An Algorithm for Automatic Integration''
-Angewandte Informatik. 15 399--401. (1973)
+\bibitem[Lautrup 71]{Lau71} Lautrup B.
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+Proc. 2nd Coll. on Advanced Methods in Theoretical Physics, Marseille. (1971)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Piessens 74]{PMB74} Piessens R.;; Mertens I.; Branders M.
-``Integration of Functions having End-point Singularities''
-Angewandte Informatik. 16 65--68. (1974)
+\bibitem[Lawson 77]{Law77} Lawson C L.
+``Software for C Surface Interpolation''
+Mathematical Software III. (ed J R Rice) Academic Press. 161--194. (1977)
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-\bibitem[Piessens 75]{PB75} Piessens R.; Branders M.
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-J. Comput. Appl. Math. 1 153--164. (1975)
+\bibitem[Lawson 74]{LH74} Lawson C L.; Hanson R J.
+``Solving Least-squares Problems''
+Prentice-Hall. (1974)
\end{chunk}
-\begin{chunk}{ignore}
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+\begin{chunk}{axiom.bib}
+@article{Laws79,
+ author = "Lawson, C.L. and Hanson R.J. and Kincaid, D.R. and Krogh, F.T.",
+ title = "Algorithm 539: Basic linear algebra subprograms for FORTRAN usage",
+ journal = "ACM Transactions on Mathematical Software",
+ volume = "5",
+ number = "3",
+ month = "September",
+ year = "1979",
+ pages = "308-323"
+}
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+\bibitem[Lawson 79]{LHKK79} Lawson C L; Hanson R J; Kincaid D R;
+ Krogh F T
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+ACM Trans. Math. Softw. 5 308--325. (1979)
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+\bibitem[Lazard92]{Laz92} Lazard, D.
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+Journal of Symbolic Computation, 1992, 13, 117-131
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@article{Laza90,
+ author = "Lazard, Daniel and Rioboo, Renaud",
+ title = "Integration of rational functions: Rational computation of the
+ logarithmic part",
+ journal = "Journal of Symbolic Computation",
+ volume = "9",
+ number = "2",
+ year = "1990",
+ month = "February",
+ pages = "113-115",
+ keywords = "axiomref",
+ paper = "Laza90.pdf",
+ abstract = "
+ A new formula is given for the logarithmic part of the integral of a
+ rational function, one that strongly improves previous algorithms and
+ does not need any computation in an algebraic extension of the field
+ of constants, nor any factorisation since only polynomial arithmetic
+ and GCD computations are used. This formula was independently found
+ and implemented in SCRATCHPAD by B.M. Trager."
+}
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@article{LeBr88,
+ author = "Le Brigand, D.; Risler, J.J.",
+ title = "Algorithme de Brill-Noether et codes de Goppa",
+ journal = "Bull. Soc. Math. France",
+ volume = "116",
+ year = "1988",
+ pages = "231--253"
+}
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-(ed P E Gill and W Murray) Academic Press. pp1-28. 1974
+\begin{chunk}{axiom.bib}
+@book{Lege11,
+ author = "Legendre, George L. and Grazini, Stefano",
+ title = "Pasta by Design",
+ publisher = "Thames and Hudson",
+ isbn = "978-0-500-51580-8",
+ year = "2011"
+}
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+Math. Comp. 48, 1987, pp. 217-231
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\begin{chunk}{axiom.bib}
-@inproceedings{Prat73,
- author = "Pratt, Vaughan R.",
- title = "Top down operator precedence",
- booktitle = "Proc. 1st annual ACM SIGACT-SIGPLAN Symposium on Principles
- of Programming Languages",
- series = "POPL'73",
- pages = "41-51",
- year = "1973",
- url = "http://hall.org.ua/halls/wizzard/pdf/Vaughan.Pratt.TDOP.pdf",
- keywords = "axiomref",
- paper = "Prat73.pdf"
+@misc{Leop03,
+ author = "Leopardi, Paul",
+ title = "A quick introduction to Clifford Algebras",
+ publisher = "School of Mathematics, University of New South Wales",
+ year = "2003",
+ paper = "Leop03.pdf"
}
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-Vetterling, William T.; Flannery, Brian P.
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+``Algorithms for sparse matrix eigenvalue problems''
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+Stanford University. (1977)
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-Inst. Math. Appl., Numerical Analysis Newsletter. 1(3) (1977)
+\bibitem[Lidl 83]{LN83} Lidl, R.; Niederreiter, H.
+``Finite Field, Encycoldia of Mathematics and Its Applications''
+Vol. 20, Cambridge Univ. Press, 1983 ISBN 0-521-30240-4
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-\bibitem[Pryce 81]{Pry81} Pryce J D.
-``Two codes for Sturm-Liouville problems''
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+\bibitem[Linger 79]{LMW79} Linger, Richard C.; Mills, Harlan D.;
+Witt, Bernard I.
+``Structured Programming: Theory and Practice''
+Addison-Wesley (March 1979) ISBN 0201144611
\end{chunk}
\begin{chunk}{ignore}
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-Sturm-Liouville Problems''
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+\bibitem[Lipson 81]{Lip81} Lipson, D.
+``Elements of Algebra and Algebraic Computing''
+The Benjamin/Cummings Publishing Company, Inc.-Menlo Park, California, 1981.
\end{chunk}
\begin{chunk}{axiom.bib}
-@misc{Puff09,
- author = "Puffinware LLC",
- title = "Singular Value Decomposition (SVD) Tutorial",
- url = "http://www.puffinwarellc.com/p3a.htm"
+@misc{Loet09,
+ author = "Loetzsch, Martin and Bleys, Joris and Wellens, Pieter",
+ title = "Understanding the Dynamics of Complex Lisp Programs",
+ year = "2009",
+ url = "http://www.martin-loetzsch.de/papers/loetzsch09understanding.pdf",
+ paper = "Loet09.pdf"
}
\end{chunk}
-\subsection{Q} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{chunk}{ignore}
-\bibitem[Quintana-Orti 06]{QG06} Quintana-Orti, Gregorio;
-van de Geijn, Robert
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-
-\end{chunk}
-
-\subsection{R} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{chunk}{ignore}
-\bibitem[Rabinowitz 70]{Rab70} Rabinowitz P.
-``Numerical Methods for Nonlinear Algebraic Equations''
-Gordon and Breach. (1970)
+\begin{chunk}{axiom.bib}
+@misc{Loet00,
+ author = "Loetzsch, M.",
+ title = "GTFL - A graphical terminal for Lisp",
+ year = "2000",
+ url = "http://martin-loetzsch.de/gtfl"
+}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Ralston 65]{Ral65} Ralston A.
-``A First Course in Numerical Analysis''
-McGraw-Hill. 87--90. (1965)
+\begin{chunk}{axiom.bib}
+@book{Losc60,
+ author = {L\"osch, Friedrich},
+ title = "Tables of Higher Functions",
+ publisher = "McGraw-Hill Book Company",
+ year = "1960"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Ramakrishnan 03]{Ram03} Ramakrishnan, Maya
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+\bibitem[LTU10]{LTU10}.
+``Lambda the Ultimate''
+\verb|lambda-the-ultimate.org/node/3663#comment-62440|
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Ramsey 03]{Ra03} Ramsey, Norman
-``Noweb--A Simple, Extensible Tool for Literate Programming''
-\verb|www.eecs.harvard.edu/~nr/noweb|
+\begin{chunk}{axiom.bib}
+@book{Luke69a,
+ author = "Luke, Yudell L.",
+ title = "The Special Functions and their Approximations",
+ volume = "1",
+ publisher = "Academic Press",
+ year = "1969",
+ booktitle = "Mathematics in Science and Engineering Volume 53-I"
+}
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-\begin{chunk}{ignore}
-\bibitem[Redfield 27]{Red27} Redfield, J.H.
-``The Theory of Group-Reduced Distributions''
-American J. Math., 49 (1927) 433-455.
+\begin{chunk}{axiom.bib}
+@book{Luke69b,
+ author = "Luke, Yudell L.",
+ title = "The Special Functions and their Approximations",
+ volume = "2",
+ publisher = "Academic Press",
+ year = "1969",
+ booktitle = "Mathematics in Science and Engineering Volume 53-I"
+}
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\begin{chunk}{ignore}
-\bibitem[Reinsch 67]{Rei67} Reinsch C H.
-``Smoothing by Spline Functions''
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+\bibitem[Lyness 83]{Lyn83} Lyness J N.
+``When not to use an automatic quadrature routine''
+SIAM Review. 25 63--87. (1983)
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-\begin{chunk}{ignore}
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-Points in the Plane''
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-
-\end{chunk}
+\subsection{M} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
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+\bibitem[Mac Lane 79]{MB79} Mac Lane, Saunders; Birkhoff, Garret
+``Algebra''
+AMS Chelsea Publishing ISBN 0821816462
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\begin{chunk}{ignore}
-\bibitem[Reutenauer 93]{Re93} Reutenauer, Christophe
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\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Reznick 93]{Rezn93} Reznick, Bruce
-``An Inequality for Products of Polynomials''
-Proc. AMS Vol 117 No 4 April 1993
-%\verb|axiom-developer.org/axiom-website/papers/Rezn93.pdf|
+\bibitem[Malcolm 76]{MS76} Malcolm M A.; Simpson R B.
+``Local Versus Global Strategies for Adaptive Quadrature''
+ACM Trans. Math. Softw. 1 129--146. (1976)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Rich xx]{Rixx} Rich, A.D.; Jeffrey, D.J.
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-\verb|www.apmaths.uwo.ca/~djeffrey/Offprints/IntegrationRules.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Rixx.pdf|
- abstract = "
- We describe the development of a repository of mathematical knowledge
- based on transformation rules. The specific mathematical problem is
- indefinite integration. It is important that the repository be not
- confused with a look-up table. The database of transformation rules is
- at present encoded in Mathematica, but this is only one convenient
- form of the repository, and it could be readily translated into other
- formats. The principles upon which the set of rules is compiled is
- described. One important principle is minimality. The benefits of the
- approach are illustrated with examples, and with the results of
- comparisons with other approaches."
+\bibitem[Marden 66]{Mar66} Marden M.
+``Geometry of Polynomials''
+Mathematical Surveys. 3 Am. Math. Soc., Providence, RI. (1966)
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Rich 10]{Ri10} Rich, Albert D.
-``Rule-based Mathematics''
-\verb|www.apmaths.uwo.ca/~arich|
+\begin{chunk}{axiom.bib}
+@misc{Mars07,
+ author = "Marshak, U.",
+ title = "HT-AJAX - AJAX framework for Hunchentoot",
+ year = "2007",
+ url = "http://common-lisp.net/project/ht-ajax/ht-ajax.html"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Richardson 94]{RF94} Richardson, Dan; Fitch, John
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+\bibitem[Maza 95]{MR95} Maza, M. Moreno; Rioboo, R.
+``Computations of gcd over algebraic towers of simple extensions''
+In proceedings of AAECC11 Paris, 1995.
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+\bibitem[Maza 97]{Maz97} Maza, M. Moreno
+``Calculs de pgcd au-dessus des tours
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+These, Universite P.etM. Curie, Paris, 1997.
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-In proceedings of the ISSAC'92 Conference, Berkeley 1992 pp. 206-215.
