diff --git a/books/bookvol10.4.pamphlet b/books/bookvol10.4.pamphlet index e764baf..3155219 100644 --- a/books/bookvol10.4.pamphlet +++ b/books/bookvol10.4.pamphlet @@ -266,6 +266,8 @@ in the bootstrap set. Thus, AffineAlgebraicSetComputeWithGroebnerBasis examples ==================================================================== +The following is part of the PAFF package + See Also: o )show AffineAlgebraicSetComputeWithGroebnerBasis @@ -496,6 +498,8 @@ AffineAlgebraicSetComputeWithGroebnerBasis(K,symb,PolyRing,E,ProjPt):Exports_ AffineAlgebraicSetComputeWithResultant examples ==================================================================== +The following is part of the PAFF package + See Also: o )show AffineAlgebraicSetComputeWithResultant @@ -687,7 +691,6 @@ AffineAlgebraicSetComputeWithResultant(K,symb,PolyRing,E,ProjPt):Ex==Impl where ++ Author: Manuel Bronstein ++ Date Created: 21 March 1988 ++ Date Last Updated: 11 November 1993 -++ Keywords: algebraic, function. ++ Description: ++ This package provides algebraic functions over an integral domain. @@ -1461,7 +1464,6 @@ AlgebraicIntegration(R, F): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 28 Mar 1988 ++ Date Last Updated: 5 August 1993 -++ Keywords: algebraic, manipulation. ++ Description: ++ AlgebraicManipulations provides functions to simplify and expand ++ expressions involving algebraic operators. @@ -1680,13 +1682,6 @@ AlgebraicManipulations(R, F): Exports == Implementation where )abbrev package ALGMFACT AlgebraicMultFact ++ Author: P. Gianni ++ Date Created: 1990 -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package factors multivariate polynomials over the ++ domain of \spadtype{AlgebraicNumber} by allowing the user @@ -1771,11 +1766,6 @@ AlgebraicMultFact(OV,E,P) : C == T ++ Authors: J. Grabmeier, R. Wisbauer ++ Date Created: 04 March 1991 ++ Date Last Updated: 04 April 1992 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: rank, nucleus, nucloid, structural constants ++ Reference: ++ R.S. Pierce: Associative Algebras ++ Graduate Texts in Mathematics 88 @@ -2250,8 +2240,6 @@ AlgebraPackage(R:IntegralDomain, A: FramedNonAssociativeAlgebra(R)): _ \begin{chunk}{package ALGFACT AlgFactor} )abbrev package ALGFACT AlgFactor ++ Author: Manuel Bronstein -++ Date Created: ??? -++ Date Last Updated: ??? ++ Description: ++ Factorization of univariate polynomials with coefficients in ++ \spadtype{AlgebraicNumber}. @@ -2372,12 +2360,6 @@ AlgFactor(UP): Exports == Implementation where ++ Author: Brian Dupee ++ Date Created: August 1994 ++ Date Last Updated: December 1997 -++ Basic Operations: integrate, measure -++ Related Constructors: Result, RoutinesTable -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ \axiomType{AnnaNumericalIntegrationPackage} is a \axiom{package} ++ of functions for the \axiom{category} @@ -2895,7 +2877,6 @@ AnnaNumericalIntegrationPackage(): EE == II where ++ Author: Brian Dupee ++ Date Created: February 1995 ++ Date Last Updated: December 1997 -++ Basic Operations: measure, optimize, goodnessOfFit. ++ Description: ++ \axiomType{AnnaNumericalOptimizationPackage} is a \axiom{package} of ++ functions for the \axiomType{NumericalOptimizationCategory} @@ -3308,7 +3289,6 @@ AnnaNumericalOptimizationPackage(): EE == II where ++ Author: Brian Dupee ++ Date Created: February 1995 ++ Date Last Updated: December 1997 -++ Basic Operations: solve, measure ++ Description: ++ \axiomType{AnnaOrdinaryDifferentialEquationPackage} is a \axiom{package} ++ of functions for the \axiom{category} @@ -3731,7 +3711,6 @@ AnnaOrdinaryDifferentialEquationPackage(): EE == II where ++ Author: Brian Dupee ++ Date Created: June 1996 ++ Date Last Updated: December 1997 -++ Basic Operations: ++ Description: ++ AnnaPartialDifferentialEquationPackage is an uncompleted ++ package for the interface to NAG PDE routines. It has been realised that @@ -4001,14 +3980,6 @@ AnnaPartialDifferentialEquationPackage(): EE == II where \begin{chunk}{package ANY1 AnyFunctions1} )abbrev package ANY1 AnyFunctions1 -++ Author: -++ Date Created: -++ Change History: -++ Basic Functions: coerce, retractIfCan, retractable?, retract -++ Related Constructors: Any -++ Also See: -++ AMS Classification: -++ Keywords: ++ Description: ++ \spadtype{AnyFunctions1} implements several utility functions for ++ working with \spadtype{Any}. These functions are used to go back @@ -4231,6 +4202,8 @@ getAncestors 'IndexedAggregate ApplicationProgramInterface examples ==================================================================== +This package contains useful functions that expose Axiom system internals + The ApplicationProgramInterface exposes Axiom internal functions which might be useful for understanding, debugging, or creating tools. @@ -4348,7 +4321,6 @@ ApplicationProgramInterface(): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 20 Mar 1990 ++ Date Last Updated: 5 Jul 1990 -++ Keywords: pattern, matching, rule. ++ Description: ++ This package apply rewrite rules to expressions, calling ++ the pattern matcher. @@ -4648,7 +4620,6 @@ AssociatedEquations(R, L):Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 21 Mar 1989 ++ Date Last Updated: 23 May 1990 -++ Keywords: pattern, matching. ++ Description: ++ Attaching predicates to symbols for pattern matching. @@ -5465,7 +5436,6 @@ BalancedFactorisation(R, UP): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 28 Mar 1988 ++ Date Last Updated: 15 May 1990 -++ Keywords: operator. ++ Description: ++ This package exports functions to set some commonly used properties ++ of operators, including properties which contain functions. @@ -6331,6 +6301,11 @@ graphs:=[[point(((functions.i)(j/100.0))::LIST(DFLOAT)) _ Bezier Curve examples ==================================================================== +Provide linear, quadratic, and cubic spline bezier curves + +BezoutMatrix contains functions for computing resultants and +discriminants using Bezout matrices. + A linear Bezier curve is a simple interpolation between the starting point and the ending point based on a parameter t. @@ -6515,12 +6490,6 @@ Bezier(R:Ring): with ++ Author: Clifton J. Williamson ++ Date Created: 2 August 1988 ++ Date Last Updated: 3 November 1993 -++ Basic Operations: bezoutMatrix, bezoutResultant, bezoutDiscriminant -++ Related Domains -++ Also See: -++ AMS Classifiactions: -++ Keywords: Bezout matrix, resultant, discriminant -++ Examples: ++ Reference: Knuth, The Art of Computer Programming, 2nd edition, ++ Vol. 2, p. 619, problem 12. ++ Description: @@ -6703,6 +6672,8 @@ BezoutMatrix(R,UP,M,Row,Col): Exports == Implementation where BlowUpPackage examples ==================================================================== +The following is part of the PAFF package + See Also: o )show BlowUpPackage @@ -7090,12 +7061,6 @@ BoundIntegerRoots(F, UP): Exports == Implementation where ++ Author: Frederic Lehobey, James H. Davenport ++ Date Created: 28 June 1994 ++ Date Last Updated: 11 July 1997 -++ Basic Operations: brillhartIrreducible? -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: factorization -++ Examples: ++ References: ++ [1] John Brillhart, Note on Irreducibility Testing, ++ Mathematics of Computation, vol. 35, num. 35, Oct. 1980, 1379-1381 @@ -7217,13 +7182,6 @@ BrillhartTests(UP): Exports == Implementation where ++ Author: Stephen M. Watt ++ Date Created: December 1986 ++ Date Last Updated: May 30, 1991 -++ Basic Operations: reshape, map -++ Related Domains: CartesianTensor -++ Also See: -++ AMS Classifications: -++ Keywords: tensor -++ Examples: -++ References: ++ Description: ++ This package provides functions to enable conversion of tensors ++ given conversion of the components. @@ -7416,13 +7374,6 @@ ChangeOfVariable(F, UP, UPUP): Exports == Implementation where )abbrev package CPIMA CharacteristicPolynomialInMonogenicalAlgebra ++ Author: Claude Quitte ++ Date Created: 10/12/93 -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package implements characteristicPolynomials for monogenic algebras ++ using resultants @@ -7466,14 +7417,6 @@ CharacteristicPolynomialInMonogenicalAlgebra(R : CommutativeRing, \begin{chunk}{package CHARPOL CharacteristicPolynomialPackage} )abbrev package CHARPOL CharacteristicPolynomialPackage ++ Author: Barry Trager -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package provides a characteristicPolynomial function ++ for any matrix over a commutative ring. @@ -7523,13 +7466,6 @@ CharacteristicPolynomialPackage(R:CommutativeRing):C == T where ++ Author: Clifton Williamson ++ Date Created: 9 August 1993 ++ Date Last Updated: 3 December 1993 -++ Basic Operations: chineseRemainder, factorList -++ Related Domains: PAdicWildFunctionFieldIntegralBasis(K,R,UP,F) -++ Also See: WildFunctionFieldIntegralBasis, FunctionFieldIntegralBasis -++ AMS Classifications: -++ Keywords: function field, finite field, integral basis -++ Examples: -++ References: ++ Description: ++ This package has no description @@ -7718,9 +7654,6 @@ ChineseRemainderToolsForIntegralBases(K,R,UP): Exports == Implementation where ++ Author: J. Grabmeier ++ Date Created: 26 June 1991 ++ Date Last Updated: 26 June 1991 -++ Related Constructors: GenericNonAssociativeAlgebra -++ Keywords: -++ Reference: ++ Description: ++ CoerceVectorMatrixPackage is an unexposed, technical package ++ for data conversions @@ -7844,6 +7777,8 @@ for i in 0..7 repeat displayRow i CombinatorialFunction examples ==================================================================== +Provides combinatorial functions over an integral domain. + f := operator 'f @@ -8050,7 +7985,6 @@ we return zero for convenience. )abbrev package COMBF CombinatorialFunction ++ Author: Manuel Bronstein, Martin Rubey ++ Date Created: 2 Aug 1988 -++ Keywords: combinatorial, function, factorial. ++ Date Last Updated: 30 October 2005 ++ Description: ++ Provides combinatorial functions over an integral domain. @@ -8519,7 +8453,6 @@ CombinatorialFunction(R, F): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 2 May 1988 ++ Date Last Updated: 22 Nov 1989 -++ Keywords: gcd, quotient, common, denominator. ++ Description: ++ CommonDenominator provides functions to compute the ++ common denominator of a finite linear aggregate of elements of @@ -8579,7 +8512,6 @@ CommonDenominator(R, Q, A): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 25 Mar 1988 ++ Date Last Updated: 2 December 1994 -++ Keywords: operator. ++ Description: ++ This package exports the elementary operators, with some semantics ++ already attached to them. The semantics that is attached here is not @@ -8793,14 +8725,6 @@ CommonOperators(): Exports == Implementation where \begin{chunk}{package COMMUPC CommuteUnivariatePolynomialCategory} )abbrev package COMMUPC CommuteUnivariatePolynomialCategory ++ Author: Manuel Bronstein -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A package for swapping the order of two variables in a tower of two ++ UnivariatePolynomialCategory extensions. @@ -8852,15 +8776,6 @@ CommuteUnivariatePolynomialCategory(R, UP, UPUP): Exports == Impl where \begin{chunk}{package COMPFACT ComplexFactorization} )abbrev package COMPFACT ComplexFactorization -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: Complex, UnivariatePolynomial -++ Also See: -++ AMS Classifications: -++ Keywords: complex, polynomial factorization, factor -++ References: ++ Description: ++ This package has no description @@ -8957,15 +8872,6 @@ ComplexFactorization(RR,PR) : C == T where \begin{chunk}{package COMPLEX2 ComplexFunctions2} )abbrev package COMPLEX2 ComplexFunctions2 -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package extends maps from underlying rings to maps between ++ complex over those rings. @@ -8995,13 +8901,6 @@ ComplexFunctions2(R:CommutativeRing, S:CommutativeRing): with )abbrev package CINTSLPE ComplexIntegerSolveLinearPolynomialEquation ++ Author: James Davenport ++ Date Created: 1990 -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package provides the generalized euclidean algorithm which is ++ needed as the basic step for factoring polynomials. @@ -9060,13 +8959,6 @@ ComplexIntegerSolveLinearPolynomialEquation(R,CR): C == T )abbrev package COMPLPAT ComplexPattern ++ Author: Barry Trager ++ Date Created: 30 Nov 1995 -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: complex, patterns -++ References: ++ Description: ++ This package supports converting complex expressions to patterns @@ -9105,13 +8997,6 @@ ComplexPattern(R, S, CS) : C == T where )abbrev package CPMATCH ComplexPatternMatch ++ Author: Barry Trager ++ Date Created: 30 Nov 1995 -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: complex, pattern matching -++ References: ++ Description: ++ This package supports matching patterns involving complex expressions @@ -9184,11 +9069,6 @@ ComplexPatternMatch(R, S, CS) : C == T where ++ Author: J. Grabmeier ++ Date Created: 31 January 1991 ++ Date Last Updated: 12 April 1991 -++ Basic Operations: factor, pleskenSplit -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: complex zeros, roots ++ References: J. Grabmeier: On Plesken's root finding algorithm, ++ in preparation ++ A. Schoenhage: The fundamental theorem of algebra in terms of computational @@ -9783,14 +9663,6 @@ ComplexRootFindingPackage(R, UP): public == private where \begin{chunk}{package CMPLXRT ComplexRootPackage} )abbrev package CMPLXRT ComplexRootPackage ++ Author: P. Gianni -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: Complex, Float, Fraction, UnivariatePolynomial -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package provides functions complexZeros ++ for finding the complex zeros @@ -9862,7 +9734,6 @@ ComplexRootPackage(UP,Par) : T == C where ++ Author: Manuel Bronstein ++ Date Created: 11 June 1993 ++ Date Last Updated: 14 June 1993 -++ Keywords: complex, function, manipulation. ++ Description: ++ \spadtype{ComplexTrigonometricManipulations} provides function that ++ compute the real and imaginary parts of complex functions. @@ -10098,11 +9969,6 @@ ConstantLODE(R, F, L): Exports == Implementation where ++ Author: Jim Wen ++ Date Created: 12 March 1990 ++ Date Last Updated: 19 June 1990, Clifton J. Williamson -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ CoordinateSystems provides coordinate transformation functions ++ for plotting. Functions in this package return conversion functions @@ -10860,6 +10726,8 @@ eval(Integers, sf3221) CycleIndicators examples ==================================================================== +Polya-Redfield enumeration by cycle indices. + This section is based upon the paper J. H. Redfield, ``The Theory of Group-Reduced Distributions'', American J. Math.,49 (1927) 433-455, and is an application of group theory to enumeration problems. It is @@ -11321,8 +11189,6 @@ o )show CycleIndicators ++ Author: William H. Burge ++ Date Created: 1986 ++ Date Last Updated: 11 Feb 1992 -++ Keywords:Polya, Redfield, enumeration -++ Examples: ++ References: J.H.Redfield, 'The Theory of Group-Reduced Distributions', ++ American J. Math., 49 (1927) 433-455. ++ G.Polya, 'Kombinatorische Anzahlbestimmungen fur Gruppen, @@ -11562,7 +11428,6 @@ CycleIndicators: Exports == Implementation where ++ Author: Clifton J. Williamson ++ Date Created: 5 December 1989 ++ Date Last Updated: 5 December 1989 -++ Keywords: stream, cyclic ++ Description: ++ This package provides tools for working with cyclic streams. @@ -12084,6 +11949,10 @@ DegreeReductionPackage(R1, R2): Cat == Capsule where DesingTreePackage examples ==================================================================== +The following is all the categories, domains and package +used for the desingularisation be means of +monoidal transformation (Blowing-up) + See Also: o )show DesingTreePackage @@ -12114,8 +11983,8 @@ o )show DesingTreePackage ++ Author: Gaetan Hache ++ Date Created: 17 nov 1992 ++ Date Last Updated: 31 jan 95 -++ Keywords: -++ Description: The following is all the categories, domains and package +++ Description: +++ The following is all the categories, domains and package ++ used for the desingularisation be means of ++ monoidal transformation (Blowing-up) DesingTreePackage(K, @@ -12552,12 +12421,6 @@ DesingTreePackage(K, ++ Author: A. Fortenbacher ++ Date Created: 29 March 1991 ++ Date Last Updated: 29 March 1991 -++ Basic Operations: dioSolve -++ Related Constructors: Equation, Vector -++ Also See: -++ AMS Classifications: -++ Keywords: Diophantine equation, nonnegative solutions, -++ basis, depth-first-search ++ Reference: ++ M. Clausen, A. Fortenbacher: Efficient Solution of ++ Linear Diophantine Equations. in JSC (1989) 8, 201-216 @@ -12744,15 +12607,6 @@ DiophantineSolutionPackage(): Cat == Capsule where \begin{chunk}{package DIRPROD2 DirectProductFunctions2} )abbrev package DIRPROD2 DirectProductFunctions2 -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package provides operations which all take as arguments direct ++ products of elements of some type \spad{A} and functions from \spad{A} @@ -12809,11 +12663,6 @@ DirectProductFunctions2(dim, A, B): Exports == Implementation where ++ Author: J. Grabmeier, A. Scheerhorn ++ Date Created: 12 March 1991 ++ Date Last Updated: 31 March 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: discrete logarithm ++ References: ++ J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM. ++ AXIOM Technical Report Series, ATR/5 NP2522. @@ -12910,13 +12759,6 @@ DiscreteLogarithmPackage(M): public == private where )abbrev package DISPLAY DisplayPackage ++ Author: Robert S. Sutor ++ Date Created: September 1986 -++ Date Last Updated: -++ Basic Operations: bright, newLine, copies, center, say, sayLength -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ DisplayPackage allows one to print strings in a nice manner, ++ including highlighting substrings. @@ -13053,12 +12895,6 @@ DisplayPackage: public == private where ++ Author: P. Gianni, B.Trager ++ Date Created: 1983 ++ Date Last Updated: 22 November 1993 -++ Basic Functions: factor, irreducible? -++ Related Constructors: PrimeField, FiniteField -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Package for the factorization of a univariate polynomial with ++ coefficients in a finite field. The algorithm used is the @@ -14014,6 +13850,9 @@ pearceyS:=_ DoubleFloatSpecialFunctions examples ==================================================================== +This package provides special functions for double precision +real and complex floating point. + The formula used will agree with the Table of the Fresnel Integral by Pearcey (1959) to 6 decimal places up to an argument of about 35.0. After that the summation gets slowly worse, agreeing to only 2 digits @@ -14066,13 +13905,6 @@ o )show DoubleFloatSpecialFunctions ++ Author: Bruce W. Char, Timothy Daly, Stephen M. Watt ++ Date Created: 1990 ++ Date Last Updated: Jan 19, 2008 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ This package provides special functions for double precision ++ real and complex floating point. @@ -16554,12 +16386,6 @@ DrawComplex(): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 21 Feb 1990 ++ Date Last Updated: 21 Feb 1990 -++ Basic Operations: coerce -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Hack for the draw interface. DrawNumericHack provides ++ a "coercion" from something of the form \spad{x = a..b} where \spad{a} @@ -16851,8 +16677,6 @@ DrawOptionFunctions1(S:Type): Exports == Implementation where ++ Author: Brian Dupee ++ Date Created: March 1994 ++ Date Last Updated: December 1997 -++ Basic Operations: rangeIsFinite, functionIsContinuousAtEndPoints, -++ functionIsOscillatory ++ Description: ++ \axiomType{d01AgentsPackage} is a package of numerical agents to be used ++ to investigate attributes of an input function so as to decide the @@ -17144,9 +16968,7 @@ d01AgentsPackage(): E == I where )abbrev package D01WGTS d01WeightsPackage ++ Author: Brian Dupee ++ Date Created: July 1994 -++ Date Last Updated: January 1998 (Bug fix - exprHasListOfWeightsCosWXorSinWX) -++ Basic Operations: exprHasWeightCosWXorSinWX, exprHasAlgebraicWeight, -++ exprHasLogarithmicWeights +++ Date Last Updated: January 1998 ++ Description: ++ \axiom{d01WeightsPackage} is a package for functions used to investigate ++ whether a function can be divided into a simpler function and a weight @@ -17427,7 +17249,6 @@ d01WeightsPackage(): E == I where ++ Author: Brian Dupee ++ Date Created: May 1994 ++ Date Last Updated: January 1997 -++ Basic Operations: stiffnessFactor, jacobian ++ Description: ++ \axiom{d02AgentsPackage} contains a set of computational agents ++ for use with Ordinary Differential Equation solvers. @@ -17736,7 +17557,6 @@ d02AgentsPackage(): E == I where ++ Author: Brian Dupee ++ Date Created: May 1994 ++ Date Last Updated: December 1997 -++ Basic Operations: ++ Description: ++ \axiom{d03AgentsPackage} contains a set of computational agents ++ for use with Partial Differential Equation solvers. @@ -17852,13 +17672,6 @@ d03AgentsPackage(): E == I where ++ Author: P. Gianni ++ Date Created: summer 1986 ++ Date Last Updated: October 1992 -++ Basic Functions: -++ Related Constructors: NumericRealEigenPackage, NumericComplexEigenPackage, -++ RadicalEigenPackage -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This is a package for the exact computation of eigenvalues and eigenvectors. ++ This package can be made to work for matrices with coefficients which are @@ -18519,6 +18332,8 @@ D(acsch(3*x+6*y),x) ElementaryFunction examples ==================================================================== +This package provides elementary functions over an integral domain. + D(sin(3*x+6*y),x) 3cos(6y + 3x) @@ -18715,8 +18530,6 @@ o )show ElementaryFunction ++ Author: Manuel Bronstein ++ Date Created: 1987 ++ Date Last Updated: 10 April 1995 -++ Keywords: elementary, function, logarithm, exponential. -++ Examples: )r EF INPUT ++ Description: ++ Provides elementary functions over an integral domain. @@ -19818,7 +19631,6 @@ ElementaryFunctionDefiniteIntegration(R, F): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 3 February 1994 ++ Date Last Updated: 9 March 1994 -++ Keywords: differential equation, ODE ++ Description: ++ \spad{ElementaryFunctionLODESolver} provides the top-level ++ functions for finding closed form solutions of linear ordinary @@ -20089,7 +19901,6 @@ ElementaryFunctionLODESolver(R, F, L): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 18 March 1991 ++ Date Last Updated: 8 March 1994 -++ Keywords: differential equation, ODE ++ Description: ++ \spad{ElementaryFunctionODESolver} provides the top-level ++ functions for finding closed form solutions of ordinary @@ -20366,13 +20177,6 @@ ElementaryFunctionODESolver(R, F): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 25 Aug 1989 ++ Date Last Updated: 4 May 1992 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: elementary function, sign -++ Examples: -++ References: ++ Description: ++ This package provides functions to determine the sign of an ++ elementary function around a point or infinity. @@ -20580,7 +20384,6 @@ ElementaryFunctionSign(R,F): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 1987 ++ Date Last Updated: 16 August 1995 -++ Keywords: elementary, function, structure. ++ Description: ++ ElementaryFunctionStructurePackage provides functions to test the ++ algebraic independence of various elementary functions, using the @@ -21045,9 +20848,6 @@ ElementaryFunctionStructurePackage(R,F): Exports == Implementation where ++ Author: Clifton J. Williamson ++ Date Created: 6 February 1990 ++ Date Last Updated: 25 February 1990 -++ Keywords: elementary function, Laurent series -++ Examples: -++ References: ++ Description: ++ This package provides elementary functions on any Laurent series ++ domain over a field which was constructed from a Taylor series @@ -21460,7 +21260,6 @@ ElementaryFunctionsUnivariateLaurentSeries(Coef,UTS,ULS):_ ++ Author: Clifton J. Williamson ++ Date Created: 20 February 1990 ++ Date Last Updated: 20 February 1990 -++ References: ++ Description: ++ This package provides elementary functions on any Laurent series ++ domain over a field which was constructed from a Taylor series @@ -21739,7 +21538,6 @@ ElementaryFunctionsUnivariatePuiseuxSeries(Coef,ULS,UPXS,EFULS):_ ++ Author: Manuel Bronstein ++ Date Created: 1 February 1988 ++ Date Last Updated: 24 October 1995 -++ Keywords: elementary, function, integration. ++ Description: ++ This package provides functions for integration, limited integration, ++ extended integration and the risch differential equation for @@ -22158,7 +21956,6 @@ See the above discussion for why this causes an infinite loop. ++ Author: Manuel Bronstein ++ Date Created: 1 February 1988 ++ Date Last Updated: 2 November 1995 -++ Keywords: elementary, function, integration. ++ Description: ++ Risch differential equation, elementary case. @@ -22549,7 +22346,6 @@ ElementaryRischDE(R, F): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 12 August 1992 ++ Date Last Updated: 17 August 1992 -++ Keywords: elementary, function, integration. ++ Description: ++ Risch differential equation, elementary case. @@ -22664,9 +22460,6 @@ ElementaryRischDESystem(R, F): Exports == Implementation where ++ Author: Bill Burge, Clifton J. Williamson ++ Date Created: 1986 ++ Date Last Updated: 17 February 1992 -++ Keywords: elliptic function, Taylor series -++ Examples: -++ References: ++ Description: ++ The elliptic functions sn, sc and dn are expanded as Taylor series. @@ -22730,16 +22523,7 @@ EllipticFunctionsUnivariateTaylorSeries(Coef,UTS): \begin{chunk}{package EQ2 EquationFunctions2} )abbrev package EQ2 EquationFunctions2 -++ Author: -++ Date Created: ++ Date Last Updated: June 3, 1991 -++ Basic Operations: -++ Related Domains: Equation -++ Also See: -++ AMS Classifications: -++ Keywords: equation -++ Examples: -++ References: ++ Description: ++ This package provides operations for mapping the sides of equations. @@ -23927,6 +23711,22 @@ euclideanGroebner(cn,"info","redcrit") euclideanGroebner examples ==================================================================== +EuclideanGroebnerBasisPackage computes groebner bases for polynomial +ideals over euclidean domains. The basic computation provides a +distinguished set of generators for these ideals. This basis allows +an easy test for membership: the operation euclideanNormalForm returns +zero on ideal members. The string "info" and "redcrit" can be given as +additional args to provide incremental information during the computation. + +If "info" is given, a computational summary is given for each s-polynomial. +If "redcrit" is given, the reduced critical pairs are printed. + +The term ordering is determined by the polynomial type used. +Suggested types include + * DistributedMultivariatePolynomial + * HomogeneousDistributedMultivariatePolynomial + * GeneralDistributedMultivariatePolynomial + Example to call euclideanGroebner: a1:DMP([y,x],INT):= (9*x**2 + 5*x - 3)+ y*(3*x**2 + 2*x + 1) @@ -23993,12 +23793,6 @@ o )show GroebnerPackage ++ Authors: Gebauer, Moeller ++ Date Created: 12-1-86 ++ Date Last Updated: 2-28-91 -++ Basic Functions: -++ Related Constructors: Ideal, IdealDecompositionPackage, GroebnerPackage -++ Also See: -++ AMS Classifications: -++ Keywords: groebner basis, polynomial ideal, euclidean domain -++ References: ++ Description: ++ \spadtype{EuclideanGroebnerBasisPackage} computes groebner ++ bases for polynomial ideals over euclidean domains. @@ -24550,13 +24344,6 @@ EuclideanGroebnerBasisPackage(Dom, Expon, VarSet, Dpol): T == C where ++ Author: William H. Burge ++ Date Created: 1986 ++ Date Last Updated: Feb 1992 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ This package is to be used in conjuction with the CycleIndicators package. ++ It provides an evaluation function for SymmetricPolynomials. @@ -24621,8 +24408,6 @@ EvaluateCycleIndicators(F):T==C where ++ Author: Brian Dupee ++ Date Created: May 1994 ++ Date Last Updated: June 1995 -++ Basic Operations: problemPoints, singularitiesOf, zerosOf -++ Related Constructors: ++ Description: ++ ExpertSystemContinuityPackage is a package of functions for the use of ++ domains belonging to the category \axiomType{NumericalIntegration}. @@ -24888,8 +24673,6 @@ ExpertSystemContinuityPackage(): E == I where ++ Author: Brian Dupee ++ Date Created: May 1994 ++ Date Last Updated: June 1995 -++ Basic Operations: problemPoints, singularitiesOf, zerosOf -++ Related Constructors: ++ Description: ++ ExpertSystemContinuityPackage1 exports a function to check range inclusion @@ -24986,7 +24769,6 @@ ExpertSystemContinuityPackage1(A:DF,B:DF): E == I where ++ Author: Brian Dupee ++ Date Created: May 1994 ++ Date Last Updated: July 1996 -++ Basic Operations: ++ Description: ++ \axiom{ExpertSystemToolsPackage} contains some useful functions for use ++ by the computational agents of numerical solvers. @@ -25373,7 +25155,6 @@ ExpertSystemToolsPackage():E == I where ++ Author: Brian Dupee ++ Date Created: February 1995 ++ Date Last Updated: February 1995 -++ Basic Operations: neglist ++ Description: ++ \axiom{ExpertSystemToolsPackage1} contains some useful functions for use ++ by the computational agents of Ordinary Differential Equation solvers. @@ -25406,8 +25187,6 @@ ExpertSystemToolsPackage1(R1:OR): E == I where ++ Author: Brian Dupee ++ Date Created: February 1995 ++ Date Last Updated: July 1996 -++ Basic Operations: map -++ Related Constructors: Matrix ++ Description: ++ \axiom{ExpertSystemToolsPackage2} contains some useful functions for use ++ by the computational agents of Ordinary Differential Equation solvers. @@ -25682,7 +25461,6 @@ ExpressionSpaceFunctions2(E:ExpressionSpace, F:ExpressionSpace): with ++ Author: Manuel Bronstein ++ Date Created: 5 Mar 1990 ++ Date Last Updated: 30 September 1993 -++ Keywords: differential equation, ODE, Taylor series ++ Description: ++ Taylor series solutions of explicit ODE's; @@ -25896,14 +25674,6 @@ ExpressionSpaceODESolver(R, F): Exports == Implementation where \begin{chunk}{package OMEXPR ExpressionToOpenMath} )abbrev package OMEXPR ExpressionToOpenMath ++ Author: Mike Dewar & Vilya Harvey -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ \spadtype{ExpressionToOpenMath} provides support for ++ converting objects of type \spadtype{Expression} into OpenMath. @@ -26195,14 +25965,6 @@ ExpressionToOpenMath(R: Join(OpenMath, OrderedSet, Ring)): with ++ Author: Clifton J. Williamson ++ Date Created: 9 May 1989 ++ Date Last Updated: 20 September 1993 -++ Basic Operations: taylor, laurent, puiseux, series -++ Related Domains: UnivariateTaylorSeries, UnivariateLaurentSeries, -++ UnivariatePuiseuxSeries, Expression -++ Also See: FunctionSpaceToUnivariatePowerSeries -++ AMS Classifications: -++ Keywords: Taylor series, Laurent series, Puiseux series -++ Examples: -++ References: ++ Description: ++ This package provides functions to convert functional expressions ++ to power series. @@ -26541,7 +26303,6 @@ ExpressionToUnivariatePowerSeries(R,FE): Exports == Implementation where ++ Author: Clifton J. Williamson ++ Date Created: Bastille Day 1989 ++ Date Last Updated: 5 June 1990 -++ Keywords: ++ Description: ++ Package for constructing tubes around 3-dimensional parametric curves. @@ -26749,6 +26510,9 @@ ExpressionTubePlot(): Exports == Implementation where Export3D examples ==================================================================== +This package provides support for exporting SubSpace and +ThreeSpace structures to files. + See Also: o )show Export3D @@ -26893,7 +26657,6 @@ Export3D(): with ++ Author: Brian Dupee ++ Date Created: February 1996 ++ Date Last Updated: June 1996 -++ Basic Operations: simple? linear?, quadratic?, nonLinear? ++ Description: ++ \axiomType{e04AgentsPackage} is a package of numerical agents to be used ++ to investigate attributes of an input function so as to decide the @@ -27284,6 +27047,11 @@ nthFlag(g,1) FactoredFunctions2 examples ==================================================================== +FactoredFunctions2 contains functions that involve factored objects +whose underlying domains may not be the same. For example, map might +be used to coerce an object of type Factored(Integer) to +Factored(Complex(Integer)). + The