-0-89791-490-2 (hard cover) LCCN QA76.95.I59 1992
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+``A new algorithm for computing triangular
+decomposition of algebraic varieties''
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\begin{chunk}{ignore}
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-\bibitem[Rote 01]{Rote01} Rote, G\"unter
-``Division-free algorithms for the determinant and the Pfaffian''
-in Computational Discrete Mathematics ISBN 3-540-42775-9 pp119-135
-\verb|page.mi.fu-berlin.de/rote/Papers/pdf/Division-free+algorithms.pdf|
+\bibitem[Mie97]{Mie97} Mielenz, Klaus D.
+``Computation of Fresnel Integrals''
+J. Res. Natl. Inst. Stand. Technol. (NIST) V102 No3 May-June 1997 pp363-365
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Rubey 07]{Rub07} Rubey, Martin
-``Formula Guessing with Axiom''
-April 2007
+\bibitem[Mie00]{Mie00} Mielenz, Klaus D.
+``Computation of Fresnel Integrals II''
+J. Res. Natl. Inst. Stand. Technol. (NIST) V105 No4 July-Aug 2000 pp589-590
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Rutishauser 69]{Rut69} Rutishauser H.
-``Computational aspects of F L Bauer's simultaneous iteration method''
-Num. Math. 13 4--13. (1969)
+\bibitem[Millen 68]{Mil68} Millen, J. K.
+``CHARYBDIS: A LISP program to display mathematical expressions on
+typewriter-like devices''
+Interactive Systems for Experimental and Applied Mathematics
+M. Klerer and J. Reinfelds, eds., Academic Press, New York 1968, pp79-90
+%\verb|axiom-developer.org/axiom-website/papers/Mil68.pdf|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Rutishauser 70]{Rut70} Rutishauser H.
-``Simultaneous iteration method for symmetric matrices''
-Num. Math. 16 205--223. (1970)
+\bibitem[Minc 79]{Min79} Henryk Minc
+``Evaluation of Permanents''
+Proc. of the Edinburgh Math. Soc.(1979), 22/1 pp 27-32.
\end{chunk}
-\subsection{S} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Schafer 66]{Sch66} Schafer, R.D.
-``An Introduction to Nonassociative Algebras''
-Academic Press, New York, 1966
+\bibitem[More 74]{MGH74} More J J.; Garbow B S.; Hillstrom K E.
+``User Guide for Minpack-1''
+ANL-80-74 Argonne National Laboratory. (1974)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Schoenberg 53]{SW53} Schoenberg I J.; Whitney A.
-``On Polya Frequency Functions III''
-Trans. Amer. Math. Soc. 74 246--259. (1953)
+\bibitem[Mikhlin 67]{MS67} Mikhlin S G.; Smolitsky K L.
+``Approximate Methods for the Solution of Differential and
+Integral Equations''
+Elsevier. (1967)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Schoenhage 82]{Sch82} Schoenhage, A.
-``The fundamental theorem of algebra in terms of computational complexity''
-preliminary report, Univ. Tuebingen, 1982
+\bibitem[Mitchell 80]{MG80} Mitchell A R.; Griffiths D F.
+``The Finite Difference Method in Partial Differential Equations''
+Wiley. (1980)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Schonfelder 76]{Sch76} Schonfelder J L.
-``The Production of Special Function Routines for a Multi-Machine Library''
-Software Practice and Experience. 6(1) (1976)
+\bibitem[Moler 73]{MS73} Moler C B.; Stewart G W.
+``An Algorithm for Generalized Matrix Eigenproblems''
+SIAM J. Numer. Anal. 10 241--256. 1973
\end{chunk}
\begin{chunk}{axiom.bib}
-@book{Segg93,
- author = "{von Seggern}, David Henry",
- title = "CRC Standard Curves and Surfaces",
- publisher = "CRC Press",
- year = "1993",
- isbn = "0-8493-0196-3"
+@article{Muld97,
+ author = "Mulders, Thom",
+ title = "A Note on Subresultants and the Lazard/Rioboo/Trager Formula in
+ Rational Function Integration",
+ journal = "Journal of Symbolic Computation",
+ year = "1997",
+ volume = "24",
+ number = "1",
+ month = "July",
+ pages = "45-50",
+ paper = "Muld97.pdf",
+ abstract = "
+ An ambiguity in a formula of Lazard, Rioboo and Trager, connecting
+ subresultants and rational function integration, is indicated and
+ examples of incorrect interpretations are given."
}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Seiler 95a]{Sei95a} Seiler, W.M.; Calmet, J.
-``JET -- An Axiom Environment for Geometric Computations with Differential
-Equations''
-%\verb|axiom-developer.org/axiom-website/papers/Sei95a.pdf|
+\bibitem[Munksgaard 80]{Mun80} Munksgaard N.
+``Solving Sparse Symmetric Sets of Linear Equations by Pre-conditioned
+Conjugate Gradients''
+ACM Trans. Math. Softw. 6 206--219. (1980)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Shepard 68]{She68} Shepard D.
-``A Two-dimensional Interpolation Function for Irregularly Spaced Data''
-Proc. 23rd Nat. Conf. ACM. Brandon/Systems Press Inc.,
-Princeton. 517--523. 1968
+\bibitem[Murray 72]{Mur72} Murray W, (ed)
+``Numerical Methods for Unconstrained Optimization''
+Academic Press. (1972)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Shirayanagi 96]{Shir96} Shirayanagi, Kiyoshi
-``Floating point Gr\"obner bases''
-Mathematics and Computers in Simulation 42 pp 509-528 (1996)
-%\verb|axiom-developer.org/axiom-website/papers/Shir96.pdf|
- abstract = "
- Bracket coefficients for polynomials are introduced. These are like
- specific precision floating point numbers together with error
- terms. Working in terms of bracket coefficients, an algorithm that
- computes a Gr{\"o}bner basis with floating point coefficients is
- presented, and a new criterion for determining whether a bracket
- coefficient is zero is proposed. Given a finite set $F$ of polynomials
- with real coefficients, let $G_\mu$ be the result of the algorithm for
- $F$ and a precision $\mu$, and $G$ be a true Gr{\"o}bner basis of
- $F$. Then, as $\mu$ approaches infinity, $G_\mu$ converges to $G$
- coefficientwise. Moreover, there is a precision $M$ such that if
- $\mu \ge M$, then the sets of monomials with non-zero coefficients of
- $G_\mu$ and $G$ are exactly the same. The practical usefulness of the
- algorithm is suggested by experimental results."
+\bibitem[Murtagh 83]{MS83} Murtagh B A.; Saunders M A
+``MINOS 5.0 User's Guide''
+Report SOL 83-20. Department of Operations Research, Stanford University 1983
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Sims 71]{Sims71} Sims, C.
-``Determining the Conjugacy Classes of a Permutation Group''
-Computers in Algebra and Number Theory, SIAM-AMS Proc., Vol. 4,
-American Math. Soc., 1991, pp191-195
+\bibitem[Musser 78]{Mus78} Musser, David R.
+``On the Efficiency of a Polynomial Irreducibility Test''
+Journal of the ACM, Vol. 25, No. 2, April 1978, pp. 271-282
\end{chunk}
+\subsection{N} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Singer 89]{Sing89} Singer, M.F.
-``Formal Solutions of Differential Equations''
-J. Symbolic COmputation 10, No.1 59-94 (1990)
-%\verb|axiom-developer.org/axiom-website/papers/Sing89.pdf|
- keywords = "survey",
- abstract = "
- We give a survey of some methods for finding formal solutions of
- differential equations. These include methods for finding power series
- solutions, elementary and liouvillian solutions, first integrals, Lie
- theoretic methods, transform methods, asymptotic methods. A brief
- discussion of difference equations is also included."
+\bibitem[Nijenhuis 78]{NW78} Nijenhuis and Wilf
+``Combinatorical Algorithms''
+Academic Press, New York 1978.
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Sit 92]{REF-Sit92} Sit, William
-``An Algorithm for Parametric Linear Systems''
-J. Sym. Comp., April 1992
+\bibitem[Nikolai 79]{Nik79} Nikolai P J.
+``Algorithm 538: Eigenvectors and eigenvalues of real generalized
+symmetric matrices by simultaneous iteration''
+ACM Trans. Math. Softw. 5 118--125. (1979)
+
+\end{chunk}
+
+\subsection{O} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{axiom.bib}
+@misc{OCAM14,
+ author = "unknown",
+ title = "The OCAML website",
+ url = "http://ocaml.org"
+}
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Smith 67]{Smi67} Smith B T.
-``ZERPOL: A Zero Finding Algorithm for Polynomials Using Laguerre's Method''
-Technical Report. Department of Computer Science, University of Toronto,
-Canada. (1967)
+\bibitem[Ollagnier 94]{Olla94} Ollagnier, Jean Moulin
+``Algorithms and methods in differential algebra''
+\verb|www.lix.polytechnique.fr/~moulin/papiers/atelier.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Olla94.pdf|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Smith 85]{Smi85} Smith G D.
-``Numerical Solution of Partial Differential Equations: Finite Difference
-Methods''
-Oxford University Press (3rd Edition). (1985)
+\bibitem[Olver 10]{NIST10} Olver, Frank W.; Lozier, Daniel W.;
+Boisvert, Ronald F.; Clark, Charles W. (ed)
+``NIST Handbook of Mathematical Functions''
+(2010) Cambridge University Press ISBN 978-0-521-19225-5
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Sobol 74]{Sob74} Sobol I M.
-``The Monte Carlo Method''
-The University of Chicago Press. 1974
+\bibitem[OpenM]{OpenM}.
+``OpenMath Technical Overview''
+\verb|www.openmath.org/overview/technical.html|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Steele 90]{Ste90} Steele, Guy L.
-``Common Lisp The Language''
-Second Edition ISBN 1-55558-041-6 Digital Press (1990)
+\bibitem[Ortega 70]{OR70} Ortega J M.; Rheinboldt W C.
+``Iterative Solution of Nonlinear Equations in Several Variables''
+Academic Press. (1970)
\end{chunk}
\begin{chunk}{axiom.bib}
-@misc{Stic93,
- author = "Stichtenoth, H.",
- title = "Algebraic function fields and codes",
- publisher = "Springer-Verlag",
- year = "1993"
+@misc{Ostr1845,
+ author = "Ostrogradsky. M.W.",
+ title = "De l'int\'{e}gration des fractions rationelles.",
+ journal = "Bulletin de la Classe Physico-Math\'{e}matiques de
+ l'Acae\'{e}mie Imp\'{e}riale des Sciences de St. P\'{e}tersbourg,",
+ volume = "IV",
+ pages = "145-167,286-300",
+ year = "1845"
}
\end{chunk}
-\begin{chunk}{ignore}
-\bibitem[Stinson 90]{Stin90} Stinson, D.R.
-``Some observations on parallel Algorithms for fast exponentiation
-in $GF(2^n)$''
-Siam J. Comp., Vol.19, No.4, pp.711-717, August 1990
-%\verb|axiom-developer.org/axiom-website/Stin90.pdf|
- abstract = "
- A normal basis represention in $GF(2^n)$ allows squaring to be
- accomplished by a cyclic shift. Algorithms for multiplication in
- $GF(2^n)$ using a normal basis have been studied by several
- researchers. In this paper, algorithms for performing exponentiation
- in $GF(2^n)$ using a normal basis, and how they can be speeded up by
- using parallelization, are investigated."
-
-\end{chunk}
+\subsection{P} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{chunk}{ignore}
-\bibitem[Stroud 66]{SS66} Stroud A H.; Secrest D.
-``Gaussian Quadrature Formulas''
-Prentice-Hall. (1966)
+\bibitem[Paige 75]{PS75} Paige C C.; Saunders M A.
+``Solution of Sparse Indefinite Systems of Linear Equations''
+SIAM J. Numer. Anal. 12 617--629. (1975)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Stroud 71]{Str71} Stroud A H.
-``Approximate Calculation of Multiple Integrals''
-Prentice-Hall 1971
+\bibitem[Paige 82a]{PS82a} Paige C C.; Saunders M A.
+``LSQR: An Algorithm for Sparse Linear Equations and Sparse Least-squares''
+ACM Trans. Math. Softw. 8 43--71. (1982)
\end{chunk}
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-\bibitem[Swarztrauber 79]{SS79} Swarztrauber P N.; Sweet R A.
-``Efficient Fortran Subprograms for the Solution of Separable Elliptic Partial
-Differential Equations''
-ACM Trans. Math. Softw. 5 352--364. (1979)
+\bibitem[Paige 82b]{PS82b} Paige C C.; Saunders M A.
+``ALGORITHM 583 LSQR: Sparse Linear Equations and Least-squares Problems''
+ACM Trans. Math. Softw. 8 195--209. (1982)
\end{chunk}
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-\bibitem[Swarztrauber 84]{SS84} Swarztrauber P N.
-``Fast Poisson Solvers''
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-Mathematical Association of America. (1984)
-
-\end{chunk}
-
-\subsection{T} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{chunk}{axiom.bib}
-@book{Tait1890,
- author = "Tait, P.G.",
- title = "An Elementary Treatise on Quaternions",
- publisher = "C.J. Clay and Sons, Cambridge University Press Warehouse,
- Ave Maria Lane",
- year = "1890"
-}
+\bibitem[Parker 84]{Par84} Parker, R. A.
+``The Computer Calculation of Modular Characters (The Meat-Axe)''
+M. D. Atkinson (Ed.), Computational Group Theory
+Academic Press, Inc., London 1984
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Taivalsaari 96]{Tai96} Taivalsaari, Antero
-``On the Notion of Inheritance''
-ACM Computing Surveys, Vol 28 No 3 Sept 1996 pp438-479
+\bibitem[Parlett 80]{Par80} Parlett B N.
+``The Symmetric Eigenvalue Problem''
+Prentice-Hall. 1980
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Temme 87]{Tem87} Temme N M.
-``On the Computation of the Incomplete Gamma Functions for Large Values of
-the Parameters''
-Algorithms for Approximation. (ed J C Mason and M G Cox)
-Oxford University Press. (1987)
+\bibitem[Parnas 10]{PJ10} Parnas, David Lorge; Jin, Ying
+``Defining the meaning of tabular mathematical expressions''
+Science of Computer Programming V75 No.11 Nov 2010 pp980-1000 Elesevier
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-\bibitem[Temperton 83a]{Tem83a} Temperton C.
-``Self-sorting Mixed-radix Fast Fourier Transforms''
-J. Comput. Phys. 52 1--23. (1983)
+\bibitem[Parnas 95]{PM95} Parnas, David Lorge; Madey, Jan
+``Functional Documents for Computer Systems''
+Science of Computer Programming V25 No.1 Oct 1995 pp41-61 Elesevier
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-\bibitem[Temperton 83b]{Tem83b} Temperton C.
-``Fast Mixed-Radix Real Fourier Transforms''
-J. Comput. Phys. 52 340--350. (1983)
+\bibitem[Paul 81]{Paul81} Paul, Richard
+``Robot Manipulators''
+MIT Press 1981
\end{chunk}
\begin{chunk}{axiom.bib}
-@article{Thur94,
- author = "Thurston, William P.",
- title = "On Proof and Progress in Mathematics",
- journal = "Bulletin AMS",
- volume = "30",
- number = "2",
- month = "April",
- year = "1994",
- url = "http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf",
- paper = "Thur94.pdf"
+@book{Pear56,
+ author = "Pearcey, T.",
+ title = "Table of the Fresnel Integral",
+ publisher = "Cambridge University Press",
+ year = "1956"
}
\end{chunk}
-\subsection{U} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Unknown 61]{Unk61} Unknown
-``Chebyshev-series''
-Modern Computing Methods
-Chapter 8. NPL Notes on Applied Science (2nd Edition). 16 HMSO. 1961
+\bibitem[Pereyra 79]{Per79} Pereyra V.
+``PASVA3: An Adaptive Finite-Difference Fortran Program for First Order
+Nonlinear, Ordinary Boundary Problems''
+Codes for Boundary Value Problems in Ordinary Differential Equations.
+Lecture Notes in Computer Science.
+(ed B Childs, M Scott, J W Daniel, E Denman and P Nelson) 76
+Springer-Verlag. (1979)
\end{chunk}
-\subsection{V} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Van Dooren 76]{vDDR76} Van Dooren P.; De Ridder L.
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-Cube''
-J. Comput. Appl. Math. 2 207--217. (1976)
+\bibitem[Peters 67a]{Pet67a} Peters G.
+``NPL Algorithms Library''
+Document No. F2/03/A. (1967)
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\begin{chunk}{ignore}
-\bibitem[van Hoeij 94]{REF-vH94} van Hoeij, M.
-``An algorithm for computing an integral
-basis in an algebraic function field''
-{\sl J. Symbolic Computation}
-18(4):353-364, October 1994
+\bibitem[Peters 67b]{Pet67b} Peters G.
+``NPL Algorithms Library''
+Document No.F1/04/A (1967)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Van Loan 92]{Van92} Van Loan, C.
-``Computational Frameworks for the Fast Fourier Transform''
-SIAM Philadelphia. (1992)
+\bibitem[Peters 70]{PW70} Peters G.; Wilkinson J H.
+``The Least-squares Problem and Pseudo-inverses''
+Comput. J. 13 309--316. (1970)
\end{chunk}
-\subsection{W} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Wait 85]{WM85} Wait R.; Mitchell A R.
-``Finite Element Analysis and Application''
-Wiley. (1985)
+\bibitem[Peters 71]{PW71} Peters G.; Wilkinson J H.
+``Practical Problems Arising in the Solution of Polynomial Equations''
+J. Inst. Maths Applics. 8 16--35. (1971)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Wang 92]{Wang92} Wang, D.M.
-``An implementation of the characteristic set method in Maple''
-Proc. DISCO'92 Bath, England
+\bibitem[Pierce 82]{Pie82} R.S. Pierce
+``Associative Algebras''
+Graduate Texts in Mathematics 88
+Springer-Verlag, Heidelberg, 1982, ISBN 0-387-90693-2
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Ward 75]{War75} Ward, R C.
-``The Combination Shift QZ Algorithm''
-SIAM J. Numer. Anal. 12 835--853. 1975
-
-\end{chunk}
-
-\begin{chunk}{axiom.bib}
-@misc{Watt03,
- author = "Watt, Stephen",
- title = "Aldor",
- url = "http://www.aldor.org",
- year = "2003"
-}
+\bibitem[Piessens 73]{Pie73} Piessens R.
+``An Algorithm for Automatic Integration''
+Angewandte Informatik. 15 399--401. (1973)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@misc{Weil71,
- author = "Weil, Andr\'{e}",
- title = "Courbes alg\'{e}briques et vari\'{e}t\'{e}s Abeliennes",
- year = "1971"
-}
+\begin{chunk}{ignore}
+\bibitem[Piessens 74]{PMB74} Piessens R.;; Mertens I.; Branders M.
+``Integration of Functions having End-point Singularities''
+Angewandte Informatik. 16 65--68. (1974)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Weisstein]{Wein} Weisstein, Eric W.
-``Hypergeometric Function''
-MathWorld - A Wolfram Web Resource
-\verb|mathworld.wolfram.com/HypergeometricFunction.html|
+\bibitem[Piessens 75]{PB75} Piessens R.; Branders M.
+``Algorithm 002. Computation of Oscillating Integrals''
+J. Comput. Appl. Math. 1 153--164. (1975)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@misc{Weit03,
- author = "Weitz, E.",
- title = "CL-WHO -Yet another Lisp markup language",
- year = "2003",
- url = "http://www.weitz.de/cl-who/"
-}
+\begin{chunk}{ignore}
+\bibitem[Piessens 76]{PVRBM76} Piessens R.; Van Roy-Branders M.; Mertens I.
+``The Automatic Evaluation of Cauchy Principal Value Integrals''
+Angewandte Informatik. 18 31--35. (1976)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@misc{Weit06,
- author = "Weitz, E.",
- title = "HUNCHENTOOT - The Common Lisp web server formerly known as TBNL",
- year = "2006",
- url = "http://www.weitz.de/hunchentoot"
-}
+\begin{chunk}{ignore}
+\bibitem[Piessens 83]{PDUK83} Piessens R.; De Doncker-Kapenga E.;
+Uberhuber C.; Kahaner D.
+``QUADPACK, A Subroutine Package for Automatic Integration''
+Springer-Verlag.(1983)
\end{chunk}
\begin{chunk}{ignore}
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-``MGD1 - A Robust and Efficient Multigrid Method''
-Multigrid Methods. Lecture Notes in Mathematics. 960
-Springer-Verlag. 614--630. (1982)
+\bibitem[Polya 37]{Pol37} Polya, G.
+``Kombinatorische Anzahlbestimmungen fur Gruppen,
+Graphen und chemische Verbindungen''
+Acta Math. 68 (1937) 145-254.
\end{chunk}
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+\bibitem[Powell 70]{Pow70} Powell M J D.
+``A Hybrid Method for Nonlinear Algebraic Equations''
+Numerical Methods for Nonlinear Algebraic Equations.
+(ed P Rabinowitz) Gordon and Breach. (1970)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Wicks 89]{Wic89} Wicks, Mark; Carlisle, David, Rahtz, Sebastian
-``dvipdfm.def''
-\verb|web.mit.edu/texsrc/source/latex/graphics/dvipdfm.def|
+\bibitem[Powell 74]{Pow74} Powell M J D.
+``Introduction to Constrained Optimization''
+Numerical Methods for Constrained Optimization.
+(ed P E Gill and W Murray) Academic Press. pp1-28. 1974
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Wiki 3]{Wiki3}.
-``Givens Rotations''
-\verb|en.wikipedia.org/wiki/Givens_rotation|
+\bibitem[Powell 83]{Pow83} Powell M J D.
+``Variable Metric Methods in Constrained Optimization''
+Mathematical Programming: The State of the Art.
+(ed A Bachem, M Groetschel and B Korte) Springer-Verlag. pp288--311. 1983
\end{chunk}
\begin{chunk}{axiom.bib}
-@misc{Wiki14a,
- author = "ProofWiki",
- title = "Euclidean Algorithm",
- url = "http://proofwiki.org/wiki/Euclidean_Algorithm"
+@inproceedings{Prat73,
+ author = "Pratt, Vaughan R.",
+ title = "Top down operator precedence",
+ booktitle = "Proc. 1st annual ACM SIGACT-SIGPLAN Symposium on Principles
+ of Programming Languages",
+ series = "POPL'73",
+ pages = "41-51",
+ year = "1973",
+ url = "http://hall.org.ua/halls/wizzard/pdf/Vaughan.Pratt.TDOP.pdf",
+ keywords = "axiomref",
+ paper = "Prat73.pdf"
}
\end{chunk}
-\begin{chunk}{axiom.bib}
-@misc{Wiki14b,
- author = "ProofWiki",
- title = "Division Theorem",
- url = "http://proofwiki.org/wiki/Division_Theorem"
-}
+\begin{chunk}{ignore}
+\bibitem[Press 95]{PTVF95} Press, William H.; Teukolsky, Saul A.;
+Vetterling, William T.; Flannery, Brian P.