FactoredFunctions2 package implements one operation, map, for applying an operation to every base in a factored object and to the unit. @@ -27342,13 +27110,6 @@ o )show FactoredFunctions2 )abbrev package FR2 FactoredFunctions2 ++ Author: Robert S. Sutor ++ Date Created: 1987 -++ Change History: -++ Basic Operations: map -++ Related Constructors: Factored -++ Also See: -++ AMS Classifications: 11A51, 11Y05 -++ Keywords: map, factor -++ References: ++ Description: ++ \spadtype{FactoredFunctions2} contains functions that involve ++ factored objects whose underlying domains may not be the same. @@ -27392,15 +27153,6 @@ FactoredFunctions2(R, S): Exports == Implementation where \begin{chunk}{package FRUTIL FactoredFunctionUtilities} )abbrev package FRUTIL FactoredFunctionUtilities -++ Author: -++ Date Created: -++ Change History: -++ Basic Operations: refine, mergeFactors -++ Related Constructors: Factored -++ Also See: -++ AMS Classifications: 11A51, 11Y05 -++ Keywords: factor -++ References: ++ Description: ++ \spadtype{FactoredFunctionUtilities} implements some utility ++ functions for manipulating factored objects. @@ -27474,13 +27226,6 @@ FactoredFunctionUtilities(R): Exports == Implementation where )abbrev package FACUTIL FactoringUtilities ++ Author: Barry Trager ++ Date Created: March 12, 1992 -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package provides utilities used by the factorizers ++ which operate on polynomials represented as univariate polynomials @@ -27604,6 +27349,8 @@ FactoringUtilities(E,OV,R,P) : C == T where FactorisationOverPseudoAlgebraicClosureOfAlgExtOfRationalNumber examples ==================================================================== +Part of the Package for Algebraic Function Fields in one variable PAFF + See Also: o )show FactorisationOverPseudoAlgebraicClosureOfAlgExtOfRationalNumber @@ -27784,6 +27531,8 @@ FactorisationOverPseudoAlgebraicClosureOfAlgExtOfRationalNumber(K):Exports == Im FactorisationOverPseudoAlgebraicClosureOfRationalNumber examples ==================================================================== +Part of the Package for Algebraic Function Fields in one variable PAFF + See Also: o )show FactorisationOverPseudoAlgebraicClosureOfRationalNumber @@ -28202,6 +27951,16 @@ FiniteDivisorFunctions2(R1, UP1, UPUP1, F1, R2, UP2, UPUP2, F2): FiniteFieldFactorizationWithSizeParseBySideEffect examples ==================================================================== +Part of the package for Algebraic Function Fields in one variable (PAFF) + +It has been modified (very slightly) so that each time the "factor" +function is used, the variable related to the size of the field +over which the polynomial is factorized is reset. This is done in +order to be used with a "dynamic extension field" which size is not +fixed but set before calling the "factor" function and which is +parse by side effect to this package via the function "size". See +the local function "initialize" of this package. + See Also: o )show FiniteFieldFactorizationWithSizeParseBySideEffect @@ -28465,11 +28224,6 @@ FiniteFieldFactorizationWithSizeParseBySideEffect(K : FiniteFieldCategory, ++ Author: J. Grabmeier, A. Scheerhorn ++ Date Created: 21 March 1991 ++ Date Last Updated: 31 March 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: finite field, Zech logarithm, Jacobi logarithm, normal basis ++ References: ++ Lidl, R. & Niederreiter, H., "Finite Fields", ++ Encycl. of Math. 20, Addison-Wesley, 1983 @@ -28719,12 +28473,6 @@ FiniteFieldFunctions(GF): Exports == Implementation where )abbrev package FFHOM FiniteFieldHomomorphisms ++ Authors: J.Grabmeier, A.Scheerhorn ++ Date Created: 26.03.1991 -++ Date Last Updated: -++ Basic Operations: -++ Related Constructors: FiniteFieldCategory, FiniteAlgebraicExtensionField -++ Also See: -++ AMS Classifications: -++ Keywords: finite field, homomorphism, isomorphism ++ References: ++ R.Lidl, H.Niederreiter: Finite Field, Encycoldia of Mathematics and ++ Its Applications, Vol. 20, Cambridge Univ. Press, 1983, ISBN 0 521 30240 4 @@ -29118,12 +28866,6 @@ FiniteFieldHomomorphisms(F1,GF,F2): Exports == Implementation where ++ Author: A. Bouyer, J. Grabmeier, A. Scheerhorn, R. Sutor, B. Trager ++ Date Created: January 1991 ++ Date Last Updated: 1 June 1994 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: finite field, polynomial, irreducible polynomial, normal -++ polynomial, primitive polynomial, random polynomials ++ References: ++ [LS] Lenstra, H. W. & Schoof, R. J., "Primitivive Normal Bases ++ for Finite Fields", Math. Comp. 48, 1987, pp. 217-231 @@ -30107,12 +29849,6 @@ FiniteFieldPolynomialPackage GF : Exports == Implementation where )abbrev package FFPOLY2 FiniteFieldPolynomialPackage2 ++ Authors: J.Grabmeier, A.Scheerhorn ++ Date Created: 26.03.1991 -++ Date Last Updated: -++ Basic Operations: rootOfIrreduciblePoly -++ Related Constructors: FiniteFieldCategory -++ Also See: -++ AMS Classifications: -++ Keywords: finite field, zeros of polynomials, Berlekamp's trace algorithm ++ References: ++ R.Lidl, H.Niederreiter: Finite Field, Encycoldia of Mathematics and ++ Its Applications, Vol. 20, Cambridge Univ. Press, 1983, ISBN 0 521 30240 4 @@ -30239,13 +29975,6 @@ FiniteFieldPolynomialPackage2(F,GF):Exports == Implementation where )abbrev package FFSLPE FiniteFieldSolveLinearPolynomialEquation ++ Author: Davenport ++ Date Created: 1991 -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package solves linear diophantine equations for Bivariate polynomials ++ over finite fields @@ -30321,6 +30050,8 @@ FiniteFieldSolveLinearPolynomialEquation(F:FiniteFieldCategory, FiniteFieldSquareFreeDecomposition examples ==================================================================== +Part of the package for Algebraic Function Fields in one variable (PAFF) + See also: o )show FiniteFieldSquareFreeDecomposition @@ -30454,9 +30185,6 @@ FiniteFieldSquareFreeDecomposition (K : FiniteFieldCategory, \begin{chunk}{package FLAGG2 FiniteLinearAggregateFunctions2} )abbrev package FLAGG2 FiniteLinearAggregateFunctions2 -++ Author: ??? -++ Date Created: ??? -++ Date Last Updated: ??? ++ Description: ++ FiniteLinearAggregateFunctions2 provides functions involving two ++ FiniteLinearAggregates where the underlying domains might be @@ -30586,7 +30314,6 @@ FiniteLinearAggregateFunctions2(S, A, R, B): \begin{chunk}{package FLASORT FiniteLinearAggregateSort} )abbrev package FLASORT FiniteLinearAggregateSort ++ Author: Michael Monagan Sep/88 -++ RelatedOperations: sort ++ Description: ++ This package exports 3 sorting algorithms which work over ++ FiniteLinearAggregates. @@ -30768,14 +30495,6 @@ FiniteSetAggregateFunctions2(S, A, R, B): Exports == Implementation where )abbrev package FLOATCP FloatingComplexPackage ++ Author: P. Gianni ++ Date Created: January 1990 -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: SystemSolvePackage, RadicalSolvePackage, -++ FloatingRealPackage -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This is a package for the approximation of complex solutions for ++ systems of equations of rational functions with complex rational @@ -30905,14 +30624,6 @@ FloatingComplexPackage(Par): Cat == Cap where )abbrev package FLOATRP FloatingRealPackage ++ Author: P. Gianni ++ Date Created: January 1990 -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: SystemSolvePackage, RadicalSolvePackage, -++ FloatingComplexPackage -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This is a package for the approximation of real solutions for ++ systems of polynomial equations over the rational numbers. @@ -31042,13 +30753,6 @@ FloatingRealPackage(Par): Cat == Cap where )abbrev package FCPAK1 FortranCodePackage1 ++ Author: Grant Keady and Godfrey Nolan ++ Date Created: April 1993 -++ Date Last Updated: -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ \spadtype{FortranCodePackage1} provides some utilities for ++ producing useful objects in FortranCode domain. @@ -31207,14 +30911,6 @@ FortranCodePackage1: Exports == Implementation where ++ Author: Mike Dewar ++ Date Created: October 1992 -++ Date Last Updated: -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ Code to manipulate Fortran Output Stack @@ -31310,12 +31006,6 @@ FortranOutputStackPackage() : specification == implementation where ++ Author: Mike Dewar ++ Date Created: October 6 1991 ++ Date Last Updated: 13 July 1994 -++ Basic Operations: linkToFortran -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ provides an interface to the boot code for calling Fortran @@ -31416,7 +31106,6 @@ FortranPackage(): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 1 Feb 1989 ++ Date Last Updated: 27 Feb 1990 -++ Keywords: ideal, algebra, module. ++ Description: ++ Lifting of morphisms to fractional ideals. @@ -32085,15 +31774,6 @@ FractionFreeFastGaussianFractions(D, V, VF): Exports == Implementation where \begin{chunk}{package FRAC2 FractionFunctions2} )abbrev package FRAC2 FractionFunctions2 -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package extends a map between integral domains to ++ a map between Fractions over those domains by applying the map to the @@ -32133,12 +31813,6 @@ FractionFunctions2(A, B): Exports == Impl where ++ Author: Johannes Grabmeier ++ Date Created: 28 February 1992 ++ Date Last Updated: 28 February 1992 -++ Basic Operations: map -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: non-associative algebra -++ References: ++ Description: ++ FramedNonAssociativeAlgebraFunctions2 implements functions between ++ two framed non associative algebra domains defined over different rings. @@ -32219,7 +31893,6 @@ Gregory Vanuxem supplied the fix below. ++ Author: Manuel Bronstein ++ Date Created: 18 Apr 1989 ++ Date Last Updated: 4 October 1993 -++ Keywords: special, function. ++ Description: ++ Provides some special functions over an integral domain. @@ -32665,9 +32338,6 @@ FunctionFieldCategoryFunctions2(R1, UP1, UPUP1, F1, R2, UP2, UPUP2, F2): ++ Author: Victor Miller ++ Date Created: 9 April 1990 ++ Date Last Updated: 20 September 1994 -++ Keywords: -++ Examples: -++ References: ++ Description: ++ Integral bases for function fields of dimension one ++ In this package R is a Euclidean domain and F is a framed algebra @@ -32805,7 +32475,6 @@ FunctionFieldIntegralBasis(R,UP,F): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 21 Mar 1989 ++ Date Last Updated: 23 May 1990 -++ Keywords: pattern, matching. ++ Description: ++ Attaching assertions to symbols for pattern matching; @@ -32892,7 +32561,6 @@ FunctionSpaceAssertions(R, F): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 21 Mar 1989 ++ Date Last Updated: 23 May 1990 -++ Keywords: pattern, matching. ++ Description: ++ Attaching predicates to symbols for pattern matching. @@ -32960,7 +32628,6 @@ FunctionSpaceAttachPredicates(R, F, D): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 4 February 1988 ++ Date Last Updated: 11 June 1993 -++ Keywords: function, integration. ++ Description: ++ Top-level complex function integration ++ \spadtype{FunctionSpaceComplexIntegration} provides functions for the @@ -33102,7 +32769,6 @@ FunctionSpaceFunctions2(R, A, S, B): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 4 February 1988 ++ Date Last Updated: 11 June 1993 -++ Keywords: function, integration. ++ Description: ++ Top-level real function integration ++ \spadtype{FunctionSpaceIntegration} provides functions for the @@ -33248,7 +32914,6 @@ FunctionSpaceIntegration(R, F): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 6 Jun 1990 ++ Date Last Updated: 25 April 1991 -++ Keywords: algebraic, extension, primitive. ++ Description: ++ FunctionsSpacePrimitiveElement provides functions to compute ++ primitive elements in functions spaces; @@ -33373,7 +33038,6 @@ FunctionSpacePrimitiveElement(R, F): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 1988 ++ Date Last Updated: 11 Jul 1990 -++ Keywords: function, space, redcution. ++ Description: ++ Reduction from a function space to the rational numbers ++ This package provides function which replaces transcendental kernels @@ -33445,7 +33109,6 @@ FunctionSpaceReduce(R, F): Exports == Implementation where \begin{chunk}{package SUMFS FunctionSpaceSum} )abbrev package SUMFS FunctionSpaceSum ++ Author: Manuel Bronstein -++ Date Created: ??? ++ Date Last Updated: 19 April 1991 ++ Description: ++ Computes sums of top-level expressions @@ -33523,13 +33186,6 @@ FunctionSpaceSum(R, F): Exports == Implementation where ++ Author: Clifton J. Williamson ++ Date Created: 17 August 1992 ++ Date Last Updated: 2 December 1994 -++ Basic Operations: -++ Related Domains: ExponentialExpansion, UnivariatePuiseuxSeries(FE,x,cen) -++ Also See: FunctionSpaceToUnivariatePowerSeries -++ AMS Classifications: -++ Keywords: elementary function, power series -++ Examples: -++ References: ++ Description: ++ This package converts expressions in some function space to exponential ++ expansions. @@ -34085,13 +33741,6 @@ FunctionSpaceToExponentialExpansion(R,FE,x,cen):_ ++ Author: Clifton J. Williamson ++ Date Created: 21 March 1989 ++ Date Last Updated: 2 December 1994 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: elementary function, power series -++ Examples: -++ References: ++ Description: ++ This package converts expressions in some function space to power ++ series in a variable x with coefficients in that function space. @@ -34868,7 +34517,6 @@ FunctionSpaceToUnivariatePowerSeries(R,FE,Expon,UPS,TRAN,x):_ ++ Author: Manuel Bronstein ++ Date Created: 12 May 1988 ++ Date Last Updated: 22 September 1993 -++ Keywords: function, space, polynomial, factoring ++ Description: ++ This package is used internally by IR2F @@ -35036,11 +34684,6 @@ FunctionSpaceUnivariatePolynomialFactor(R, F, UP): ++ Author: Frederic Lehobey ++ Date Created: 30 June 1994 ++ Date Last Updated: 19 October 1995 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: ++ References: ++ [1] Bernard Beauzamy, Products of polynomials and a priori estimates for ++ coefficients in polynomial decompositions: a sharp result, @@ -35228,12 +34871,6 @@ GaloisGroupFactorizationUtilities(R,UP,F): Exports == Implementation where ++ Author: Frederic Lehobey ++ Date Created: 28 June 1994 ++ Date Last Updated: 11 July 1997 -++ Basic Operations: factor -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: factorization -++ Examples: ++ References: ++ [1] Bernard Beauzamy, Vilmar Trevisan and Paul S. Wang, Polynomial ++ Factorization: Sharp Bounds, Efficient Algorithms, @@ -36055,13 +35692,6 @@ GaloisGroupFactorizer(UP): Exports == Implementation where ++ Author: Frederic Lehobey ++ Date Created: 30 June 1994 ++ Date Last Updated: 15 July 1994 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ \spadtype{GaloisGroupPolynomialUtilities} provides useful ++ functions for univariate polynomials which should be added to @@ -36178,12 +35808,6 @@ GaloisGroupPolynomialUtilities(R,UP): Exports == Implementation where ++ Author: Frederic Lehobey ++ Date Created: 29 June 1994 ++ Date Last Updated: 30 June 1994 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ \spadtype{GaloisGroupUtilities} provides several useful functions. @@ -36313,13 +35937,6 @@ GaloisGroupUtilities(R): Exports == Implementation where )abbrev package GAUSSFAC GaussianFactorizationPackage ++ Author: Patrizia Gianni ++ Date Created: Summer 1986 -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Package for the factorization of complex or gaussian integers. @@ -36674,13 +36291,6 @@ GeneralHenselPackage(RP,TP):C == T where ++ Author: P. Gianni ++ Date Created: 1983 ++ Date Last Updated: Sept. 1990 -++ Basic Functions: -++ Related Constructors: MultFiniteFactorize, AlgebraicMultFact, -++ MultivariateFactorize -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This is the top level package for doing multivariate factorization ++ over basic domains like \spadtype{Integer} or \spadtype{Fraction Integer}. @@ -36786,6 +36396,9 @@ GeneralizedMultivariateFactorize(OV,E,S,R,P) : C == T GeneralPackageForAlgebraicFunctionField examples ==================================================================== +A package that implements the Brill-Noether algorithm. +Part of the PAFF package. + See Also: o )show GeneralPackageForAlgebraicFunctionField @@ -38173,13 +37786,6 @@ GeneralPolynomialGcdPackage(E,OV,R,P):C == T where ++ Author: Clifton J. Williamson ++ Date Created: 29 April 1990 ++ Date Last Updated: 31 May 1990 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: series, Taylor, Laurent, Puiseux -++ Examples: -++ References: ++ Description: ++ \spadtype{GenerateUnivariatePowerSeries} provides functions that create ++ power series from explicit formulas for their \spad{n}th coefficient. @@ -38988,6 +38594,12 @@ GenusZeroIntegration(R, F, L): Exports == Implementation where GnuDraw examples ==================================================================== +This package provides support for gnuplot. These routines +generate output files contain gnuplot scripts that may be +processed directly by gnuplot. This is especially convenient +in the axiom-wiki environment where gnuplot is called from +LaTeX via gnuplottex. + See Also: o )show GnuDraw @@ -39101,7 +38713,6 @@ GnuDraw(): Exports == Implementation where \begin{chunk}{package GOSPER GosperSummationMethod} )abbrev package GOSPER GosperSummationMethod ++ Author: SMW -++ Date Created: ??? ++ Date Last Updated: 19 August 1991 ++ Description: ++ Gosper's summation algorithm. @@ -39338,13 +38949,6 @@ GosperSummationMethod(E, V, R, P, Q): Exports == Impl where ++ Author: Clifton J. Williamson ++ Date Created: 8 January 1990 ++ Date Last Updated: 8 January 1990 -++ Basic Operations: clipPointsDefault, drawToScale, adaptive, maxPoints, -++ minPoints, screenResolution -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ TwoDimensionalPlotSettings sets global flags and constants ++ for 2-dimensional plotting. @@ -39444,11 +39048,6 @@ GraphicsDefaults(): Exports == Implementation where ++ Authors: Johannes Grabmeier, Oswald Gschnitzer ++ Date Created: 7 August 1989 ++ Date Last Updated: 23 August 1990 -++ Basic Operations: nextSubsetGray -++ Related Constructors: Permanent -++ Also See: SymmetricGroupCombinatoric Functions -++ AMS Classifications: -++ Keywords: gray code, subsets of finite sets ++ References: ++ Henryk Minc: Evaluation of Permanents, ++ Proc. of the Edinburgh Math. Soc.