+``Numerical Recipes in C''
+Cambridge University Press (1995) ISBN 0-521-43108-5
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Williamson 85]{Wil85} Williamson, S.G.
-``Combinatorics for Computer Science''
-Computer Science Press, 1985.
+\bibitem[Pryce 77]{PH77} Pryce J D.; Hargrave B A.
+``The Scale Pruefer Method for one-parameter and multi-parameter eigenvalue
+problems in ODEs''
+Inst. Math. Appl., Numerical Analysis Newsletter. 1(3) (1977)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Wilkinson 71]{WR71} Wilkinson J H.; Reinsch C.
-``Handbook for Automatic Computation II, Linear Algebra''
-Springer-Verlag. 1971
+\bibitem[Pryce 81]{Pry81} Pryce J D.
+``Two codes for Sturm-Liouville problems''
+Technical Report CS-81-01. Dept of Computer Science, Bristol University (1981)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Wilkinson 63]{Wil63} Wilkinson J H.
-``Rounding Errors in Algebraic Processes''
- Chapter 2. HMSO. (1963)
+\bibitem[Pryce 86]{Pry86} Pryce J D.
+``Error Estimation for Phase-function Shooting Methods for
+Sturm-Liouville Problems''
+J. Num. Anal. 6 103--123. (1986)
+
+\end{chunk}
+
+\begin{chunk}{axiom.bib}
+@misc{Puff09,
+ author = "Puffinware LLC",
+ title = "Singular Value Decomposition (SVD) Tutorial",
+ url = "http://www.puffinwarellc.com/p3a.htm"
+}
\end{chunk}
+\subsection{Q} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Wilkinson 65]{Wil65} Wilkinson J H.
-``The Algebraic Eigenvalue Problem''
- Oxford University Press. (1965)
+\bibitem[Quintana-Orti 06]{QG06} Quintana-Orti, Gregorio;
+van de Geijn, Robert
+``Improving the performance of reduction to Hessenberg form''
+ACM Transactions on Mathematical Software, 32(2):180-194, June 2006.
\end{chunk}
+\subsection{R} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{ignore}
-\bibitem[Wilkinson 78]{Wil78} Wilkinson J H.
-``Singular Value Decomposition -- Basic Aspects''
-Numerical Software -- Needs and Availability.
-(ed D A H Jacobs) Academic Press. (1978)
+\bibitem[Rabinowitz 70]{Rab70} Rabinowitz P.
+``Numerical Methods for Nonlinear Algebraic Equations''
+Gordon and Breach. (1970)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Wilkinson 79]{Wil79} Wilkinson J H.
-``Kronecker's Canonical Form and the QZ Algorithm''
-Linear Algebra and Appl. 28 285--303. 1979
+\bibitem[Ralston 65]{Ral65} Ralston A.
+``A First Course in Numerical Analysis''
+McGraw-Hill. 87--90. (1965)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Wisbauer 91]{Wis91} Wisbauer, R.
-``Bimodule Structure of Algebra''
-Lecture Notes Univ. Duesseldorf 1991
+\bibitem[Ramakrishnan 03]{Ram03} Ramakrishnan, Maya
+``A Gentle Introduction to Lyapunov Functions''
+ORSUM August 2003
+\verb|www.or.ms.unimelb.edu.au/handouts/lyaptalk.1.pdf|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Woerz-Busekros 80]{Woe80} Woerz-Busekros, A.
-``Algebra in Genetics''
-Lectures Notes in Biomathematics 36, Springer-Verlag, Heidelberg, 1980
+\bibitem[Ramsey 03]{Ra03} Ramsey, Norman
+``Noweb--A Simple, Extensible Tool for Literate Programming''
+\verb|www.eecs.harvard.edu/~nr/noweb|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Wolberg 67]{Wol67} Wolberg J R.
-``Prediction Analysis''
-Van Nostrand. (1967)
+\bibitem[Redfield 27]{Red27} Redfield, J.H.
+``The Theory of Group-Reduced Distributions''
+American J. Math., 49 (1927) 433-455.
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Wolfram 09]{Wo09} Wolfram Research
-\verb|mathworld.wolfram.com/Quaternion.html|
+\bibitem[Reinsch 67]{Rei67} Reinsch C H.
+``Smoothing by Spline Functions''
+Num. Math. 10 177--183. (1967)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Wu 87]{WU87} Wu, W.T.
-``A Zero Structure Theorem for polynomial equations solving''
-MM Research Preprints, 1987
+\bibitem[Renka 84]{Ren84} Renka R L.
+``Algorithm 624: Triangulation and Interpolation of Arbitrarily Distributed
+Points in the Plane''
+ACM Trans. Math. Softw. 10 440--442. (1984)
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Wynn 56]{Wynn56} Wynn P.
-``On a Device for Computing the $e_m(S_n )$ Transformation''
-Math. Tables Aids Comput. 10 91--96. (1956)
+\bibitem[Renka 84]{RC84} Renka R L.; Cline A K.
+``A Triangle-based C Interpolation Method''
+Rocky Mountain J. Math. 14 223--237. (1984)
\end{chunk}
-\subsection{Y} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\subsection{Z} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
\begin{chunk}{ignore}
-\bibitem[Zakrajsek 02]{Zak02} Zakrajsek, Helena
-``Applications of Hermite transform in computer algebra''
-\verb|www.imfm.si/preprinti/PDF/00835.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Zak02.pdf|
- abstract = "
- let $L$ be a linear differential operator with polynomial
- coefficients. We show that there is an isomorphism of differential
- operators ${\bf D_\alpha}$ and an integral transform ${\bf H_\alpha}$
- (called the Hermite transform) on functions for which $({\bf
- D_\alpha}{\bf L})f(x)=0$ implies ${\bf L}{\bf H_alpha}(f)(x)=0$. We
- present an algorithm that computes the Hermite transform of a rational
- function and use it to find $n+1$ linearly independent solutions of
- ${\bf L}y=0$ when $({\bf D_\alpha}{\bf L})f(x)=0$ has a rational
- solution with $n$ distinct finite poles."
+\bibitem[Reutenauer 93]{Re93} Reutenauer, Christophe
+``Free Lie Algebras''
+Oxford University Press, June 1993 ISBN 0198536798
\end{chunk}
-\begin{chunk}{axiom.bib}
-@misc{Zdan14,
- author = "Zdancewic, Steve and Martin, Milo M.K.",
- title = "Vellvm: Verifying the LLVM",
- url = "http://www.cis.upenn.edu/~stevez/vellvm"
-}
+\begin{chunk}{ignore}
+\bibitem[Reznick 93]{Rezn93} Reznick, Bruce
+``An Inequality for Products of Polynomials''
+Proc. AMS Vol 117 No 4 April 1993
+%\verb|axiom-developer.org/axiom-website/papers/Rezn93.pdf|
\end{chunk}
\begin{chunk}{ignore}
-\bibitem[Zhi 97]{Zhi97} Zhi, Lihong
-``Optimal Algorithm for Algebraic Factoring''
-\verb|www.mmrc.iss.ac.cn/~lzhi/Publications/zopfac.pdf|
-%\verb|axiom-developer.org/axiom-website/papers/Zhi97.pdf|
+\bibitem[Rich xx]{Rixx} Rich, A.D.; Jeffrey, D.J.
+``Crafting a Repository of Knowledge Based on Transformation''
+\verb|www.apmaths.uwo.ca/~djeffrey/Offprints/IntegrationRules.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Rixx.pdf|
abstract = "
- This paper presents an optimized method for factoring multivariate
- polynomials over algebraic extension fields which defined by an
- irreducible ascending set. The basic idea is to convert multivariate
- polynomials to univariate polynomials and algebraic extensions fields
- to algebraic number fields by suitable integer substitutions, then
- factorize the univariate polynomials over the algebraic number fields.
- Finally, construct multivariate factors of the original polynomial by
- Hensel lemma and TRUEFACTOR test. Some examples with timing are
- included."
+ We describe the development of a repository of mathematical knowledge
+ based on transformation rules. The specific mathematical problem is
+ indefinite integration. It is important that the repository be not
+ confused with a look-up table. The database of transformation rules is
+ at present encoded in Mathematica, but this is only one convenient
+ form of the repository, and it could be readily translated into other
+ formats. The principles upon which the set of rules is compiled is
+ described. One important principle is minimality. The benefits of the
+ approach are illustrated with examples, and with the results of
+ comparisons with other approaches."
\end{chunk}
-\subsection{To Be Classified} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{chunk}{axiom.bib}
-@PhdThesis{Kalt82,
- author = "Kaltofen, E.",
- title = "On the complexity of factoring polynomials with integer
- coefficients",
- school = "RPI",
- address = "Troy, N. Y.",
- year = "1982",
- month = "December",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/82/Ka82_thesis.pdf",
- paper = "Kalt82.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Rich 10]{Ri10} Rich, Albert D.
+``Rule-based Mathematics''
+\verb|www.apmaths.uwo.ca/~arich|
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt82a,
- author = "Kaltofen, E.",
- title = "A polynomial-time reduction from bivariate to univariate
- integral polynomial factorization",
- booktitle = "Proc. 23rd Annual Symp. Foundations of Comp. Sci.",
- year = "1982",
- pages = "57--64",
- organization = "IEEE",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/82/Ka82_focs.pdf",
- paper = "Kalt82a.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Richardson 94]{RF94} Richardson, Dan; Fitch, John
+``The identity problem for elementary functions and constants''
+ACM Proc. of ISSAC 94 pp285-290 ISBN 0-89791-638-7
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InCollection{Kalt82b,
- author = "Kaltofen, E.",
- title = "Polynomial factorization",
- editor = "B. Buchberger and G. Collins and R. Loos",
- booktitle = "Computer Algebra",
- edition = "2",
- pages = "95--113",
- publisher = "Springer-Verlag",
- year = "1982",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/82/Ka82_survey.ps.gz",
- keywords = "survey",
- paper = "Kalt82b.ps"
-}
+\begin{chunk}{ignore}
+\bibitem[Richtmyer 67]{RM67} Richtmyer R D.; Morton K W.
+``Difference Methods for Initial-value Problems''
+Interscience (2nd Edition). (1967)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt83,
- author = "Kaltofen, E.",
- title = "On the complexity of finding short vectors in integer lattices",
- booktitle = "Proc. EUROCAL '83",
- series = "Lect. Notes Comput. Sci.",
- year = "1983",
- volume = "162",
- pages = "236--244",
- publisher = "Springer-Verlag",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/83/Ka83_eurocal.pdf",
- paper = "Kalt83.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Rioboo 92]{REF-Rio92} Rioboo, R.
+``Real algebraic closure of an ordered field, implementation in Axiom''
+In Wang [Wan92], pp206-215 ISBN 0-89791-489-9 (soft cover)
+In proceedings of the ISSAC'92 Conference, Berkeley 1992 pp. 206-215.
+0-89791-490-2 (hard cover) LCCN QA76.95.I59 1992
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt84,
- author = "Kaltofen, E.",
- title = "A Note on the {Risch} Differential Equation",
- booktitle = "Proc. EUROSAM '84",
- pages = "359--366",
- crossref = "EUROSAM84",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/84/Ka84_risch.ps.gz",
- paper = "Kalt84.ps"
-}
+\begin{chunk}{ignore}
+\bibitem[Rioboo 96]{Rio96} Rioboo, R.
+``Generic computation of the real closure of an ordered field''
+In Mathematics and Computers in Simulation Volume 42, Issue 4-6,
+November 1996.
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt84a,
- author = "Kaltofen, E. and Yui, N.",
- title = "Explicit construction of the {Hilbert} class field of imaginary
- quadratic fields with class number 7 and 11",
- booktitle = "Proc. EUROSAM '84",
- pages = "310--320",
- crossref = "EUROSAM84",
- url =
- "http://www.math.ncsu.edu/~kaltofen/bibliography/84/KaYui84_eurosam.ps.gz",
- paper = "Kalt84a.ps"
-}
+\begin{chunk}{ignore}
+\bibitem[Ritt 50]{Ritt50} Ritt, Joseph Fels
+``Differential Algebra''
+AMS Colloquium Publications Volume 33 ISBN 978-0-8218-4638-4
\end{chunk}
-\begin{chunk}{axiom.bib}
-@TechReport{Kalt84b,
- author = "Kaltofen, E.",
- title = "The Algebraic Theory of Integration",
- institution = "RPI",
- address = "Dept. Comput. Sci., Troy, New York",
- year = "1984",
- url =
- "http://www.math.ncsu.edu/~kaltofen/bibliography/84/Ka84_integration.pdf",
- paper = "Kalt84b.pdf"
-}
-
-\end{chunk}
+\begin{chunk}{ignore}
+\bibitem[Rote 01]{Rote01} Rote, G\"unter
+``Division-free algorithms for the determinant and the Pfaffian''
+in Computational Discrete Mathematics ISBN 3-540-42775-9 pp119-135
+\verb|page.mi.fu-berlin.de/rote/Papers/pdf/Division-free+algorithms.pdf|
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt85,
- author = "Kaltofen, E.",
- title = "Effective {Hilbert} Irreducibility",
- booktitle = "Proc. EUROSAM '84",
- pages = "275--284",
- crossref = "EUROSAM84",
- url =
- "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_infcontr.ps.gz",
- paper = "Kalt85.ps"
-}
-
\end{chunk}
-\begin{chunk}{axiom.bib}
-@Article{Kalt85a,
- author = "Kaltofen, E.",
- title = "Fast parallel absolute irreducibility testing",
- journal = "Journal of Symbolic Computation",
- year = "1985",
- volume = "1",
- number = "1",
- pages = "57--67",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_jsc.pdf",
- paper = "Kalt85a.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Rubey 07]{Rub07} Rubey, Martin
+``Formula Guessing with Axiom''
+April 2007
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt85b,
- author = "Kaltofen, E.",
- title = "Computing with polynomials given by straight-line programs {II};
- sparse factorization",
- booktitle = "Proc. 26th Annual Symp. Foundations of Comp. Sci.",
- year = "1985",
- pages = "451--458",
- organization = "IEEE",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_focs.ps.gz",
- paper = "Kalt85b.ps"
-}
+\begin{chunk}{ignore}
+\bibitem[Rutishauser 69]{Rut69} Rutishauser H.
+``Computational aspects of F L Bauer's simultaneous iteration method''
+Num. Math. 13 4--13. (1969)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@TechReport{Kalt85c,
- author = "E. Kaltofen",
- title = "Sparse Hensel lifting",
- institution = "RPI",
- address = "Dept. Comput. Sci., Troy, N. Y.",
- year = "1985",
- number = "85-12",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_techrep.pdf",
- paper = "Kalt85c.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Rutishauser 70]{Rut70} Rutishauser H.
+``Simultaneous iteration method for symmetric matrices''
+Num. Math. 16 205--223. (1970)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt85d,
- author = "Kaltofen, E.",
- title = "Sparse Hensel lifting",
- booktitle = "EUROCAL 85 European Conf. Comput. Algebra Proc. Vol. 2",
- crossref = "EUROCAL85",
- pages = "4--17",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_eurocal.pdf",
- paper = "Kalt85d.pdf"
-}
+\subsection{S} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{ignore}
+\bibitem[Schafer 66]{Sch66} Schafer, R.D.
+``An Introduction to Nonassociative Algebras''
+Academic Press, New York, 1966
\end{chunk}
-\begin{chunk}{axiom.bib}
-@Article{Kalt85e,
- author = "Kaltofen, E.",
- title = "Polynomial-time reductions from multivariate to bi- and univariate
- integral polynomial factorization",
- journal = "{SIAM} J. Comput.",
- year = "1985",
- volume = "14",
- number = "2",
- pages = "469--489",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/85/Ka85_sicomp.pdf",
- paper = "Kalt85e.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Schoenberg 53]{SW53} Schoenberg I J.; Whitney A.
+``On Polya Frequency Functions III''
+Trans. Amer. Math. Soc. 74 246--259. (1953)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@Article{Gath85,
- author = "Gathen, Joachim von zur; Kaltofen, E.",
- title = "Factoring sparse multivariate polynomials",
- journal = "J. Comput. System Sci.",
- year = "1985",
- volume = "31",
- pages = "265--287",
- url =
- "http://www.math.ncsu.edu/~kaltofen/bibliography/85/GaKa85_mathcomp.ps.gz",
- paper = "Gath85.ps"
-}
+\begin{chunk}{ignore}
+\bibitem[Schoenhage 82]{Sch82} Schoenhage, A.
+``The fundamental theorem of algebra in terms of computational complexity''
+preliminary report, Univ. Tuebingen, 1982
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt86,
- author = "Kaltofen, E.",
- title = "Uniform closure properties of p-computable functions",
- booktitle = "Proc. 18th Annual ACM Symp. Theory Comput.",
- year = "1986",
- pages = "330--337",
- organization = "ACM",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/86/Ka86_stoc.pdf",
- paper = "Kalt86.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Schonfelder 76]{Sch76} Schonfelder J L.
+``The Production of Special Function Routines for a Multi-Machine Library''
+Software Practice and Experience. 6(1) (1976)
\end{chunk}
\begin{chunk}{axiom.bib}
-@Article{Kalt87,
- author = "Kaltofen, E. and Krishnamoorthy, M.S. and Saunders, B.D.",
- title = "Fast parallel computation of Hermite and Smith forms of
- polynomial matrices",
- journal = "SIAM J. Alg. Discrete Math.",
- year = "1987",
- volume = "8",
- pages = "683--690",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/KKS87.pdf",
- paper = "Kalt87.pdf"
+@book{Segg93,
+ author = "{von Seggern}, David Henry",
+ title = "CRC Standard Curves and Surfaces",
+ publisher = "CRC Press",
+ year = "1993",
+ isbn = "0-8493-0196-3"
}
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InCollection{Kalt87a,
- author = "Kaltofen, E.",
- editor = "J. F. Traub",
- title = "Computer algebra algorithms",
- booktitle = "Annual Review in Computer Science",
- pages = "91--118",
- publisher = "Annual Reviews Inc.",
- year = "1987",
- volume = "2",
- address = "Palo Alto, California",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_annrev.pdf",
- paper = "Kalt87a.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Seiler 95a]{Sei95a} Seiler, W.M.; Calmet, J.
+``JET -- An Axiom Environment for Geometric Computations with Differential
+Equations''
+%\verb|axiom-developer.org/axiom-website/papers/Sei95a.pdf|
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt87b,
- author = "Kaltofen, E.",
- title = "Single-factor Hensel lifting and its application to the
- straight-line complexity of certain polynomials",
- booktitle = "Proc. 19th Annual ACM Symp. Theory Comput.",
- year = "1987",
- pages = "443--452",
- organization = "ACM",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_stoc.pdf",
- paper = "Kalt87b.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Shepard 68]{She68} Shepard D.
+``A Two-dimensional Interpolation Function for Irregularly Spaced Data''
+Proc. 23rd Nat. Conf. ACM. Brandon/Systems Press Inc.,
+Princeton. 517--523. 1968
\end{chunk}
-\begin{chunk}{axiom.bib}
-@Article{Kalt87c,
- author = "Kaltofen, E.",
- title = "Deterministic irreducibility testing of polynomials over
- large finite fields",
- journal = "Journal of Symbolic Computation",
- year = "1987",
- volume = "4",
- pages = "77--82",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/87/Ka87_jsc.ps.gz",
- paper = "Kalt87c.ps"
-}
+\begin{chunk}{ignore}
+\bibitem[Shirayanagi 96]{Shir96} Shirayanagi, Kiyoshi
+``Floating point Gr\"obner bases''
+Mathematics and Computers in Simulation 42 pp 509-528 (1996)
+%\verb|axiom-developer.org/axiom-website/papers/Shir96.pdf|
+ abstract = "
+ Bracket coefficients for polynomials are introduced. These are like
+ specific precision floating point numbers together with error
+ terms. Working in terms of bracket coefficients, an algorithm that
+ computes a Gr{\"o}bner basis with floating point coefficients is
+ presented, and a new criterion for determining whether a bracket
+ coefficient is zero is proposed. Given a finite set $F$ of polynomials
+ with real coefficients, let $G_\mu$ be the result of the algorithm for
+ $F$ and a precision $\mu$, and $G$ be a true Gr{\"o}bner basis of
+ $F$. Then, as $\mu$ approaches infinity, $G_\mu$ converges to $G$
+ coefficientwise. Moreover, there is a precision $M$ such that if
+ $\mu \ge M$, then the sets of monomials with non-zero coefficients of
+ $G_\mu$ and $G$ are exactly the same. The practical usefulness of the
+ algorithm is suggested by experimental results."
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt88,
- author = "Kaltofen, E. and Trager, B.",
- title = "Computing with polynomials given by black boxes for their
- evaluations: Greatest common divisors, factorization, separation of
- numerators and denominators",
- booktitle = "Proc. 29th Annual Symp. Foundations of Comp. Sci.",
- pages = "296--305",
- year = "1988",
- organization = "IEEE",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/focs88.ps.gz",
- paper = "Kalt88.ps"
-}
+\begin{chunk}{ignore}
+\bibitem[Sims 71]{Sims71} Sims, C.
+``Determining the Conjugacy Classes of a Permutation Group''
+Computers in Algebra and Number Theory, SIAM-AMS Proc., Vol. 4,
+American Math. Soc., 1991, pp191-195
\end{chunk}
-\begin{chunk}{axiom.bib}
-@Article{Mill88,
- author = "Miller, G.L. and Ramachandran, V. and Kaltofen, E.",
- title = "Efficient parallel evaluation of straight-line code and
- arithmetic circuits",
- journal = "SIAM J. Comput.",
- year = "1988",
- volume = "17",
- number = "4",
- pages = "687--695",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/MRK88.pdf",
- paper = "Mill88.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Singer 89]{Sing89} Singer, M.F.