(1979), 22/1 pp 27-32. @@ -39607,6 +39206,25 @@ groebnerFactorize [eq,eval(eq, [x,y,z],[y,z,x]), eval(eq,[x,y,z],[z,x,y])] GroebnerFactorizationPackage examples ==================================================================== +GroebnerFactorizationPackage provides the function "groebnerFactor" which +uses the factorization routines of Axiom to factor each polynomial under +consideration while doing the groebner basis algorithm. Then it writes +the ideal as an intersection of ideals determined by the irreducible +factors. Note that the whole ring may occur as well as other redundancies. +We also use the fact, that from the second factor on we can assume that +the preceding factors are not equal to 0 and we divide all polynomials +under considerations by the elements of this list of "nonZeroRestrictions". + +The result is a list of groebner bases, whose union of solutions of the +corresponding systems of equations is the solution of the system of +equation corresponding to the input list. The term ordering is determined +by the polynomial type used. + +Suggested types include + * DistributedMultivariatePolynomial + * HomogeneousDistributedMultivariatePolynomial + * GeneralDistributedMultivariatePolynomial + Solving systems of polynomial equations with the Groebner basis algorithm can often be very time consuming because, in general, the algorithm has exponential run-time. These systems, which often come @@ -39738,12 +39356,6 @@ o )show EuclideanGroebnerBasisPackage ++ Author: H. Michael Moeller, Johannes Grabmeier ++ Date Created: 24 August 1989 ++ Date Last Updated: 01 January 1992 -++ Basic Operations: groebnerFactorize factorGroebnerBasis -++ Related Constructors: -++ Also See: GroebnerPackage, Ideal, IdealDecompositionPackage -++ AMS Classifications: -++ Keywords: groebner basis, groebner factorization, ideal decomposition -++ References: ++ Description: ++ \spadtype{GroebnerFactorizationPackage} provides the function ++ groebnerFactor" which uses the factorization routines of \Language{} to @@ -40112,10 +39724,6 @@ GroebnerFactorizationPackage(Dom, Expon, VarSet, Dpol): T == C where \begin{chunk}{package GBINTERN GroebnerInternalPackage} )abbrev package GBINTERN GroebnerInternalPackage -++ Author: -++ Date Created: -++ Date Last Updated: -++ Keywords: ++ Description: ++ This package provides low level tools for Groebner basis computations @@ -41826,6 +41434,26 @@ groebner(hsn7,"redcrit","info") groebner examples ==================================================================== +GroebnerPackage computes groebner bases for polynomial ideals. The basic +computation provides a distinguished set of generators for polynomial +ideals over fields. This basis allows an easy test for membership: +the operation normalForm returns zero on ideal members. When the provided +coefficient domain, Dom, is not a field, the result is equivalent to +considering the extended ideal with Fraction(Dom) as coefficients, but +considerably more efficient since all calculations are performed in Dom. + +Additional arguments "info" and "redcrit" can be given to provide incremental +information during computation. + +Argument "info" produces a computational summary for each s-polynomial. +Argument "redcrit" prints out the reduced critical pairs. + +The term ordering is determined by the polynomial type used. +Suggested types include + * DistributedMultivariatePolynomial + * HomogeneousDistributedMultivariatePolynomial + * GeneralDistributedMultivariatePolynomial + Example to call groebner: s1:DMP[w,p,z,t,s,b]RN:= 45*p + 35*s - 165*b - 36 @@ -41885,12 +41513,6 @@ o )show EuclideanGroebnerBasisPackage ++ Authors: Gebauer, Trager ++ Date Created: 12-1-86 ++ Date Last Updated: 2-28-91 -++ Basic Functions: groebner normalForm -++ Related Constructors: Ideal, IdealDecompositionPackage -++ Also See: -++ AMS Classifications: -++ Keywords: groebner basis, polynomial ideal -++ References: ++ Description: ++ \spadtype{GroebnerPackage} computes groebner ++ bases for polynomial ideals. The basic computation provides a distinguished @@ -44479,9 +44101,6 @@ GuessUnivariatePolynomial(q: Symbol): Exports == Implementation where ++ Author : Larry Lambe ++ Date Created : August 1988 ++ Date Last Updated : March 9 1990 -++ Related Constructors: OrderedSetInts, Commutator, FreeNilpotentLie -++ AMS Classification: Primary 17B05, 17B30; Secondary 17A50 -++ Keywords: free Lie algebra, Hall basis, basic commutators ++ Description: ++ Generate a basis for the free Lie algebra on n ++ generators over a ring R with identity up to basic commutators @@ -44591,14 +44210,7 @@ HallBasis() : Export == Implement where \begin{chunk}{package HEUGCD HeuGcd} )abbrev package HEUGCD HeuGcd ++ Author: P.Gianni -++ Date Created: ++ Date Last Updated: 13 September 94 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package provides the functions for the heuristic integer gcd. ++ Geddes's algorithm,for univariate polynomials with integer coefficients @@ -44837,13 +44449,6 @@ HeuGcd (BP):C == T )abbrev package IDECOMP IdealDecompositionPackage ++ Author: P. Gianni ++ Date Created: summer 1986 -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: PolynomialIdeals -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package provides functions for the primary decomposition of ++ polynomial ideals over the rational numbers. The ideals are members @@ -45241,16 +44846,7 @@ IdealDecompositionPackage(vl,nv) : C == T -- take away nv, now doesn't \begin{chunk}{package INCRMAPS IncrementingMaps} )abbrev package INCRMAPS IncrementingMaps -++ Author: -++ Date Created: ++ Date Last Updated: June 4, 1991 -++ Basic Operations: -++ Related Domains: UniversalSegment -++ Also See: -++ AMS Classifications: -++ Keywords: equation -++ Examples: -++ References: ++ Description: ++ This package provides operations to create incrementing functions. @@ -45292,14 +44888,6 @@ IncrementingMaps(R:Join(Monoid, AbelianSemiGroup)): with ++ Author: Clifton J. Williamson ++ Date Created: 22 February 1990 ++ Date Last Updated: 23 February 1990 -++ Basic Operations: infiniteProduct, evenInfiniteProduct, oddInfiniteProduct, -++ generalInfiniteProduct -++ Related Domains: UnivariateTaylorSeriesCategory -++ Also See: -++ AMS Classifications: -++ Keywords: Taylor series, infinite product -++ Examples: -++ References: ++ Description: ++ This package computes infinite products of univariate Taylor series ++ over an integral domain of characteristic 0. @@ -45362,14 +44950,6 @@ InfiniteProductCharacteristicZero(Coef,UTS):_ ++ Author: Clifton J. Williamson ++ Date Created: 22 February 1990 ++ Date Last Updated: 23 February 1990 -++ Basic Operations: infiniteProduct, evenInfiniteProduct, oddInfiniteProduct, -++ generalInfiniteProduct -++ Related Domains: UnivariateTaylorSeriesCategory -++ Also See: -++ AMS Classifications: -++ Keywords: Taylor series, infinite product -++ Examples: -++ References: ++ Description: ++ This package computes infinite products of univariate Taylor series ++ over an arbitrary finite field. @@ -45492,14 +45072,6 @@ InfiniteProductFiniteField(K,UP,Coef,UTS):_ ++ Author: Clifton J. Williamson ++ Date Created: 22 February 1990 ++ Date Last Updated: 23 February 1990 -++ Basic Operations: infiniteProduct, evenInfiniteProduct, oddInfiniteProduct, -++ generalInfiniteProduct -++ Related Domains: UnivariateTaylorSeriesCategory -++ Also See: -++ AMS Classifications: -++ Keywords: Taylor series, infinite product -++ Examples: -++ References: ++ Description: ++ This package computes infinite products of univariate Taylor series ++ over a field of prime order. @@ -45681,9 +45253,7 @@ Infinity(): with \begin{chunk}{package IALGFACT InnerAlgFactor} )abbrev package IALGFACT InnerAlgFactor ++ Author: Patrizia Gianni -++ Date Created: ??? ++ Date Last Updated: 20 Jul 1988 -++ Keywords: factorization, algebraic extension, univariate polynomial ++ Description: ++ Factorisation in a simple algebraic extension ++ Factorization of univariate polynomials with coefficients in an @@ -45789,7 +45359,6 @@ InnerAlgFactor(F, UP, AlExt, AlPol): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 2 May 1988 ++ Date Last Updated: 22 Nov 1989 -++ Keywords: gcd, quotient, common, denominator. ++ Description: ++ InnerCommonDenominator provides functions to compute ++ the common denominator of a finite linear aggregate of elements @@ -45850,14 +45419,6 @@ InnerCommonDenominator(R, Q, A, B): Exports == Implementation where ++ Author: Clifton J. Williamson, P.Gianni ++ Date Created: 13 November 1989 ++ Date Last Updated: September 1993 -++ Basic Operations: -++ Related Domains: IndexedMatrix(R,minRow,minCol), Matrix(R), -++ RectangularMatrix(n,m,R), SquareMatrix(n,R) -++ Also See: -++ AMS Classifications: -++ Keywords: matrix, canonical forms, linear algebra -++ Examples: -++ References: ++ Description: ++ \spadtype{InnerMatrixLinearAlgebraFunctions} is an internal package ++ which provides standard linear algebra functions on domains in @@ -46092,13 +45653,6 @@ InnerMatrixLinearAlgebraFunctions(R,Row,Col,M):_ ++ Author: Clifton J. Williamson ++ Date Created: 22 November 1989 ++ Date Last Updated: 22 November 1989 -++ Basic Operations: -++ Related Domains: IndexedMatrix(R,minRow,minCol), Matrix(R), RectangularMatrix(n,m,R), SquareMatrix(n,R) -++ Also See: -++ AMS Classifications: -++ Keywords: matrix, inverse, integral domain -++ Examples: -++ References: ++ Description: ++ \spadtype{InnerMatrixQuotientFieldFunctions} provides functions on matrices ++ over an integral domain which involve the quotient field of that integral @@ -46446,13 +46000,6 @@ InnerModularGcd(R,BP,pMod,nextMod):C == T ++ Author: P. Gianni ++ Date Created: 1983 ++ Date Last Updated: Sept. 1990 -++ Additional Comments: JHD Aug 1997 -++ Basic Functions: -++ Related Constructors: MultivariateFactorize, AlgebraicMultFact -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This is an inner package for factoring multivariate polynomials ++ over various coefficient domains in characteristic 0. @@ -46917,11 +46464,6 @@ InnerMultFact(OV,E,R,P) : C == T ++ Authors: J.Grabmeier, A.Scheerhorn ++ Date Created: 26.03.1991 ++ Date Last Updated: 31 March 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: finite field, normal basis ++ References: ++ R.Lidl, H.Niederreiter: Finite Field, Encyclopedia of Mathematics and ++ Its Applications, Vol. 20, Cambridge Univ. Press, 1983, ISBN 0 521 30240 4 @@ -47287,12 +46829,6 @@ InnerNormalBasisFieldFunctions(GF): Exports == Implementation where ++ Author:P. Gianni ++ Date Created: Summer 1990 ++ Date Last Updated:Spring 1991 -++ Basic Functions: -++ Related Constructors: ModularField -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package is the inner package to be used by NumericRealEigenPackage ++ and NumericComplexEigenPackage for the computation of numeric @@ -47503,13 +47039,6 @@ InnerNumericEigenPackage(K,F,Par) : C == T )abbrev package INFSP InnerNumericFloatSolvePackage ++ Author: P. Gianni ++ Date Created: January 1990 -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This is an internal package ++ for computing approximate solutions to systems of polynomial equations. @@ -47768,7 +47297,6 @@ InnerPolySign(R, UP): Exports == Implementation where \begin{chunk}{package ISUMP InnerPolySum} )abbrev package ISUMP InnerPolySum ++ Author: SMW -++ Date Created: ??? ++ Date Last Updated: 19 April 1991 ++ Description: ++ Tools for the summation packages of polynomials @@ -47847,7 +47375,6 @@ InnerPolySum(E, V, R, P): Exports == Impl where ++ Author: Manuel Bronstein ++ Date Created: 4 April 1988 ++ Date Last Updated: 9 October 1993 -++ Keywords: trigonometric, function, manipulation. ++ Description: ++ This package provides transformations from trigonometric functions ++ to exponentials and logarithms, and back. @@ -48031,7 +47558,6 @@ InnerTrigonometricManipulations(R,F,FG): Exports == Implementation where --)boot $noSubsumption := false ++ Author: Manuel Bronstein -++ Date Created: ??? ++ Date Last Updated: 19 April 1991 ++ Description: ++ Tools for manipulating input forms. @@ -48090,6 +47616,8 @@ InputFormFunctions1(R:Type):with InterfaceGroebnerPackage examples ==================================================================== +Part of the Package for Algebraic Function Fields in one variable PAFF + See Also: o )show InterfaceGroebnerPackage @@ -48273,6 +47801,9 @@ for i in 0..7 repeat displayRow i IntegerCombinatoricFunctions examples ==================================================================== +IntegerCombinatoricFunctions package provides some standard functions +in combinatorics. + The binomial(n, r) returns the number of subsets of r objects taken among n objects, i.e. n!