+``Formal Solutions of Differential Equations''
+J. Symbolic COmputation 10, No.1 59-94 (1990)
+%\verb|axiom-developer.org/axiom-website/papers/Sing89.pdf|
+ keywords = "survey",
+ abstract = "
+ We give a survey of some methods for finding formal solutions of
+ differential equations. These include methods for finding power series
+ solutions, elementary and liouvillian solutions, first integrals, Lie
+ theoretic methods, transform methods, asymptotic methods. A brief
+ discussion of difference equations is also included."
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt88a,
- author = "Kaltofen, E. and Yagati, Lakshman",
- title = "Improved sparse multivariate polynomial interpolation algorithms",
- booktitle = "Symbolic Algebraic Comput. Internat. Symp. ISSAC '88 Proc.",
- crossref = "ISSAC88",
- pages = "467--474",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/KaLa88.pdf",
- paper = "Kalt88a.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Sit 92]{REF-Sit92} Sit, William
+``An Algorithm for Parametric Linear Systems''
+J. Sym. Comp., April 1992
\end{chunk}
-\begin{chunk}{axiom.bib}
-@Article{Kalt88b,
- author = "Kaltofen, E.",
- title = "Greatest common divisors of polynomials given by
- straight-line programs",
- journal = "J. ACM",
- year = "1988",
- volume = "35",
- number = "1",
- pages = "231--264",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/Ka88_jacm.pdf",
- paper = "Kalt88b.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Smith 67]{Smi67} Smith B T.
+``ZERPOL: A Zero Finding Algorithm for Polynomials Using Laguerre's Method''
+Technical Report. Department of Computer Science, University of Toronto,
+Canada. (1967)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@Article{Free88,
- author = "Freeman, T.S. and Imirzian, G. and Kaltofen, E. and
- Yagati, Lakshman",
- title = "DAGWOOD: A system for manipulating polynomials given by
- straight-line programs",
- journal = "ACM Trans. Math. Software",
- year = "1988",
- volume = "14",
- number = "3",
- pages = "218--240",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/FIKY88.pdf",
- paper = "Free88.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Smith 85]{Smi85} Smith G D.
+``Numerical Solution of Partial Differential Equations: Finite Difference
+Methods''
+Oxford University Press (3rd Edition). (1985)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@Article{Greg88,
- author = "Gregory, B.; Kaltofen, E.",
- title = "Analysis of the binary complexity of asymptotically fast
- algorithms for linear system solving",
- journal = "SIGSAM Bulletin",
- year = "1988",
- month = "April",
- volume = "22",
- number = "2",
- pages = "41--49",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/88/GrKa88.pdf",
- paper = "Grey88.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Sobol 74]{Sob74} Sobol I M.
+``The Monte Carlo Method''
+The University of Chicago Press. 1974
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InCollection{Kalt89,
- author = "Kaltofen, E.",
- editor = "S. Micali",
- title = "Factorization of polynomials given by straight-line programs",
- booktitle = "Randomness and Computation",
- pages = "375--412",
- publisher = "JAI Press Inc.",
- year = "1989",
- volume = "5",
- series = "Advances in Computing Research",
- address = "Greenwhich, Connecticut",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_slpfac.pdf",
- paper = "Kalt89.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Steele 90]{Ste90} Steele, Guy L.
+``Common Lisp The Language''
+Second Edition ISBN 1-55558-041-6 Digital Press (1990)
\end{chunk}
\begin{chunk}{axiom.bib}
-@Article{Kalt89a,
- author = "Kaltofen, E.; Rolletschek, H.",
- title = "Computing greatest common divisors and factorizations in
- quadratic number fields",
- journal = "Math. Comput.",
- year = "1989",
- volume = "53",
- number = "188",
- pages = "697--720",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/KaRo89.pdf",
- paper = "Kalt89a.pdf"
+@misc{Stic93,
+ author = "Stichtenoth, H.",
+ title = "Algebraic function fields and codes",
+ publisher = "Springer-Verlag",
+ year = "1993"
}
\end{chunk}
-\begin{chunk}{axiom.bib}
-@Unpublished{Kalt89b,
- author = "Kaltofen, E.",
- title = "Processor efficient parallel computation of polynomial greatest
- common divisors",
- year = "1989",
- month = "July",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_gcd.ps.gz",
- paper = "Kalt89b.ps"
-}
+\begin{chunk}{ignore}
+\bibitem[Stinson 90]{Stin90} Stinson, D.R.
+``Some observations on parallel Algorithms for fast exponentiation
+in $GF(2^n)$''
+Siam J. Comp., Vol.19, No.4, pp.711-717, August 1990
+%\verb|axiom-developer.org/axiom-website/Stin90.pdf|
+ abstract = "
+ A normal basis represention in $GF(2^n)$ allows squaring to be
+ accomplished by a cyclic shift. Algorithms for multiplication in
+ $GF(2^n)$ using a normal basis have been studied by several
+ researchers. In this paper, algorithms for performing exponentiation
+ in $GF(2^n)$ using a normal basis, and how they can be speeded up by
+ using parallelization, are investigated."
\end{chunk}
-\begin{chunk}{axiom.bib}
-@TechReport{Kalt89c,
- author = "Kaltofen, E.",
- title = "Parallel Algebraic Algorithm Design",
- institution = "RPI",
- address = "Dept. Comput. Sci., Troy, New York",
- year = "1989",
- month = "July",
- url =
- "http://www.math.ncsu.edu/~kaltofen/bibliography/89/Ka89_parallel.ps.gz",
- paper = "Kalt89c.ps"
-}
+\begin{chunk}{ignore}
+\bibitem[Stroud 66]{SS66} Stroud A H.; Secrest D.
+``Gaussian Quadrature Formulas''
+Prentice-Hall. (1966)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Cann89,
- author = "Canny, J. and Kaltofen, E. and Yagati, Lakshman",
- title = "Solving systems of non-linear polynomial equations faster",
- booktitle = "Proc. 1989 Internat. Symp. Symbolic Algebraic Comput.",
- crossref = "ISSAC89",
- pages = "121--128",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/CKL89.pdf",
- paper = "Cann89.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Stroud 71]{Str71} Stroud A H.
+``Approximate Calculation of Multiple Integrals''
+Prentice-Hall 1971
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt89d,
- author = "Kaltofen, E. and Valente, T. and Yui, N.",
- title = "An improved {Las Vegas} primality test",
- booktitle = "Proc. 1989 Internat. Symp. Symbolic Algebraic Comput.",
- crossref = "ISSAC89",
- pages = "26--33",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/89/KVY89.pdf",
- paper = "Kalt89d.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Swarztrauber 79]{SS79} Swarztrauber P N.; Sweet R A.
+``Efficient Fortran Subprograms for the Solution of Separable Elliptic Partial
+Differential Equations''
+ACM Trans. Math. Softw. 5 352--364. (1979)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt90,
- author = "Kaltofen, E. and Lakshman, Y.N. and Wiley, J.M.",
- editor = "S. Watanabe and M. Nagata",
- title = "Modular rational sparse multivariate polynomial interpolation",
- booktitle = "Proc. 1990 Internat. Symp. Symbolic Algebraic Comput.",
- pages = "135--139",
- publisher = "ACM Press",
- year = "1990",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/KLW90.pdf",
- paper = "Kalt90.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Swarztrauber 84]{SS84} Swarztrauber P N.
+``Fast Poisson Solvers''
+Studies in Numerical Analysis. (ed G H Golub)
+Mathematical Association of America. (1984)
\end{chunk}
+\subsection{T} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
\begin{chunk}{axiom.bib}
-@Article{Kalt90a,
- author = "Kaltofen, E. and Krishnamoorthy, M.S. and Saunders, B.D.",
- title = "Parallel algorithms for matrix normal forms",
- journal = "Linear Algebra and Applications",
- year = "1990",
- volume = "136",
- pages = "189--208",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/KKS90.pdf",
- paper = "Kalt90a.pdf"
+@book{Tait1890,
+ author = "Tait, P.G.",
+ title = "An Elementary Treatise on Quaternions",
+ publisher = "C.J. Clay and Sons, Cambridge University Press Warehouse,
+ Ave Maria Lane",
+ year = "1890"
}
\end{chunk}
-\begin{chunk}{axiom.bib}
-@Article{Kalt90b,
- author = "Kaltofen, E.",
- title = "Computing the irreducible real factors and components of an
- algebraic curve",
- journal = "Applic. Algebra Engin. Commun. Comput.",
- year = "1990",
- volume = "1",
- number = "2",
- pages = "135--148",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/Ka90_aaecc.pdf",
- paper = "Kalt90b.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Taivalsaari 96]{Tai96} Taivalsaari, Antero
+``On the Notion of Inheritance''
+ACM Computing Surveys, Vol 28 No 3 Sept 1996 pp438-479
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InCollection{Kalt90c,
- author = "Kaltofen, E.",
- editor = "D. V. Chudnovsky and R. D. Jenks",
- title = "Polynomial Factorization 1982-1986",
- booktitle = "Computers in Mathematics",
- pages = "285--309",
- publisher = "Marcel Dekker, Inc.",
- year = "1990",
- volume = "125",
- series = "Lecture Notes in Pure and Applied Mathematics",
- address = "New York, N. Y.",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/Ka90_survey.ps.gz",
- keywords = "survey",
- paper = "Kalt90c.ps"
-}
+\begin{chunk}{ignore}
+\bibitem[Temme 87]{Tem87} Temme N M.
+``On the Computation of the Incomplete Gamma Functions for Large Values of
+the Parameters''
+Algorithms for Approximation. (ed J C Mason and M G Cox)
+Oxford University Press. (1987)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@Article{Kalt90d,
- author = "Kaltofen, E.; Trager, B.",
- title = "Computing with polynomials given by black boxes for their
- evaluations: Greatest common divisors, factorization, separation of
- numerators and denominators",
- journal = "J. Symbolic Comput.",
- year = "1990",
- volume = "9",
- number = "3",
- pages = "301--320",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/90/KaTr90.pdf",
- paper = "Kalt90d.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Temperton 83a]{Tem83a} Temperton C.
+``Self-sorting Mixed-radix Fast Fourier Transforms''
+J. Comput. Phys. 52 1--23. (1983)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt91,
- author = "Kaltofen, E. and Saunders, B.D.",
- editor = "H. F. Mattson and T. Mora and T. R. N. Rao",
- title = "On {Wiedemann's} method of solving sparse linear systems",
- booktitle = "Proc. AAECC-9",
- series = "Lect. Notes Comput. Sci.",
- volume = "539",
- pages = "29--38",
- publisher = "Springer-Verlag",
- year = "1991",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaSa91.pdf",
- paper = "Kalt91.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Temperton 83b]{Tem83b} Temperton C.
+``Fast Mixed-Radix Real Fourier Transforms''
+J. Comput. Phys. 52 340--350. (1983)
\end{chunk}
\begin{chunk}{axiom.bib}
-@InProceedings{Kalt91a,
- author = "Kaltofen, E. and Singer, M.F.",
- editor = "D. V. Shirkov and V. A. Rostovtsev and V. P. Gerdt",
- title = "Size efficient parallel algebraic circuits for partial derivatives",
- booktitle =
- "IV International Conference on Computer Algebra in Physical Research",
- pages = "133--145",
- publisher = "World Scientific Publ. Co.",
- year = "1991",
- address = "Singapore",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaSi91.pdf",
- paper = "Kalt91a.pdf"
+@article{Thur94,
+ author = "Thurston, William P.",
+ title = "On Proof and Progress in Mathematics",
+ journal = "Bulletin AMS",
+ volume = "30",
+ number = "2",
+ month = "April",
+ year = "1994",
+ url = "http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf",
+ paper = "Thur94.pdf"
}
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InCollection{Kalt91b,
- author = "Kaltofen, E. and Yui, N.",
- editor = "D. V. Chudnovsky and G. V. Chudnovsky and H. Cohn and
- M. B. Nathanson",
- title = "Explicit construction of {Hilbert} class fields of imaginary
- quadratic fields by integer lattice reduction",
- booktitle = "Number Theory New York Seminar 1989--1990",
- pages = "150--202",
- publisher = "Springer-Verlag",
- year = "1991",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaYui91.pdf",
- paper = "Kalt91b.pdf"
-}
+\subsection{U} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{ignore}
+\bibitem[Unknown 61]{Unk61} Unknown
+``Chebyshev-series''
+Modern Computing Methods
+Chapter 8. NPL Notes on Applied Science (2nd Edition). 16 HMSO. 1961
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Diaz91,
- author = "Diaz, A.; Kaltofen,E.; Schmitz, K.; Valente, T.",
- title = "DSC A System for Distributed Symbolic Computation",
- booktitle = "Proc. 1991 Internat. Symp. Symbolic Algebraic Comput.",
- crossref = "ISSAC91",
- pages = "323--332",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/DKSV91.pdf",
- paper = "Diaz91.pdf"
-}
+\subsection{V} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{ignore}
+\bibitem[Van Dooren 76]{vDDR76} Van Dooren P.; De Ridder L.
+``An Adaptive Algorithm for Numerical Integration over an N-dimensional
+Cube''
+J. Comput. Appl. Math. 2 207--217. (1976)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt91c,
- author = "Kaltofen, E. and Pan, V.",
- title = "Processor efficient parallel solution of linear systems over
- an abstract field",
- booktitle = "Proc. SPAA '91 3rd Ann. ACM Symp. Parallel Algor. Architecture",
- pages = "180--191",
- publisher = "ACM Press",
- year = "1991",
- address = "New York, N.Y.",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/KaPa91.pdf",
- paper = "Kalt91c.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[van Hoeij 94]{REF-vH94} van Hoeij, M.
+``An algorithm for computing an integral
+basis in an algebraic function field''
+{\sl J. Symbolic Computation}
+18(4):353-364, October 1994
\end{chunk}
-\begin{chunk}{axiom.bib}
-@Article{Cant91,
- author = "Cantor, D.G. and Kaltofen, E.",
- title = "On fast multiplication of polynomials over arbitrary algebras",
- journal = "Acta Inform.",
- year = "1991",
- volume = "28",
- number = "7",
- pages = "693--701",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/91/CaKa91.pdf",
- paper = "Cant91.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Van Loan 92]{Van92} Van Loan, C.
+``Computational Frameworks for the Fast Fourier Transform''
+SIAM Philadelphia. (1992)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt92,
- author = "Kaltofen, E. and Pan, V.",
- title = "Processor-efficient parallel solution of linear systems {II}:
- the positive characteristic and singular cases",
- booktitle = "Proc. 33rd Annual Symp. Foundations of Comp. Sci.",
- year = "1992",
- pages = "714--723",
- publisher = "IEEE Computer Society Press",
- address = "Los Alamitos, California",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/92/KaPa92.pdf",
- paper = "Kalt92.pdf"
-}
+\subsection{W} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{ignore}
+\bibitem[Wait 85]{WM85} Wait R.; Mitchell A R.
+``Finite Element Analysis and Application''
+Wiley. (1985)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt92a,
- author = "Kaltofen, E.",
- title = "On computing determinants of matrices without divisions",
- booktitle = "Proc. 1992 Internat. Symp. Symbolic Algebraic Comput.",
- crossref = "ISSAC92",
- pages = "342--349",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/92/Ka92_issac.pdf",
- paper = "Kalt92a.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Wang 92]{Wang92} Wang, D.M.
+``An implementation of the characteristic set method in Maple''
+Proc. DISCO'92 Bath, England
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt92b,
- author = "Kaltofen, E.",
- title = "Polynomial factorization 1987-1991",
- booktitle = "Proc. LATIN '92",
- editor = "I. Simon",
- series = "Lect. Notes Comput. Sci.",
- volume = "583",
- pages = "294--313",
- publisher = "Springer-Verlag",
- year = "1992",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/92/Ka92_latin.pdf",
- keywords = "survey",
- paper = "Kalt92b.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Ward 75]{War75} Ward, R C.
+``The Combination Shift QZ Algorithm''
+SIAM J. Numer. Anal. 12 835--853. 1975
\end{chunk}
\begin{chunk}{axiom.bib}
-@InProceedings{Kalt93,
- author = "Kaltofen, E.",
- title = "Computational Differentiation and Algebraic Complexity Theory",
- booktitle = "Workshop Report on First Theory Institute on Computational
- Differentiation",
- editor = "C. H. Bischof and A. Griewank and P. M. Khademi",
- publisher = "Argonne National Laboratory",
- address = "Argonne, Illinois",
- series = "Tech. Rep.",
- volume = "ANL/MCS-TM-183",
- month = "December",
- year = "1993",
- pages = "28--30",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_diff.pdf",
- paper = "Kalt93.pdf"
+@misc{Watt03,
+ author = "Watt, Stephen",
+ title = "Aldor",
+ url = "http://www.aldor.org",
+ year = "2003"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@InCollection{Kalt93a,
- author = "Kaltofen, E.",
- editor = "J. Reif",
- title = "Dynamic parallel evaluation of computation {DAG}s",
- booktitle = "Synthesis of Parallel Algorithms",
- pages = "723--758",
- publisher = "Morgan Kaufmann Publ.",
- year = "1993",
- address = "San Mateo, California",
- url =
- "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_synthesis.ps.gz",
- paper = "Kalt93a.ps"
+@misc{Weil71,
+ author = "Weil, Andr\'{e}",
+ title = "Courbes alg\'{e}briques et vari\'{e}t\'{e}s Abeliennes",
+ year = "1971"
}
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Diaz93,
- author = "Diaz, A. and Kaltofen, E. and Lobo, A. and Valente, T.",
- editor = "A. Miola",
- title = "Process scheduling in {DSC} and the large sparse linear
- systems challenge",
- booktitle = "Proc. DISCO '93",
- series = "Lect. Notes Comput. Sci.",
- pages = "66--80",
- year = "1993",
- volume = "722",
- publisher = "Springer-Verlag",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/93/DHKLV93.pdf",
- paper = "Diaz93.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Weisstein]{Wein} Weisstein, Eric W.
+``Hypergeometric Function''
+MathWorld - A Wolfram Web Resource
+\verb|mathworld.wolfram.com/HypergeometricFunction.html|
\end{chunk}
\begin{chunk}{axiom.bib}
-@Article{Kalt93b,
- author = "Kaltofen, E.",
- title = "Direct proof of a theorem by Kalkbrener, Sweedler, and Taylor",
- journal = "SIGSAM Bulletin",
- year = "1993",
- volume = "27",
- number = "4",
- pages = "2",
- url =
- "http://www.math.ncsu.edu/~kaltofen/bibliography/93/Ka93_sambull.ps.gz",
- paper = "Kalt93b.ps"
+@misc{Weit03,
+ author = "Weitz, E.",
+ title = "CL-WHO -Yet another Lisp markup language",
+ year = "2003",
+ url = "http://www.weitz.de/cl-who/"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@InProceedings{Kalt94,
- author = "Kaltofen, E. and Pan, V.",
- title = "Parallel solution of Toeplitz and Toeplitz-like linear
- systems over fields of small positive characteristic",
- booktitle = "Proc. First Internat. Symp. Parallel Symbolic Comput.",
- crossref = "PASCO94",
- pages = "225--233",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/KaPa94.pdf",
- paper = "Kalt94.pdf"
+@misc{Weit06,
+ author = "Weitz, E.",
+ title = "HUNCHENTOOT - The Common Lisp web server formerly known as TBNL",
+ year = "2006",
+ url = "http://www.weitz.de/hunchentoot"
}
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Chan94,
- author = "Chan, K.C. and Diaz, A. and Kaltofen, E.",
- editor = "R. J. Lopez",
- title = "A distributed approach to problem solving in Maple",
- booktitle = "Maple V: Mathematics and its Application",
- pages = "13--21",
- publisher = {Birkh\"auser},
- year = "1994",
- series = "Proceedings of the Maple Summer Workshop and Symposium (MSWS'94)",
- address = "Boston",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/CDK94.ps.gz",
- paper = "Chan94.ps"
-}
+\begin{chunk}{ignore}
+\bibitem[Wesseling 82a]{Wes82a} Wesseling, P.
+``MGD1 - A Robust and Efficient Multigrid Method''
+Multigrid Methods. Lecture Notes in Mathematics. 960
+Springer-Verlag. 614--630. (1982)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt94a,
- author = "Kaltofen, E. and Lobo, A.",
- title = "Factoring high-degree polynomials by the black box
- Berlekamp algorithm",
- booktitle = "Proc. 1994 Internat. Symp. Symbolic Algebraic Comput.",
- crossref = "ISSAC94",
- pages = "90--98",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/KaLo94.ps.gz",
- paper = "Kalt94a.ps"
-}
+\begin{chunk}{ignore}
+\bibitem[Wesseling 82b]{Wes82b} Wesseling, P.
+``Theoretical Aspects of a Multigrid Method''
+SIAM J. Sci. Statist. Comput. 3 387--407. (1982)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt94b,
- author = "Kaltofen, E.",
- title = "Asymptotically fast solution of {Toeplitz}-like singular
- linear systems",
- booktitle = "Proc. 1994 Internat. Symp. Symbolic Algebraic Comput.",
- pages = "297--304",
- crossref = "ISSAC94",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/94/Ka94_issac.pdf",
- paper = "Kalt94b.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Wicks 89]{Wic89} Wicks, Mark; Carlisle, David, Rahtz, Sebastian
+``dvipdfm.def''
+\verb|web.mit.edu/texsrc/source/latex/graphics/dvipdfm.def|
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Sama95,
- author = "Samadani, M. and Kaltofen, E.",
- title = "Prediction based task scheduling in distributed computing",
- booktitle = "Languages, Compilers and Run-Time Systems for Scalable
- Computers",
- editor = "B. K. Szymanski and B. Sinharoy",
- publisher = "Kluwer Academic Publ.",
- address = "Boston",
- pages = "317--320",
- year = "1996",
- url =
- "http://www.math.ncsu.edu/~kaltofen/bibliography/95/SaKa95_poster.ps.gz",
- paper = "Sama95.ps"
-}
+\begin{chunk}{ignore}
+\bibitem[Wiki 3]{Wiki3}.