/(r! * (n-r)!) @@ -48342,14 +47873,6 @@ o )help set )abbrev package COMBINAT IntegerCombinatoricFunctions ++ Authors: Martin Brock, Robert Sutor, Michael Monagan ++ Date Created: June 1987 -++ Date Last Updated: -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: integer, combinatoric function -++ Examples: -++ References: ++ Description: ++ The \spadtype{IntegerCombinatoricFunctions} package provides some ++ standard functions in combinatorics. @@ -48923,6 +48446,8 @@ solveLinearlyOverQ(vector [m1, m3], m2) IntegerLinearDependence examples ==================================================================== +This packages tests for linear dependence over the integers. + The elements v1,...,vN of a module M over a ring R are said to be linearly dependent over R if there exist c1,...,cN in R, not all 0, such that c1 v1 + ... cN vNn = 0. If such ci's exist, they form @@ -49009,7 +48534,6 @@ o )show IntegerLinearDependence \begin{chunk}{package ZLINDEP IntegerLinearDependence} )abbrev package ZLINDEP IntegerLinearDependence ++ Author: Manuel Bronstein -++ Date Created: ??? ++ Date Last Updated: 14 May 1991 ++ Description: ++ Test for linear dependence over the integers. @@ -49319,6 +48843,8 @@ result:=chineseRemainder(x1,m1,x2,m2) IntegerNumberTheoryFunctions examples ==================================================================== +This package provides various number theoretic functions on the integers. + The IntegerNumberTheoryFunctions package contains a variety of operations of interest to number theorists. Many of these operations deal with divisibility properties of integers. (Recall that an integer a divides @@ -49734,13 +49260,6 @@ o )show IntegerNumberTheoryFunctions )abbrev package INTHEORY IntegerNumberTheoryFunctions ++ Author: Michael Monagan, Martin Brock, Robert Sutor, Timothy Daly ++ Date Created: June 1987 -++ Date Last Updated: -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: number theory, integer -++ Examples: ++ References: Knuth, The Art of Computer Programming Vol.2 ++ Description: ++ This package provides various number theoretic functions on the integers. @@ -50002,20 +49521,13 @@ We've expanded the list of small primes to include those between 1 and 10000. \begin{chunk}{package PRIMES IntegerPrimesPackage} )abbrev package PRIMES IntegerPrimesPackage -++ Author: Michael Monagan +++ Author: Michael Monagan, James Davenport ++ Date Created: August 1987 ++ Date Last Updated: 31 May 1993 -++ Updated by: James Davenport -++ Updated Because: of problems with strong pseudo-primes -++ and for some efficiency reasons. -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: integer, prime -++ Examples: -++ References: Davenport's paper in ISSAC 1992 -++ AXIOM Technical Report ATR/6 +++ References: +++ J. H. Davenport ``Primality testing revisited'' Technical Report TR2/93 +++ (ATR/6)(NP2556) Numerical Algorithms Group, Inc., Downer's Grove, IL, USA +++ and Oxford, UK, August 1993 ++ Description: ++ The \spadtype{IntegerPrimesPackage} implements a modification of ++ Rabin's probabilistic @@ -50491,14 +50003,6 @@ IntegerRetractions(S:RetractableTo(Integer)): with )abbrev package IROOT IntegerRoots ++ Author: Michael Monagan ++ Date Created: November 1987 -++ Date Last Updated: -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: integer roots -++ Examples: -++ References: ++ Description: ++ The \spadtype{IntegerRoots} package computes square roots and ++ nth roots of integers efficiently. @@ -50641,13 +50145,6 @@ IntegerRoots(I:IntegerNumberSystem): Exports == Implementation where )abbrev package INTSLPE IntegerSolveLinearPolynomialEquation ++ Author: Davenport ++ Date Created: 1991 -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package provides the implementation for the ++ \spadfun{solveLinearPolynomialEquation} @@ -50712,9 +50209,6 @@ IntegerSolveLinearPolynomialEquation(): C ==T ++ Author: Victor Miller, Barry Trager, Clifton Williamson ++ Date Created: 11 April 1990 ++ Date Last Updated: 20 September 1994 -++ Keywords: integral basis, function field, number field -++ Examples: -++ References: ++ Description: ++ This package contains functions used in the packages ++ FunctionFieldIntegralBasis and NumberFieldIntegralBasis. @@ -50871,13 +50365,6 @@ IntegralBasisTools(R,UP,F): Exports == Implementation where ++ Author: Clifton Williamson ++ Date Created: 13 August 1993 ++ Date Last Updated: 17 August 1993 -++ Basic Operations: mapUnivariate, mapBivariate -++ Related Domains: PAdicWildFunctionFieldIntegralBasis(K,R,UP,F) -++ Also See: WildFunctionFieldIntegralBasis, FunctionFieldIntegralBasis -++ AMS Classifications: -++ Keywords: function field, finite field, integral basis -++ Examples: -++ References: ++ Description: ++ IntegralBasisPolynomialTools provides functions for mapping functions ++ on the coefficients of univariate and bivariate polynomials. @@ -50975,7 +50462,6 @@ IntegralBasisPolynomialTools(K,R,UP,L): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 1987 ++ Date Last Updated: 12 August 1992 -++ Keywords: integration. ++ Description: ++ Internally used by the integration packages @@ -51146,7 +50632,6 @@ IntegrationResultRFToFunction(R): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 4 February 1988 ++ Date Last Updated: 9 October 1991 -++ Keywords: integration, expansion, function. ++ Description: ++ Conversion of integration results to top-level expressions ++ This package allows a sum of logs over the roots of a polynomial @@ -51385,7 +50870,6 @@ IntegrationResultToFunction(R, F): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 25 April 1990 ++ Date Last Updated: 9 June 1993 -++ Keywords: elementary, function, integration. ++ Description: ++ Tools for the integrator @@ -51536,12 +51020,6 @@ IntegrationTools(R:OrderedSet, F:FunctionSpace R): Exp == Impl where ++ Author: Themos Tsikas ++ Date Created: 09/09/1998 ++ Date Last Updated: 09/09/1998 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A package to print strings without line-feed nor carriage-return. @@ -51580,11 +51058,6 @@ InternalPrintPackage(): Exports == Implementation where ++ Author: Marc Moreno Maza ++ Date Created: 01/1999 ++ Date Last Updated: 23/01/1999 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: ++ References: ++ [1] D. LAZARD "Solving Zero-dimensional Algebraic Systems" ++ Journal of Symbolic Computation, 1992, 13, 117-131 @@ -51807,6 +51280,8 @@ InternalRationalUnivariateRepresentationPackage(R,E,V,P,TS): Exports == Implemen InterpolateFormsPackage examples ==================================================================== +The following is part of the PAFF package + See Also: o )show InterpolateFormsPackage @@ -52161,6 +51636,8 @@ InterpolateFormsPackage(K,symb,PolyRing,E,ProjPt,PCS,Plc,DIVISOR):_ IntersectionDivisorPackage examples ==================================================================== +The following is part of the PAFF package + See Also: o )show IntersectionDivisorPackage @@ -52294,7 +51771,6 @@ IntersectionDivisorPackage(K,symb,PolyRing,E,ProjPt, PCS,Plc,DIVISOR,_ \begin{chunk}{package IRREDFFX IrredPolyOverFiniteField} )abbrev package IRREDFFX IrredPolyOverFiniteField ++ Author: Robert S. Sutor (original) -++ Date Created: ??? ++ Date Last Updated: 29 May 1990 ++ Description: ++ This package exports the function generateIrredPoly that computes @@ -52378,13 +51854,6 @@ IrredPolyOverFiniteField(GF:FiniteFieldCategory): Exports == Impl where ++ Authors: Johannes Grabmeier, Thorsten Werther ++ Date Created: 04 August 1988 ++ Date Last Updated: 24 May 1991 -++ Basic Operations: dimensionOfIrreducibleRepresentation -++ irreducibleRepresentation -++ Related Constructors: RepresentationTheoryPackage1 -++ RepresentationTheoryPackage2 -++ Also See: SymmetricGroupCombinatoricFunctions -++ AMS Classifications: -++ Keywords: ++ References: ++ G. James, A. Kerber: The Representation Theory of the Symmetric ++ Group. Encycl. of Math. and its Appl. Vol 16., Cambr. Univ Press 1981; @@ -52851,7 +52320,6 @@ KernelFunctions2(R:OrderedSet, S:OrderedSet): with ++ Author: Manuel Bronstein ++ Date Created: 14 January 1992 ++ Date Last Updated: 3 February 1994 -++ Keywords: differential equation, ODE ++ Description: ++ \spadtype{Kovacic} provides a modified Kovacic's algorithm for ++ solving explicitely irreducible 2nd order linear ordinary @@ -53747,6 +53215,17 @@ zeroSetSplit(lf,false)$pack LazardSetSolvingPackage examples ==================================================================== +A package for solving polynomial systems by means of Lazard triangular +sets. This package provides two operations. One for solving in the sense +of the regular zeros, and the other for solving in the sense of +the Zariski closure. Both produce square-free regular sets. +Moreover, the decompositions do not contain any redundant component. +However, only zero-dimensional regular sets are normalized, since +normalization may be time consumming in positive dimension. +The decomposition process is that of + M. MORENO MAZA "A new algorithm for computing triangular + decomposition of algebraic varieties" NAG Tech. Rep. 4/98. + The LazardSetSolvingPackage package constructor solves polynomial systems by means of Lazard triangular sets. However one condition is relaxed: Regular triangular sets whose saturated ideals have positive @@ -54197,11 +53676,6 @@ o )show ZeroDimensionalSolvePackage ++ Author: Marc Moreno Maza ++ Date Created: 10/02/1998 ++ Date Last Updated: 12/16/1998 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: ++ References : ++ [1] D. LAZARD "A new method for solving algebraic systems of ++ positive dimension" Discr. App. Math. 33:147-160,1991 @@ -56213,6 +55687,26 @@ concat [realSolve(ts)$zdpack for ts in lts] LexTriangularPackage examples ==================================================================== +A package for solving polynomial systems with finitely many solutions. + +The decompositions are given by means of regular triangular sets. +The computations use lexicographical Groebner bases. The main operations +are lexTriangular and squareFreeLexTriangular. The second one provide +decompositions by means of square-free regular triangular sets. + +Both are based on the lexTriangular method described in + D. LAZARD "Solving Zero-dimensional Algebraic Systems" + published in the J. of Symbol. Comput. (1992) 13, 117-131. + +They differ from the algorithm described in + M. MORENO MAZA and R. RIOBOO "Computations of gcd over + algebraic towers of simple extensions" + In proceedings of AAECC11, Paris, 1995. +by the fact that multiciplities of the roots are not kept. With the +squareFreeLexTriangular operation all multiciplities are removed. +With the other operation some multiciplities may remain. Both operations + admit an optional argument to produce normalized triangular sets. + The LexTriangularPackage package constructor provides an implementation of the lexTriangular algorithm (D. Lazard "Solving Zero-dimensional Algebraic Systems", J. of Symbol. Comput., 1992). @@ -57908,11 +57402,6 @@ o )show LexTriangularPackage ++ Author: Marc Moreno Maza ++ Date Created: 08/02/1999 ++ Date Last Updated: 08/02/1999 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: ++ References: ++ [1] D. LAZARD "Solving Zero-dimensional Algebraic Systems" ++ published in the J. of Symbol. Comput. (1992) 13, 117-131. @@ -58148,7 +57637,6 @@ LexTriangularPackage(R,ls): Exports == Implementation where \begin{chunk}{package LINDEP LinearDependence} )abbrev package LINDEP LinearDependence ++ Author: Manuel Bronstein -++ Date Created: ??? ++ Date Last Updated: 14 May 1991 ++ Description: ++ Test for linear dependence. @@ -58242,7 +57730,6 @@ LinearDependence(S, R): Exports == Implementation where ++ Author: Fritz Schwarz, Manuel Bronstein ++ Date Created: 1988 ++ Date Last Updated: 3 February 1994 -++ Keywords: differential equation, ODE, LODO, factoring ++ Description: ++ \spadtype{LinearOrdinaryDifferentialOperatorFactorizer} provides a ++ factorizer for linear ordinary differential operators whose coefficients @@ -58501,14 +57988,6 @@ LinearOrdinaryDifferentialOperatorsOps(A, L): Exports == Implementation where \begin{chunk}{package LPEFRAC LinearPolynomialEquationByFractions} )abbrev package LPEFRAC LinearPolynomialEquationByFractions ++ Author: James Davenport -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Given a PolynomialFactorizationExplicit ring, this package ++ provides a defaulting rule for the \spad{solveLinearPolynomialEquation} @@ -58592,6 +58071,8 @@ LinearPolynomialEquationByFractions(R:PolynomialFactorizationExplicit): with LinearSystemFromPowerSeriesPackage examples ==================================================================== +Part of the PAFF package + See Also: o )show LinearSystemFromPowerSeriesPackage @@ -58696,12 +58177,6 @@ LinearSystemFromPowerSeriesPackage(K,PCS):P==T where ++ Author: P.Gianni, S.Watt ++ Date Created: Summer 1985 ++ Date Last Updated:Summer 1990 -++ Basic Functions: solve, particularSolution, hasSolution?, rank -++ Related Constructors: LinearSystemMatrixPackage1 -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package solves linear system in the matrix form \spad{AX = B}. @@ -58807,13 +58282,6 @@ LinearSystemMatrixPackage(F, Row, Col, M): Cat == Capsule where )abbrev package LSMP1 LinearSystemMatrixPackage1 ++ Author: R. Sutor ++ Date Created: June, 1994 -++ Date Last Updated: -++ Basic Functions: solve, particularSolution, hasSolution?, rank -++ Related Constructors: LinearSystemMatrixPackage -++ Also See: -++ AMS Classifications: -++ Keywords: solve -++ References: ++ Description: ++ This package solves linear system in the matrix form \spad{AX = B}. ++ It is essentially a particular instantiation of the package @@ -58888,12 +58356,6 @@ LinearSystemMatrixPackage1(F): Cat == Capsule where ++ Author: P.Gianni ++ Date Created: Summer 1985 ++ Date Last Updated: Summer 1993 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: SystemSolvePackage ++ Description: ++ This package finds the solutions of linear systems presented as a ++ list of polynomials. @@ -59325,6 +58787,9 @@ LinGroebnerPackage(lv,F) : C == T LinesOpPack examples ==================================================================== +A package that exports several linear algebra operations over lines +of matrices. Part of the PAFF package. + See Also: o )show LinesOpPack @@ -59467,8 +58932,6 @@ LinesOpPack(K):P==T where ++ Author: Manuel Bronstein ++ Date Created: 1987 ++ Date Last Updated: 10 August 1994 -++ Keywords: liouvillian, function, primitive, exponential. -++ Examples: )r LF INPUT ++ Description: ++ This package provides liouvillian functions over an integral domain. @@ -59688,14 +59151,6 @@ LiouvillianFunction(R, F): Exports == Implementation where \begin{chunk}{package LIST2 ListFunctions2} )abbrev package LIST2 ListFunctions2 -++ Author: -++ Date Created: -++ Change History: -++ Basic Operations: map, reduce, scan -++ Related Constructors: List -++ Also See: ListFunctions3 -++ AMS Classification: -++ Keywords: list, aggregate, map, reduce ++ Description: ++ \spadtype{ListFunctions2} implements utility functions that ++ operate on two kinds of lists, each with a possibly different @@ -59750,14 +59205,6 @@ ListFunctions2(A:Type, B:Type): public == private where \begin{chunk}{package LIST3 ListFunctions3} )abbrev package LIST3 ListFunctions3 -++ Author: -++ Date Created: -++ Change History: -++ Basic Operations: map -++ Related Constructors: List -++ Also See: ListFunctions2 -++ AMS Classification: -++ Keywords: list, aggregate, map ++ Description: ++ \spadtype{ListFunctions3} implements utility functions that ++ operate on three kinds of lists, each with a possibly different @@ -59802,13 +59249,7 @@ ListFunctions3(A:Type, B:Type, C:Type): public == private where )abbrev package LIST2MAP ListToMap ++ Author: Manuel Bronstein ++ Date Created: 22 Mar 1988 -++ Change History: -++ 11 Oct 1989 MB ? -++ Basic Operations: match -++ Related Constructors: List -++ Also See: -++ AMS Classification: -++ Keywords: mapping, list +++ Change History:11 Oct 1989 ++ Description: ++ \spadtype{ListToMap} allows mappings to be described by a pair of ++ lists of equal lengths. The image of an element \spad{x}, @@ -59929,6 +59370,8 @@ ListToMap(A:SetCategory, B:Type): Exports == Implementation where LocalParametrizationOfSimplePointPackage examples ==================================================================== +This package is part of the PAFF package + See Also: o )show LocalParametrizationOfSimplePointPackage @@ -59951,7 +59394,7 @@ o )show LocalParametrizationOfSimplePointPackage ++ Date Created: 17 nov 1992 ++ Date Last Updated: May 2010 by Tim Daly ++ Description: -++ The following is part of the PAFF package +++ This package is part of the PAFF package LocalParametrizationOfSimplePointPackage(K,symb,PolyRing,E,ProjPt,PCS,Plc):_ Exports == Implementation where K:Field @@ -60517,6 +59960,9 @@ h(m1, m2) MakeFunction examples ==================================================================== +Tools for making interpreter functions from top-level expressions +Transforms top-level objects into interpreter functions. + It is sometimes useful to be able to define a function given by the result of a calculation. @@ -60755,13 +60201,6 @@ MakeUnaryCompiledFunction(S, D, I): Exports == Implementation where ++ Author: S.M.Watt and W.H.Burge ++ Date Created:Jan 87 ++ Date Last Updated:Feb 92 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ Various Currying operations. @@ -61111,6 +60550,8 @@ fibs := curry(shiftfib, fibinit) MappingPackage examples ==================================================================== +Various Currying operations. + Function are objects of type Mapping. In this section we demonstrate some library operations from the packages MappingPackage1, MappingPackage2, and MappingPackage3 that manipulate and create functions. Some terminology: @@ -61301,13 +60742,6 @@ o )help MappingPackage4 ++ Author: S.M.Watt and W.H.Burge ++ Date Created:Jan 87 ++ Date Last Updated:Feb 92 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ Various Currying operations. @@ -61625,6 +61059,8 @@ fibs := curry(shiftfib, fibinit) MappingPackage examples ==================================================================== +Various Currying operations. + Function are objects of type Mapping. In this section we demonstrate some library operations from the packages MappingPackage1, MappingPackage2, and MappingPackage3 that manipulate and create functions. Some terminology: @@ -62103,6 +61539,8 @@ fibs := curry(shiftfib, fibinit) MappingPackage examples ==================================================================== +Various Currying operations. + Function are objects of type Mapping. In this section we demonstrate some library operations from the packages MappingPackage1, MappingPackage2, and MappingPackage3 that manipulate and create functions. Some terminology: @@ -62542,6 +61980,8 @@ h:=(x:EXPR(INT)):EXPR(INT)+->1 MappingPackage examples ==================================================================== +Given functions f and g, returns the applicable closure + We can construct some simple maps that take a variable x into an equation: @@ -62702,15 +62142,6 @@ MappingPackage4(A:SetCategory, B:Ring): ++ Author: Clifton J. Williamson ++ Date Created: 21 November 1989 ++ Date Last Updated: 21 March 1994 -++ Basic Operations: -++ Related Domains: IndexedMatrix(R,minRow,minCol), Matrix(R), -++ RectangularMatrix(n,m,R), SquareMatrix(n,R) -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Keywords: matrix, map, reduce -++ Examples: -++ References: ++ Description: ++ \spadtype{MatrixCategoryFunctions2} provides functions between two matrix ++ domains. The functions provided are \spadfun{map} and \spadfun{reduce}. @@ -62787,7 +62218,6 @@ MatrixCategoryFunctions2(R1,Row1,Col1,M1,R2,Row2,Col2,M2):_ ++ Author: Manuel Bronstein ++ Date Created: 2 May 1988 ++ Date Last Updated: 20 Jul 1990 -++ Keywords: gcd, quotient, matrix, common, denominator. ++ Description: ++ MatrixCommonDenominator provides functions to ++ compute the common denominator of a matrix of elements of the @@ -62863,14 +62293,6 @@ MatrixCommonDenominator(R, Q): Exports == Implementation where ++ Author: Clifton J. Williamson, P.Gianni ++ Date Created: 13 November 1989 ++ Date Last Updated: December 1992 -++ Basic Operations: -++ Related Domains: IndexedMatrix(R,minRow,minCol), Matrix(R), -++ RectangularMatrix(n,m,R), SquareMatrix(n,R) -++ Also See: -++ AMS Classifications: -++ Keywords: matrix, canonical forms, linear algebra -++ Examples: -++ References: ++ Description: ++ \spadtype{MatrixLinearAlgebraFunctions} provides functions to compute ++ inverses and canonical forms. @@ -63261,11 +62683,7 @@ MergeThing(S:OrderedSet): Exports == Implementation where \begin{chunk}{package MESH MeshCreationRoutinesForThreeDimensions} )abbrev package MESH MeshCreationRoutinesForThreeDimensions ++ Author: Jim Wen -++ Date Created: ?? ++ Date Last Updated: October 1991 by Jon Steinbach -++ Keywords: -++ Examples: -++ References: ++ Description: ++ This package has no description @@ -63421,14 +62839,7 @@ MeshCreationRoutinesForThreeDimensions():Exports == Implementation where \begin{chunk}{package MDDFACT ModularDistinctDegreeFactorizer} )abbrev package MDDFACT ModularDistinctDegreeFactorizer ++ Author: Barry Trager -++ Date Created: ++ Date Last Updated: 20.9.95 (JHD) -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package supports factorization and gcds ++ of univariate polynomials over the integers modulo different @@ -63669,7 +63080,6 @@ ModularDistinctDegreeFactorizer(U):C == T where ++ Author: Manuel Bronstein ++ Date Created: 22 February 1989 ++ Date Last Updated: 24 November 1993 -++ Keywords: matrix, reduction. ++ Description: ++ Modular hermitian row reduction. -- should be moved into matrix whenever possible @@ -63926,12 +63336,6 @@ ModularHermitianRowReduction(R): Exports == Implementation where ++ Author: Johannes Grabmeier ++ Date Created: 14 May 1991 ++ Date Last Updated: 14 May 1991 -++ Basic Operations: map -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: monoid ring, group ring, change of coefficient domain -++ References: ++ Description: ++ MonoidRingFunctions2 implements functions between ++ two monoid rings defined with the same monoid over different rings. @@ -64063,14 +63467,6 @@ MonomialExtensionTools(F, UP): Exports == Implementation where \begin{chunk}{package MSYSCMD MoreSystemCommands} )abbrev package MSYSCMD MoreSystemCommands -++ Author: -++ Date Created: -++ Change History: -++ Basic Operations: systemCommand -++ Related Constructors: -++ Also See: -++ AMS Classification: -++ Keywords: command ++ Description: ++ \spadtype{MoreSystemCommands} implements an interface with the ++ system command facility. These are the commands that are issued @@ -64109,14 +63505,7 @@ MoreSystemCommands: public == private where \begin{chunk}{package MPCPF MPolyCatPolyFactorizer} )abbrev package MPCPF MPolyCatPolyFactorizer ++ Author: P. Gianni -++ Date Created: ++ Date Last Updated: March 1995 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package exports a factor operation for multivariate polynomials ++ with coefficients which are polynomials over @@ -64192,14 +63581,6 @@ MPolyCatPolyFactorizer(E,OV,R,PPR) : C == T \begin{chunk}{package MPRFF MPolyCatRationalFunctionFactorizer} )abbrev package MPRFF MPolyCatRationalFunctionFactorizer ++ Author: P. Gianni -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package exports a factor operation for multivariate polynomials ++ with coefficients which are rational functions over @@ -64461,14 +63842,6 @@ MPolyCatFunctions3(Vars1,Vars2,E1,E2,R,PR1,PR2): C == T where \begin{chunk}{package MRATFAC MRationalFactorize} )abbrev package MRATFAC MRationalFactorize ++ Author: P. Gianni -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: MultivariateFactorize -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ MRationalFactorize contains the factor function for multivariate ++ polynomials over the quotient field of a ring R such that the package @@ -64530,12 +63903,6 @@ MRationalFactorize(E,OV,R,P) : C == T ++ Author: P. Gianni ++ Date Created: Summer 1990 ++ Date Last Updated: 19 March 1992 -++ Basic Functions: -++ Related Constructors: PrimeField, FiniteField, Polynomial -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Package for factorization of multivariate polynomials over finite fields. @@ -65094,12 +64461,6 @@ MultipleMap(R1,UP1,UPUP1,R2,UP2,UPUP2): Exports == Implementation where ++ Author: Themos Tsikas, Grant Keady ++ Date Created: December 1992 ++ Date Last Updated: June 1993 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ \spadtype{MultiVariableCalculusFunctions} Package provides several ++ functions for multivariable calculus. @@ -65269,12 +64630,6 @@ MultiVariableCalculusFunctions(S,F,FLAF,FLAS) : Exports == Implementation where ++ Author: P. Gianni ++ Date Created: 1983 ++ Date Last Updated: Sept. 1990 -++ Basic Functions: -++ Related Constructors: MultFiniteFactorize, AlgebraicMultFact, UnivariateFactorize -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This is the top level package for doing multivariate factorization ++ over basic domains like \spadtype{Integer} or \spadtype{Fraction Integer}. @@ -65928,6 +65283,12 @@ MultivariateSquareFree (E,OV,R,P) : C == T where \section{package NAGF02 NagEigenPackage} \begin{chunk}{NagEigenPackage.help} +This package uses the NAG Library to compute + * eigenvalues and eigenvectors of a matrix\ + * eigenvalues and eigenvectors of generalized matrix + * eigenvalue problems + * singular values and singular vectors of a matrix. + F02 -- Eigenvalues and Eigenvectors Introduction -- F02 Chapter F02 Eigenvalues and Eigenvectors @@ -69824,6 +69185,28 @@ NagEigenPackage(): Exports == Implementation where \section{package NAGE02 NagFittingPackage} \begin{chunk}{NagFittingPackage.help} +This package uses the NAG Library to find a function which +approximates a set of data points. Typically the data contain random +errors, as of experimental measurement, which need to be smoothed +out. To seek an approximation to the data, it is first necessary to +specify for the approximating function a mathematical form (a +polynomial, for example) which contains a number of unspecified +coefficients: the appropriate fitting routine then derives for the +coefficients the values which provide the best fit of that particular +form. The package deals mainly with curve and surface fitting (i.e., +fitting with functions of one and of two variables) when a polynomial +or a cubic spline is used as the fitting function, since these cover +the most common needs. However, fitting with other functions and/or +more variables can be undertaken by means of general linear or +nonlinear routines (some of which are contained in other packages) +depending on whether the coefficients in the function occur linearly +or nonlinearly. Cases where a graph rather than a set of data points +is given can be treated simply by first reading a suitable set of +points from the graph. The package also contains routines for +evaluating, differentiating and integrating polynomial and spline +curves and surfaces, once the numerical values of their coefficients +have been determined. + E02 -- Curve and Surface Fitting Introduction -- E02 Chapter E02 Curve and Surface Fitting @@ -76923,6 +76306,14 @@ NagFittingPackage(): Exports == Implementation where \section{package NAGF04 NagLinearEquationSolvingPackage} \begin{chunk}{NagLinearEquationSolvingPackage.help} +This package uses the NAG Library to solve the matrix equation + axiom{AX=B}, +where B may be a single vector or a matrix of multiple right-hand sides. + +The matrix A may be real, complex, symmetric, Hermitian positive-definite, +or sparse. It may also be rectangular, in which case a least-squares +solution is obtained. + F04 -- Simultaneous Linear Equations Introduction -- F04 Chapter F04 Simultaneous Linear Equations @@ -80607,15 +79998,7 @@ NagLinearEquationSolvingPackage(): Exports == Implementation where )abbrev package NAGSP NAGLinkSupportPackage ++ Author: Mike Dewar and Godfrey Nolan ++ Date Created: March 1993 -++ Date Last Updated: March 4 1994 -++ October 6 1994 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: +++ Date Last Updated: October 6 1994 ++ Description: ++ Support functions for the NAG Library Link functions @@ -80691,6 +80074,10 @@ NAGLinkSupportPackage() : exports == implementation where \section{package NAGD01 NagIntegrationPackage} \begin{chunk}{NagIntegrationPackage.help} +This package uses the NAG Library to calculate the numerical value of +definite integrals in one or more dimensions and to evaluate +weights and abscissae of integration rules. + D01(3NAG) Foundation Library (12/10/92) D01(3NAG) D01 -- Quadrature Introduction -- D01 @@ -84975,6 +84362,15 @@ NagIntegrationPackage(): Exports == Implementation where \section{package NAGE01 NagInterpolationPackage} \begin{chunk}{NagInterpolationPackage.help} +This package uses the NAG Library to calculate the interpolation of a +function of one or two variables. When provided with the value of the +function (and possibly one or more of its lowest-order +derivatives) at each of a number of values of the variable(s), +the routines provide either an interpolating function or an +interpolated value. For some of the interpolating functions, +there are supporting routines to evaluate, differentiate or +integrate them. + E01 -- Interpolation Introduction -- E01 Chapter E01 Interpolation @@ -87219,6 +86615,10 @@ NagInterpolationPackage(): Exports == Implementation where \section{package NAGF07 NagLapack} \begin{chunk}{NagLapack.help} +This package uses the NAG Library to compute matrix factorizations, +and to solve systems of linear equations following the matrix +factorizations. + F07 -- Linear Equations (LAPACK) Introduction -- F07 Chapter F07 Linear Equations (LAPACK) @@ -88054,6 +87454,9 @@ NagLapack(): Exports == Implementation where \section{package NAGF01 NagMatrixOperationsPackage} \begin{chunk}{NagMatrixOperationsPackage.help} +This package uses the NAG Library to provide facilities for matrix +factorizations and associated transformations. + F01 -- Matrix Factorization Introduction -- F01 Chapter F01 Matrix Factorization @@ -91179,6 +90582,15 @@ NagMatrixOperationsPackage(): Exports == Implementation where \section{package NAGE04 NagOptimisationPackage} \begin{chunk}{NagOptimisationPackage.help} +This package uses the NAG Library to perform optimization. An +optimization problem involves minimizing a function (called the +objective function) of several variables, possibly subject to +restrictions on the values of the variables defined by a set of +constraint functions. The routines in the NAG Foundation Library are +concerned with function minimization only, since the problem of +maximizing a given function can be transformed into a minimization +problem simply by multiplying the function by -1. + E04 -- Minimizing or Maximizing a Function Introduction -- E04 Chapter E04 Minimizing or Maximizing a Function @@ -99170,6 +98582,15 @@ NagOptimisationPackage(): Exports == Implementation where \section{package NAGD02 NagOrdinaryDifferentialEquationsPackage} \begin{chunk}{NagOrdinaryDifferentialEquationsPackage.help} +This package uses the NAG Library to calculate the numerical solution +of ordinary differential equations. There are two main types of +problem, those in which all boundary conditions are specified at one +point (initial-value problems), and those in which the boundary +conditions are distributed between two or more points (boundary-value +problems and eigenvalue problems). Routines are available for +initial-value problems, two-point boundary-value problems and +Sturm-Liouville eigenvalue problems. + D02(3NAG) Foundation Library (12/10/92) D02(3NAG) D02 -- Ordinary Differential Equations Introduction -- D02 @@ -104157,6 +103578,8 @@ NagOrdinaryDifferentialEquationsPackage(): Exports == Implementation where \section{package NAGD03 NagPartialDifferentialEquationsPackage} \begin{chunk}{NagPartialDifferentialEquationsPackage.help} +This package uses the NAG Library to solve partial differential equations. + D03 -- Partial Differential Equations Introduction -- D03 Chapter D03 Partial Differential Equations @@ -106118,6 +105541,10 @@ NagPartialDifferentialEquationsPackage(): Exports == Implementation where \section{package NAGC02 NagPolynomialRootsPackage} \begin{chunk}{NagPolynomialRootsPackage.help} +This package uses the NAG Library to compute the zeros of a polynomial +with real or complex coefficients. + + C02(3NAG) Foundation Library (12/10/92) C02(3NAG) C02 -- Zeros of Polynomials Introduction -- C02 @@ -106887,6 +106314,10 @@ NagPolynomialRootsPackage(): Exports == Implementation where \section{package NAGC05 NagRootFindingPackage} \begin{chunk}{NagRootFindingPackage.help} +This package uses the NAG Library to calculate real zeros of continuous +real functions of one or more variables. (Complex equations must be +expressed in terms of the equivalent larger system of real equations.) + C05(3NAG) Foundation Library (12/10/92) C05(3NAG) C05 -- Roots of One or More Transcendental Equations @@ -107853,6 +107284,11 @@ NagRootFindingPackage(): Exports == Implementation where \section{package NAGC06 NagSeriesSummationPackage} \begin{chunk}{NagSeriesSummationPackage.help} +This