+``Givens Rotations''
+\verb|en.wikipedia.org/wiki/Givens_rotation|
\end{chunk}
\begin{chunk}{axiom.bib}
-@Article{Kalt95,
- author = "Kaltofen, E.",
- title = "Analysis of {Coppersmith}'s block {Wiedemann} algorithm for the
- parallel solution of sparse linear systems",
- journal = "Math. Comput.",
- year = "1995",
- volume = "64",
- number = "210",
- pages = "777--806",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/Ka95_mathcomp.pdf",
- paper = "Kalt95.pdf"
+@misc{Wiki14a,
+ author = "ProofWiki",
+ title = "Euclidean Algorithm",
+ url = "http://proofwiki.org/wiki/Euclidean_Algorithm"
}
\end{chunk}
\begin{chunk}{axiom.bib}
-@InProceedings{Diaz95,
- author = "Diaz, A. and Kaltofen, E.",
- title = "On computing greatest common divisors with polynomials given by
- black boxes for their evaluation",
- booktitle = "Proc. 1995 Internat. Symp. Symbolic Algebraic Comput.",
- crossref = "ISSAC95",
- pages = "232--239",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/DiKa95.ps.gz",
- paper = "Diaz95.ps"
+@misc{Wiki14b,
+ author = "ProofWiki",
+ title = "Division Theorem",
+ url = "http://proofwiki.org/wiki/Division_Theorem"
}
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt95a,
- author = "Kaltofen, E. and Shoup, V.",
- title = "Subquadratic-time factoring of polynomials over finite fields",
- booktitle = "Proc. 27th Annual ACM Symp. Theory Comput.",
- year = "1995",
- publisher = "ACM Press",
- address = "New York, N.Y.",
- pages = "398--406",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/KaSh95.ps.gz",
- paper = "Kalt95a.ps"
-}
+\begin{chunk}{ignore}
+\bibitem[Williamson 85]{Wil85} Williamson, S.G.
+``Combinatorics for Computer Science''
+Computer Science Press, 1985.
\end{chunk}
-\begin{chunk}{axiom.bib}
-@Article{Hitz95,
- author = "Kitz, M.A. and Kaltofen, E.",
- title = "Integer division in residue number systems",
- journal = "IEEE Trans. Computers",
- year = "1995",
- volume = "44",
- number = "8",
- pages = "983--989",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/HiKa95.pdf",
- paper = "Hitz95.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Wilkinson 71]{WR71} Wilkinson J H.; Reinsch C.
+``Handbook for Automatic Computation II, Linear Algebra''
+Springer-Verlag. 1971
\end{chunk}
-\begin{chunk}{axiom.bib}
-@Article{Diaz95a,
- author = "Diaz, A. and Hitz, M. and Kaltofen, E. and Lobo, A. and
- Valtente, T.",
- title = "Process scheduling in {DSC} and the large sparse linear
- systems challenge",
- journal = "Journal of Symbolic Computing",
- year = "1995",
- volume = "19",
- number = "1--3",
- pages = "269--282",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/DHKLV95.pdf",
- paper = "Diaz95a.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Wilkinson 63]{Wil63} Wilkinson J H.
+``Rounding Errors in Algebraic Processes''
+ Chapter 2. HMSO. (1963)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@Article{Kalt95b,
- author = "Kaltofen, E.",
- title = "Effective {Noether} irreducibility forms and applications",
- journal = "J. Comput. System Sci.",
- year = "1995",
- volume = "50",
- number = "2",
- pages = "274--295",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/95/Ka95_jcss.pdf",
- paper = "Kalt95b.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Wilkinson 65]{Wil65} Wilkinson J H.
+``The Algebraic Eigenvalue Problem''
+ Oxford University Press. (1965)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Erli96,
- author = "Erlingsson, U. and Kaltofen, E. and Musser, D.",
- title = "Generic {Gram}-{Schmidt} Orthogonalization by Exact Division",
- booktitle = "Proc. 1996 Internat. Symp. Symbolic Algebraic Comput.",
- crossref = "ISSAC96",
- pages = "275--282",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/96/EKM96.pdf",
- paper = "Erli96.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Wilkinson 78]{Wil78} Wilkinson J H.
+``Singular Value Decomposition -- Basic Aspects''
+Numerical Software -- Needs and Availability.
+(ed D A H Jacobs) Academic Press. (1978)
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt96,
- author = "Kaltofen, E. and Lobo, A.",
- title = "On rank properties of {Toeplitz} matrices over finite fields",
- booktitle = "Proc. 1996 Internat. Symp. Symbolic Algebraic Comput.",
- crossref = "ISSAC96",
- pages = "241--249",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/96/KaLo96_issac.pdf",
- paper = "Kalt96.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Wilkinson 79]{Wil79} Wilkinson J H.
+``Kronecker's Canonical Form and the QZ Algorithm''
+Linear Algebra and Appl. 28 285--303. 1979
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt96a,
- author = "Kaltofen, E. and Lobo, A.",
- title = "Distributed matrix-free solution of large sparse linear systems
- over finite fields",
- booktitle = "Proc. High Performance Computing '96",
- year = "1996",
- editor = "A. M. Tentner",
- pages = "244--247",
- organization = "Society for Computer Simulation",
- publisher = "Simulation Councils, Inc.",
- address = "San Diego, CA",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/96/KaLo96_hpc.pdf",
- paper = "Kalt96a.pdf"
-}
+\begin{chunk}{ignore}
+\bibitem[Wisbauer 91]{Wis91} Wisbauer, R.
+``Bimodule Structure of Algebra''
+Lecture Notes Univ. Duesseldorf 1991
\end{chunk}
-\begin{chunk}{axiom.bib}
-@InProceedings{Kalt96b,
- author = "Kaltofen, E.",
- title = "Blocked iterative sparse linear system solvers for finite fields",
- booktitle = "Proc. Symp. Parallel Comput. Solving Large Scale Irregular
- Applic. (Stratagem '96)",
- editor = "C. Roucairol",
- publisher = "INRIA",
- address = "Sophia Antipolis, France",
- pages = "91--95",
- year = "1996",
- url =
- "http://www.math.ncsu.edu/~kaltofen/bibliography/96/Ka96_stratagem.ps.gz",
- paper = "Kalt96b.ps"
-}
+\begin{chunk}{ignore}
+\bibitem[Woerz-Busekros 80]{Woe80} Woerz-Busekros, A.
+``Algebra in Genetics''
+Lectures Notes in Biomathematics 36, Springer-Verlag, Heidelberg, 1980
\end{chunk}
-\begin{chunk}{axiom.bib}
-@Article{Kalt97,
- author = "E. Kaltofen",
- title = "Teaching Computational Abstract Algebra",
- journal = "Journal of Symbolic Computation",
- volume = "23",
- number = "5-6",
- pages = "503--515",
- year = "1997",
- note = "Special issue on education, L. Lambe, editor.",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/Ka97_jsc.pdf",
- keywords = "axiomref,read",
- paper = "Kalt97.pdf",
+\begin{chunk}{ignore}
+\bibitem[Wolberg 67]{Wol67} Wolberg J R.
+``Prediction Analysis''
+Van Nostrand. (1967)
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Wolfram 09]{Wo09} Wolfram Research
+\verb|mathworld.wolfram.com/Quaternion.html|
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Wu 87]{WU87} Wu, W.T.
+``A Zero Structure Theorem for polynomial equations solving''
+MM Research Preprints, 1987
+
+\end{chunk}
+
+\begin{chunk}{ignore}
+\bibitem[Wynn 56]{Wynn56} Wynn P.
+``On a Device for Computing the $e_m(S_n )$ Transformation''
+Math. Tables Aids Comput. 10 91--96. (1956)
+
+\end{chunk}
+
+\subsection{Y} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\subsection{Z} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{chunk}{ignore}
+\bibitem[Zakrajsek 02]{Zak02} Zakrajsek, Helena
+``Applications of Hermite transform in computer algebra''
+\verb|www.imfm.si/preprinti/PDF/00835.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Zak02.pdf|
abstract = "
- We report on the contents and pedagogy of a course in abstract algebra
- that was taught with the aid of educational software developed within
- the Mathematica system. We describe the topics covered and the
- didactical use of the corresponding Mathematica packages, as well as
- draw conclusions for future such courses from the students' comments
- and our own experience."
-}
+ let $L$ be a linear differential operator with polynomial
+ coefficients. We show that there is an isomorphism of differential
+ operators ${\bf D_\alpha}$ and an integral transform ${\bf H_\alpha}$
+ (called the Hermite transform) on functions for which $({\bf
+ D_\alpha}{\bf L})f(x)=0$ implies ${\bf L}{\bf H_alpha}(f)(x)=0$. We
+ present an algorithm that computes the Hermite transform of a rational
+ function and use it to find $n+1$ linearly independent solutions of
+ ${\bf L}y=0$ when $({\bf D_\alpha}{\bf L})f(x)=0$ has a rational
+ solution with $n$ distinct finite poles."
\end{chunk}
\begin{chunk}{axiom.bib}
-@InProceedings{Kalt97a,
- author = "Kaltofen, E. and Shoup, V.",
- title = "Fast polynomial factorization over high algebraic extensions of
- finite fields",
- booktitle = "Proc. 1997 Internat. Symp. Symbolic Algebraic Comput.",
- crossref = "ISSAC97",
- pages = "184--188",
- url = "http://www.math.ncsu.edu/~kaltofen/bibliography/97/KaSh97.pdf",
- paper = "Kalt97a.pdf"
+@misc{Zdan14,
+ author = "Zdancewic, Steve and Martin, Milo M.K.",
+ title = "Vellvm: Verifying the LLVM",
+ url = "http://www.cis.upenn.edu/~stevez/vellvm"
}
\end{chunk}
\begin{chunk}{ignore}
+\bibitem[Zhi 97]{Zhi97} Zhi, Lihong
+``Optimal Algorithm for Algebraic Factoring''
+\verb|www.mmrc.iss.ac.cn/~lzhi/Publications/zopfac.pdf|
+%\verb|axiom-developer.org/axiom-website/papers/Zhi97.pdf|
+ abstract = "
+ This paper presents an optimized method for factoring multivariate
+ polynomials over algebraic extension fields which defined by an
+ irreducible ascending set. The basic idea is to convert multivariate
+ polynomials to univariate polynomials and algebraic extensions fields
+ to algebraic number fields by suitable integer substitutions, then
+ factorize the univariate polynomials over the algebraic number fields.
+ Finally, construct multivariate factors of the original polynomial by
+ Hensel lemma and TRUEFACTOR test. Some examples with timing are
+ included."
+
\end{chunk}
+
\eject
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Bibliography}
diff --git a/changelog b/changelog
index 14f3319..0dac261 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,5 @@
+20140920 tpd src/axiom-website/patches.html 20140920.02.tpd.patch
+20140920 tpd books/bookvolbib add abstracts, rearrange, add new sections
20140920 tpd src/axiom-website/patches.html 20140920.01.tpd.patch
20140920 tpd books/bookvolbib add abstracts, rearrange, add new entries
20140919 tpd src/axiom-website/patches.html 20140919.01.tpd.patch
diff --git a/patch b/patch
index a644f10..73fef81 100644
--- a/patch
+++ b/patch
@@ -1,3 +1,3 @@
-books/bookvolbib add abstracts, rearrange, add new entries
+books/bookvolbib add abstracts, rearrange, add new sections
Expand and cleanup bibliography
diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html
index 5e0a8d3..6c4d579 100644
--- a/src/axiom-website/patches.html
+++ b/src/axiom-website/patches.html
@@ -4648,6 +4648,8 @@ books/bookvolbib add references
books/axiom.bst use axiom specific bib style
20140920.01.tpd.patch
books/bookvolbib add abstracts, rearrange, add new entries
+20140920.02.tpd.patch
+books/bookvolbib add abstracts, rearrange, add new sections