package uses the NAG Library to calculate the discrete Fourier +transform of a sequence of real or complex data values, and +applies it to calculate convolutions and correlations. + + C06(3NAG) Foundation Library (12/10/92) C06(3NAG) @@ -110385,6 +109821,9 @@ NagSeriesSummationPackage(): Exports == Implementation where \section{package NAGS NagSpecialFunctionsPackage} \begin{chunk}{NagSpecialFunctionsPackage.help} +This package uses the NAG Library to compute some commonly occurring +physical and mathematical functions. + S -- Approximations of Special Functions Introduction -- S Chapter S Approximations of Special Functions @@ -118505,15 +117944,6 @@ NagSpecialFunctionsPackage(): Exports == Implementation where \begin{chunk}{package NSUP2 NewSparseUnivariatePolynomialFunctions2} )abbrev package NSUP2 NewSparseUnivariatePolynomialFunctions2 -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package lifts a mapping from coefficient rings R to S to ++ a mapping from sparse univariate polynomial over R to @@ -118627,6 +118057,8 @@ NewtonInterpolation F: Exports == Implementation where NewtonPolygon examples ==================================================================== +The following is part of the PAFF package + See Also: o )show NewtonPolygon @@ -118877,13 +118309,6 @@ NewtonPolygon(K,PolyRing,E,dim):Exports == Implementation where ++ Author: Jean Della Dora, Stephen M. Watt ++ Date Created: 1986 ++ Date Last Updated: May 30, 1991 -++ Basic Operations: -++ Related Domains: LinearOrdinaryDifferentialOperator -++ Also See: -++ AMS Classifications: -++ Keywords: gcd, lcm, division, non-commutative -++ Examples: -++ References: ++ Description: ++ This package provides a division and related operations for ++ \spadtype{MonogenicLinearOperator}s over a \spadtype{Field}. @@ -118983,14 +118408,6 @@ NonCommutativeOperatorDivision(P, F): PDcat == PDdef where \begin{chunk}{package NONE1 NoneFunctions1} )abbrev package NONE1 NoneFunctions1 -++ Author: -++ Date Created: -++ Change History: -++ Basic Functions: coerce -++ Related Constructors: None -++ Also See: -++ AMS Classification: -++ Keywords: ++ Description: ++ \spadtype{NoneFunctions1} implements functions on \spadtype{None}. ++ It particular it includes a particulary dangerous coercion from @@ -119025,7 +118442,6 @@ NoneFunctions1(S:Type): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 2 September 1991 ++ Date Last Updated: 14 October 1994 -++ Keywords: differential equation, ODE ++ Description: ++ NonLinearFirstOrderODESolver provides a function ++ for finding closed form first integrals of nonlinear ordinary @@ -119312,11 +118728,6 @@ NonLinearSolvePackage(R:IntegralDomain): Exports == Implementation where ++ Author: Marc Moreno Maza ++ Date Created: 09/23/1998 ++ Date Last Updated: 12/16/1998 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: ++ References: ++ [1] D. LAZARD "A new method for solving algebraic systems of ++ positive dimension" Discr. App. Math. 33:147-160,1991 @@ -119523,7 +118934,6 @@ NormalizationPackage(R,E,V,P,TS): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 23 February 1995 ++ Date Last Updated: 23 February 1995 -++ Basic Functions: norm ++ Description: ++ This package implements the norm of a polynomial with coefficients ++ in a monogenic algebra (using resultants) @@ -119830,13 +119240,6 @@ NPCoef(BP,E,OV,R,P) : C == T where ++ Author: Victor Miller, Clifton Williamson ++ Date Created: 9 April 1990 ++ Date Last Updated: 20 September 1994 -++ Basic Operations: discriminant, integralBasis -++ Related Domains: IntegralBasisTools, TriangularMatrixOperations -++ Also See: FunctionFieldIntegralBasis, WildFunctionFieldIntegralBasis -++ AMS Classifications: -++ Keywords: number field, integral basis, discriminant -++ Examples: -++ References: ++ Description: ++ In this package F is a framed algebra over the integers (typically ++ \spad{F = Z[a]} for some algebraic integer a). The package provides @@ -120078,7 +119481,6 @@ NumberFieldIntegralBasis(UP,F): Exports == Implementation where \begin{chunk}{package NUMFMT NumberFormats} )abbrev package NUMFMT NumberFormats ++ SMW March 88 -++ Keywords: string manipulation, roman numerals, format ++ Description: ++ NumberFormats provides function to format and read arabic and ++ roman numbers, to convert numbers to strings and to read @@ -120291,13 +119693,6 @@ NumberFormats(): NFexports == NFimplementation where ++ Author: Stephen M. Watt ++ Date Created: 1990 ++ Date Last Updated: June 25, 1991 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ This package provides polynomials as functions on a ring. @@ -120379,14 +119774,7 @@ NumberTheoreticPolynomialFunctions(R: CommutativeRing): Exports == Impl where )abbrev package NUMERIC Numeric ++ Author: Manuel Bronstein ++ Date Created: 21 Feb 1990 -++ Date Last Updated: 17 August 1995, Mike Dewar -++ 24 January 1997, Miked Dewar (added partial operators) -++ Basic Operations: numeric, complexNumeric, numericIfCan, complexNumericIfCan -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: +++ Date Last Updated: 24 January 1997 ++ Description: ++ Numeric provides real and complex numerical evaluation ++ functions for various symbolic types. @@ -120835,12 +120223,6 @@ Numeric(S:ConvertibleTo Float): with ++ Author: Yurij Baransky ++ Date Created: October 90 ++ Date Last Updated: October 90 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package is a suite of functions for the numerical integration of an ++ ordinary differential equation of n variables:\br @@ -121214,12 +120596,6 @@ NumericalOrdinaryDifferentialEquations(): Exports == Implementation where ++ Author: Yurij A. Baransky ++ Date Created: October 90 ++ Date Last Updated: October 90 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This suite of routines performs numerical quadrature using ++ algorithms derived from the basic trapezoidal rule. Because @@ -121779,12 +121155,6 @@ NumericalQuadrature(): Exports == Implementation where ++ Author: P. Gianni ++ Date Created: Summer 1990 ++ Date Last Updated: Spring 1991 -++ Basic Functions: -++ Related Constructors: FloatingComplexPackage -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package computes explicitly eigenvalues and eigenvectors of ++ matrices with entries over the complex rational numbers. @@ -121873,13 +121243,6 @@ NumericComplexEigenPackage(Par) : C == T )abbrev package NCNTFRAC NumericContinuedFraction ++ Author: Clifton J. Williamson ++ Date Created: 12 April 1990 -++ Change History: -++ Basic Operations: continuedFraction -++ Related Constructors: ContinuedFraction, Float -++ Also See: Fraction -++ AMS Classifications: 11J70 11A55 11K50 11Y65 30B70 40A15 -++ Keywords: continued fraction -++ References: ++ Description: ++ \spadtype{NumericContinuedFraction} provides functions ++ for converting floating point numbers to continued fractions. @@ -121936,12 +121299,6 @@ NumericContinuedFraction(F): Exports == Implementation where ++ Author:P. Gianni ++ Date Created:Summer 1990 ++ Date Last Updated:Spring 1991 -++ Basic Functions: -++ Related Constructors: FloatingRealPackage -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package computes explicitly eigenvalues and eigenvectors of ++ matrices with entries over the Rational Numbers. @@ -122029,8 +121386,6 @@ NumericRealEigenPackage(Par) : C == T ++ Author: Clifton J. Williamson ++ Date Created: Bastille Day 1989 ++ Date Last Updated: 5 June 1990 -++ Keywords: -++ Examples: ++ Description: ++ Package for constructing tubes around 3-dimensional parametric curves. @@ -122152,12 +121507,6 @@ NumericTubePlot(Curve): Exports == Implementation where ++ Author: Johannes Grabmeier ++ Date Created: 10 September 1990 ++ Date Last Updated: 10 September 1990 -++ Basic Operations: map -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: octonion, non-associative algebra, Cayley-Dixon -++ References: ++ Description: ++ OctonionCategoryFunctions2 implements functions between ++ two octonion domains defined over different rings. @@ -122517,14 +121866,6 @@ OnePointCompletionFunctions2(R, S): Exports == Implementation where \begin{chunk}{package OMPKG OpenMathPackage} )abbrev package OMPKG OpenMathPackage ++ Author: Vilya Harvey -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ \spadtype{OpenMathPackage} provides some simple utilities ++ to make reading OpenMath objects easier. @@ -122613,14 +121954,6 @@ OpenMathPackage(): with \begin{chunk}{package OMSERVER OpenMathServerPackage} )abbrev package OMSERVER OpenMathServerPackage ++ Author: Vilya Harvey -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ \spadtype{OpenMathServerPackage} provides the necessary ++ operations to run AXIOM as an OpenMath server, reading/writing objects @@ -122775,14 +122108,6 @@ OrderedCompletionFunctions2(R, S): Exports == Implementation where \begin{chunk}{package ORDFUNS OrderingFunctions} )abbrev package ORDFUNS OrderingFunctions ++ Author: Barry Trager -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: OrderedDirectProduct -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package provides ordering functions on vectors which ++ are suitable parameters for OrderedDirectProduct. @@ -122870,13 +122195,6 @@ OrderingFunctions(dim,S) : T == C where ++ Author: Stephen M. Watt ++ Date Created: 1990 ++ Date Last Updated: June 25, 1991 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ This package provides orthogonal polynomials as functions on a ring. @@ -122993,12 +122311,6 @@ OrthogonalPolynomialFunctions(R: CommutativeRing): Exports == Impl where ++ Author: Stephen M. Watt ++ Date Created: February 1986 ++ Date Last Updated: October 27 1995 (MCD) -++ Basic Operations: output -++ Related Constructors: OutputForm -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ OutPackage allows pretty-printing from programs. @@ -123121,6 +122433,8 @@ OutputPackage: with PackageForAlgebraicFunctionField examples ==================================================================== +Part of the PAFF package + See Also: o )show PackageForAlgebraicFunctionField @@ -123524,6 +122838,8 @@ PackageForAlgebraicFunctionField(K,symb,BLMET):Exports == Implementation where PackageForAlgebraicFunctionFieldOverFiniteField examples ==================================================================== +Part of the PAFF package + See Also: o )show PackageForAlgebraicFunctionFieldOverFiniteField @@ -123965,6 +123281,8 @@ PackageForAlgebraicFunctionFieldOverFiniteField(K,symb,BLMET):Exp == Impl where PackageForPoly examples ==================================================================== +The following is part of the PAFF package + See Also: o )show PackageForPoly @@ -124310,10 +123628,10 @@ PackageForPoly(R,PolyRing,E,dim): public == private where ++ Authors: Trager,Burge, Hassner & Watt. ++ Date Created: April 1987 ++ Date Last Updated: 12 April 1990 -++ Keywords: Pade, series -++ Examples: ++ References: -++ "Pade Approximants, Part I: Basic Theory", Baker & Graves-Morris. +++ George A. Baker and Peter Graves-Morris +++ ``Pade Approximants'' +++ Cambridge University Press, March 1996 ISBN 9870521450072 ++ Description: ++ This package computes reliable Pad&ea. approximants using ++ a generalized Viskovatov continued fraction algorithm. @@ -124403,10 +123721,10 @@ PadeApproximantPackage(R: Field, x:Symbol, pt:R): Exports == Implementation wher ++ Authors: Burge, Hassner & Watt. ++ Date Created: April 1987 ++ Date Last Updated: 12 April 1990 -++ Keywords: Pade, series -++ Examples: ++ References: -++ "Pade Approximants, Part I: Basic Theory", Baker & Graves-Morris. +++ George A. Baker and Peter Graves-Morris +++ ``Pade Approximants'' +++ Cambridge University Press, March 1996 ISBN 9870521450072 ++ Description: ++ This package computes reliable Pad&ea. approximants using ++ a generalized Viskovatov continued fraction algorithm. @@ -124563,13 +123881,6 @@ PadeApproximants(R,PS,UP): Exports == Implementation where ++ Author: Clifton Williamson ++ Date Created: 5 July 1993 ++ Date Last Updated: 17 August 1993 -++ Basic Operations: integralBasis, localIntegralBasis -++ Related Domains: WildFunctionFieldIntegralBasis(K,R,UP,F) -++ Also See: FunctionFieldIntegralBasis -++ AMS Classifications: -++ Keywords: function field, finite field, integral basis -++ Examples: -++ References: ++ Description: ++ In this package K is a finite field, R is a ring of univariate ++ polynomials over K, and F is a monogenic algebra over R. @@ -124823,9 +124134,6 @@ PAdicWildFunctionFieldIntegralBasis(K,R,UP,F): Exports == Implementation where ++ Author: Burge, Watt (revised by Williamson) ++ Date Created: 1986 ++ Date Last Updated: 21 October 1989 -++ Keywords: stream, fixed point -++ Examples: -++ References: ++ Description: ++ Computation of fixed points of mappings on streams @@ -125653,6 +124961,8 @@ ParametricSurfaceFunctions2(CF1: Type, CF2:Type): with ParametrizationPackage examples ==================================================================== +The following is part of the PAFF package + See Also: o )show ParametrizationPackage @@ -125802,6 +125112,11 @@ partialFraction(b,x)$PFRPAC(INT) PartialFractionPackage examples ==================================================================== +The package PartialFractionPackage gives an easier to use interfact +the domain PartialFraction. The user gives a fraction of polynomials, +and a variable and the package converts it to the proper datatype for +the PartialFraction domain. + When Axiom is given an expression such as: a:=x+1/(y+1) @@ -125848,12 +125163,6 @@ o )show PartialFractionPackage )abbrev package PFRPAC PartialFractionPackage ++ Author: Barry M. Trager ++ Date Created: 1992 -++ BasicOperations: -++ Related Constructors: PartialFraction -++ Also See: -++ AMS Classifications: -++ Keywords: partial fraction, factorization, euclidean domain -++ References: ++ Description: ++ The package \spadtype{PartialFractionPackage} gives an easier ++ to use interfact the domain \spadtype{PartialFraction}. @@ -125929,12 +125238,6 @@ PartialFractionPackage(R): Cat == Capsule where ++ Author: William H. Burge ++ Date Created: 29 October 1987 ++ Date Last Updated: 3 April 1991 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: partition, permutation -++ References: ++ Description: ++ PartitionsAndPermutations contains functions for generating streams of ++ integer partitions, and streams of sequences of integers @@ -126064,7 +125367,6 @@ PartitionsAndPermutations: Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 28 Nov 1989 ++ Date Last Updated: 5 Jul 1990 -++ Keywords: pattern, matching. ++ Description: ++ Utilities for handling patterns @@ -126146,7 +125448,6 @@ PatternFunctions1(R:SetCategory, D:Type): with ++ Author: Manuel Bronstein ++ Date Created: 28 Nov 1989 ++ Date Last Updated: 12 Jan 1990 -++ Keywords: pattern, matching. ++ Description: ++ Lifts maps to patterns @@ -126203,7 +125504,6 @@ PatternFunctions2(R:SetCategory, S:SetCategory): with ++ Author: Manuel Bronstein ++ Date Created: 3 Dec 1989 ++ Date Last Updated: 29 Jun 1990 -++ Keywords: pattern, matching. ++ Description: ++ This package provides the top-level pattern macthing functions. @@ -126300,7 +125600,6 @@ PatternMatch(Base, Subject, Pat): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 21 Mar 1989 ++ Date Last Updated: 23 May 1990 -++ Keywords: pattern, matching. ++ Description: ++ Attaching assertions to symbols for pattern matching. @@ -126352,7 +125651,6 @@ PatternMatchAssertions(): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 15 Mar 1990 ++ Date Last Updated: 20 June 1991 -++ Keywords: pattern, matching, function, space. ++ Description: ++ This package provides pattern matching functions on function spaces. @@ -126440,7 +125738,6 @@ PatternMatchFunctionSpace(S, R, F): Exports== Implementation where ++ Author: Manuel Bronstein ++ Date Created: 29 Nov 1989 ++ Date Last Updated: 22 Mar 1990 -++ Keywords: pattern, matching, integer. ++ Description: ++ This package provides pattern matching functions on integers. @@ -126877,7 +126174,6 @@ PatternMatchIntegration(R, F): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 12 Jan 1990 ++ Date Last Updated: 4 May 1992 -++ Keywords: pattern, matching, kernel. ++ Description: ++ This package provides pattern matching functions on kernels. @@ -126987,7 +126283,6 @@ PatternMatchKernel(S, E): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 4 Dec 1989 ++ Date Last Updated: 29 Jun 1990 -++ Keywords: pattern, matching, list. ++ Description: ++ This package provides pattern matching functions on lists. @@ -127046,7 +126341,6 @@ PatternMatchListAggregate(S, R, L): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 9 Jan 1990 ++ Date Last Updated: 20 June 1991 -++ Keywords: pattern, matching, polynomial. ++ Description: ++ This package provides pattern matching functions on polynomials. @@ -127146,7 +126440,6 @@ PatternMatchPolynomialCategory(S,E,V,R,P):Exports== Implementation where ++ Author: Manuel Bronstein ++ Date Created: 1 Dec 1989 ++ Date Last Updated: 16 August 1995 -++ Keywords: pattern, matching, quotient, field. ++ Description: ++ This packages provides tools for matching recursively in type towers. @@ -127259,7 +126552,6 @@ PatternMatchPushDown(S, A, B): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 1 Dec 1989 ++ Date Last Updated: 20 June 1991 -++ Keywords: pattern, matching, quotient, field. ++ Description: ++ This package provides pattern matching functions on quotients. @@ -127310,7 +126602,6 @@ PatternMatchQuotientFieldCategory(S,R,Q):Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 1 Dec 1989 ++ Date Last Updated: 14 Dec 1989 -++ Keywords: pattern, matching. ++ Description: ++ Lifts maps to pattern matching results. @@ -127351,7 +126642,6 @@ PatternMatchResultFunctions2(R, A, B): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 9 Jan 1990 ++ Date Last Updated: 20 June 1991 -++ Keywords: pattern, matching, symbol. ++ Description: ++ This package provides pattern matching functions on symbols. @@ -127395,7 +126685,6 @@ PatternMatchSymbol(S:SetCategory): with ++ Author: Manuel Bronstein ++ Date Created: 13 Mar 1990 ++ Date Last Updated: 4 February 1992 -++ Keywords: pattern, matching, tools. ++ Description: ++ This package provides tools for the pattern matcher. @@ -127650,11 +126939,6 @@ o )show Permanent ++ Authors: Johannes Grabmeier, Oswald Gschnitzer ++ Date Created: 7 August 1989 ++ Date Last Updated: 23 August 1990 -++ Basic Operations: permanent -++ Related Constructors: GrayCode -++ Also See: MatrixLinearAlgebraFunctions -++ AMS Classifications: -++ Keywords: permanent ++ References: ++ Henryk Minc: Evaluation of Permanents, ++ Proc. of the Edinburgh Math. Soc.(1979), 22/1 pp 27-32. @@ -127865,11 +127149,6 @@ Permanent(n : PositiveInteger, R : Ring with commutative("*")): ++ Authors: M. Weller, G. Schneider, J. Grabmeier ++ Date Created: 20 February 1990 ++ Date Last Updated: 09 June 1990 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: ++ References: ++ J. Conway, R. Curtis, S. Norton, R. Parker, R. Wilson: ++ Atlas of Finite Groups, Oxford, Clarendon Press, 1987 @@ -128294,12 +127573,6 @@ PiCoercions(R:Join(OrderedSet, IntegralDomain)): with ++ Authors: R.T.M. Bronstein, C.J. Williamson ++ Date Created: Jan 1989 ++ Date Last Updated: 4 Mar 1990 -++ Basic Operations: plot, plotPolar -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ PlotFunctions1 provides facilities for plotting curves ++ where functions SF -> SF are specified by giving an expression @@ -128343,12 +127616,6 @@ PlotFunctions1(S:ConvertibleTo InputForm): with \begin{chunk}{package PLOTTOOL PlotTools} )abbrev package PLOTTOOL PlotTools -++ Author: -++ Date Created: -++ Date Last Updated: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ This package exports plotting tools @@ -128462,6 +127729,8 @@ PRJALGPK ProjectiveAlgebraicSetPackage ProjectiveAlgebraicSetPackage examples ==================================================================== +The following is part of the PAFF package + See Also: o )show ProjectiveAlgebraicSetPackage @@ -128811,7 +128080,6 @@ PointPackage(R:Ring):Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 1988 ++ Date Last Updated: 22 July 1998 -++ Keywords: divisor, algebraic, curve. ++ Description: ++ This package provides function for testing whether a divisor on a ++ curve is a torsion divisor. @@ -129109,7 +128377,6 @@ PointsOfFiniteOrder(R0, F, UP, UPUP, R): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 25 Aug 1988 ++ Date Last Updated: 3 August 1993 -++ Keywords: divisor, algebraic, curve. ++ Description: ++ This package provides function for testing whether a divisor on a ++ curve is a torsion divisor. @@ -129304,6 +128571,8 @@ PointsOfFiniteOrderTools(UP, UPUP): Exports == Implementation where PolynomialPackageForCurve examples ==================================================================== +The following is part of the PAFF package + See Also: o )show PolynomialPackageForCurve @@ -129525,13 +128794,6 @@ PolToPol(lv,R) : C == T )abbrev package PGROEB PolyGroebner ++ Author: P. Gianni ++ Date Created: Summer 1988 -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: GroebnerPackage -++ Also See: -++ AMS Classifications: -++ Keywords: groebner basis, polynomial ideals -++ References: ++ Description: ++ Groebner functions for P F ++ This package is an interface package to the groebner basis @@ -129649,14 +128911,6 @@ PolynomialAN2Expression():Target == Implementation where \begin{chunk}{package POLYLIFT PolynomialCategoryLifting} )abbrev package POLYLIFT PolynomialCategoryLifting ++ Author: Manuel Bronstein -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package provides a very general map function, which ++ given a set S and polynomials over R with maps from the @@ -129721,7 +128975,6 @@ PolynomialCategoryLifting(E,Vars,R,P,S): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: March 1988 ++ Date Last Updated: 9 July 1990 -++ Keywords: polynomial, fraction, transformation ++ Description: ++ Manipulations on polynomial quotients ++ This package transforms multivariate polynomials or fractions into @@ -129902,7 +129155,9 @@ PolynomialComposition(UP: UnivariatePolynomialCategory(R), R: Ring): with \begin{chunk}{package PDECOMP PolynomialDecomposition} )abbrev package PDECOMP PolynomialDecomposition -++ References: Kozen and Landau, Cornell University TR 86-773 +++ References: +++ Dexter Kozen and Susan Landau, ``Polynomial Decomposition Algorithms'' +++ Journal of Symbolic Computation (1989) 7, 445-456 ++ Description: ++ Polynomial composition and decomposition functions\br ++ If f = g o h then g=leftFactor(f,h) and h=rightFactor(f,g) @@ -130554,15 +129809,6 @@ PolynomialFactorizationByRecursionUnivariate(R, S): public == private where \begin{chunk}{package POLY2 PolynomialFunctions2} )abbrev package POLY2 PolynomialFunctions2 -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package takes a mapping between coefficient rings, and lifts ++ it to a mapping between polynomials over those rings. @@ -130597,17 +129843,8 @@ PolynomialFunctions2(R:Ring, S:Ring): with \begin{chunk}{package PGCD PolynomialGcdPackage} )abbrev package PGCD PolynomialGcdPackage -++ Author: Michael Lucks, P. Gianni -++ Date Created: +++ Author: Michael Lucks, P. Gianni, Frederic Lehobey ++ Date Last Updated: 17 June 1996 -++ Fix History: Moved unitCanonicals for performance (BMT); -++ Fixed a problem with gcd(x,0) (Frederic Lehobey) -++ Basic Functions: gcd, content -++ Related Constructors: Polynomial -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package computes multivariate polynomial gcd's using ++ a hensel lifting strategy. The contraint on the coefficient @@ -131097,12 +130334,6 @@ PolynomialInterpolationAlgorithms(F, P): Cat == Body where ++ Author: Michael Monagan, Clifton J. Williamson ++ Date Created: June 1987 ++ Date Last Updated: 10 November 1996 (Claude Quitte) -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: polynomial, number theory -++ Examples: ++ References: Knuth, The Art of Computer Programming Vol.2 ++ Description: ++ This package provides various polynomial number theoretic functions @@ -131503,14 +130734,6 @@ PolynomialRoots(E, V, R, P, F):Exports == Implementation where ++ Author: Marc Moreno Maza (marc@nag.co.uk) ++ Date Created: 12/01/1995 ++ Date Last Updated: 12/15/1998 -++ SPARC Version -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ This package provides modest routines for polynomial system solving. ++ The aim of many of the operations of this package is to remove certain @@ -132344,9 +131567,7 @@ PolynomialSetUtilitiesPackage (R,E,V,P) : Exports == Implementation where \begin{chunk}{package SOLVEFOR PolynomialSolveByFormulas} )abbrev package SOLVEFOR PolynomialSolveByFormulas --- Current fields with "**": (%, RationalNumber) -> % are --- ComplexFloat, RadicalExtension(K) and RationalRadical --- SMW June 86, BMT Sept 93 +++ Author: SMW June 86, BMT Sept 93 ++ Description: ++ This package factors the formulas out of the general solve code, ++ allowing their recursive use over different domains. @@ -132630,15 +131851,7 @@ PolynomialSolveByFormulas(UP, F): PSFcat == PSFdef where \begin{chunk}{package PSQFR PolynomialSquareFree} )abbrev package PSQFR PolynomialSquareFree -++ Author: -++ Date Created: ++ Date Last Updated: November 1993, (P.Gianni) -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package computes square-free decomposition of multivariate ++ polynomials over a coefficient ring which is an arbitrary gcd domain. @@ -132771,15 +131984,6 @@ PolynomialSquareFree(VarSet:OrderedSet,E,RC:GcdDomain,P):C == T where \begin{chunk}{package POLY2UP PolynomialToUnivariatePolynomial} )abbrev package POLY2UP PolynomialToUnivariatePolynomial -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package is primarily to help the interpreter do coercions. ++ It allows you to view a polynomial as a @@ -132822,14 +132026,6 @@ PolynomialToUnivariatePolynomial(x:Symbol, R:Ring): with ++ Author: Clifton J. Williamson ++ Date Created: 21 March 1989 ++ Date Last Updated: 30 March 1994 -++ Basic Operations: -++ Related Domains: UnivariateLaurentSeries(FE,x,a), -++ UnivariatePuiseuxSeries(FE,x,a),ExponentialExpansion(R,FE,x,a) -++ Also See: -++ AMS Classifications: -++ Keywords: limit, functional expression, power series -++ Examples: -++ References: ++ Description: ++ PowerSeriesLimitPackage implements limits of expressions ++ in one or more variables as one of the variables approaches a @@ -133544,7 +132740,6 @@ PrimitiveArrayFunctions2(A, B): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 6 Jun 1990 ++ Date Last Updated: 25 April 1991 -++ Keywords: algebraic, extension, primitive. ++ Description: ++ PrimitiveElement provides functions to compute primitive elements ++ in algebraic extensions; @@ -134139,13 +133334,6 @@ PrimitiveRatRicDE(F, UP, L, LQ): Exports == Implementation where )abbrev package PRINT PrintPackage ++ Author: Scott Morrison ++ Date Created: Aug. 1, 1990 -++ Date Last Updated: -++ Basic Operations: print -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: print -++ References: ++ Description: ++ PrintPackage provides a print function for output forms. @@ -134390,14 +133578,9 @@ algebraic factorization, integration). ++ Author: Ducos Lionel (Lionel.Ducos@mathlabo.univ-poitiers.fr) ++ Date Created: january 1995 ++ Date Last Updated: 5 february 1999 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: ++ References : -++ Lionel Ducos "Optimizations of the subresultant algorithm" -++ to appear in the Journal of Pure and Applied Algebra. +++ Lionel Ducos ``Optimizations of the subresultant algorithm'' +++ Journal of Pure and Applied Algebra V145 No 2 Jan 2000 pp149-163 ++ Description: ++ This package contains some functions: discriminant, resultant, ++ subResultantGcd, chainSubResultants, degreeSubResultant, lastSubResultant, @@ -135870,12 +135053,6 @@ PushVariables(R,E,OV,PPR):C == T where ++ Author: William Sit ++ Date Created: March 13, 1992 ++ Date Last Updated: June 12, 1992 -++ Basic Operations: -++ Related Constructors:GroebnerPackage, IdealDecompositionPackage, -++ PolynomialIdeals -++ See Also: QuasiAlgebraicSet -++ AMS Classifications: -++ Keywords: Zariski closed sets, quasi-algebraic sets ++ References:William Sit, "An Algorithm for Parametric Linear Systems" ++ J. Sym. Comp., April, 1992 ++ Description: @@ -136010,15 +135187,9 @@ QuasiAlgebraicSet2(vl,nv) : C == T where \begin{chunk}{package QCMPACK QuasiComponentPackage} )abbrev package QCMPACK QuasiComponentPackage -++ Author: Marc Moreno Maza -++ marc@nag.co.uk +++ Author: Marc Moreno Maza ++ Date Created: 08/30/1998 ++ Date Last Updated: 12/16/1998 -++ Basic Functions: -++ Related Constructors: -++ Also See: `tosedom.spad' -++ AMS Classifications: -++ Keywords: ++ References : ++ [1] D. LAZARD "A new method for solving algebraic systems of ++ positive dimension" Discr. App. Math. 33:147-160,1991 @@ -136408,15 +135579,6 @@ QuasiComponentPackage(R,E,V,P,TS): Exports == Implementation where \begin{chunk}{package QFCAT2 QuotientFieldCategoryFunctions2} )abbrev package QFCAT2 QuotientFieldCategoryFunctions2 -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package extends a function between integral domains ++ to a mapping between their quotient fields. @@ -136501,6 +135663,10 @@ map(f,q) QuaternionCategoryFunctions2 examples ==================================================================== +QuaternionCategoryFunctions2 implements functions between two +quaternion domains. The function map is used by the system +interpreter to coerce between quaternion types. + The package QuaternionCategoryFunctions2 provides a function map to convert an expression in Quaternion(R) to an expression in Quaternion(S) using the function f. @@ -136541,13 +135707,7 @@ o )show QuaternionCategoryFunctions2 )abbrev package QUATCT2 QuaternionCategoryFunctions2 ++ Author: Robert S. Sutor ++ Date Created: 23 May 1990 -++ Change History: -++ 23 May 1990 -++ Basic Operations: map -++ Related Constructors: QuaternionCategory, Quaternion -++ Also See: -++ AMS Classifications: 11R52 -++ Keywords: quaternions, division ring, map +++ Date Last Updated: 23 May 1990 ++ Description: ++ \spadtype{QuaternionCategoryFunctions2} implements functions between ++ two quaternion domains. The function \spadfun{map} is used by @@ -136601,12 +135761,6 @@ QuaternionCategoryFunctions2(QR,R,QS,S) : Exports == ++ Author: P.Gianni ++ Date Created: Summer 1987 ++ Date Last Updated: October 1992 -++ Basic Functions: -++ Related Constructors: EigenPackage, RadicalSolve -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Package for the computation of eigenvalues and eigenvectors. ++ This package works for matrices with coefficients which are @@ -137050,6 +136204,10 @@ radicalRoots([b,c],[x,y]) RadicalSolvePackage examples ==================================================================== +This package tries to find solutions expressed in terms of radicals +for systems of equations of rational functions with coefficients in an +integral domain R. + b:Fraction(Polynomial(Integer)):=(3*x^3+7)/(5*x^2-13) 3 3x + 7 @@ -137260,12 +136418,6 @@ o )show RealSolvePackage ++ Author: P.Gianni ++ Date Created: Summer 1990 ++ Date Last Updated: October 1991 -++ Basic Functions: -++ Related Constructors: SystemSolvePackage, FloatingRealPackage, -++ FloatingComplexPackage -++ Also See: -++ AMS Classifications: -++ Keywords: ++ References: ++ Description: ++ This package tries to find solutions @@ -137590,13 +136742,6 @@ RadicalSolvePackage(R): Cat == Capsule where ++ Author: Stephen M. Watt ++ Date Created: October 1986 ++ Date Last Updated: May 15, 1991 -++ Basic Operations: -++ Related Domains: RadixExpansion -++ Also See: -++ AMS Classifications: -++ Keywords: radix, base, repeading decimal -++ Examples: -++ References: ++ Description: ++ This package provides tools for creating radix expansions. @@ -137889,13 +137034,6 @@ RandomIntegerDistributions(): with ++ Author:S.M.Watt ++ Date Created: April 87 ++ Date Last Updated:Jan 92, May 1995 (MCD) -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ Random number generators. ++ All random numbers used in the system should originate from @@ -137967,9 +137105,7 @@ RandomNumberSource(): with \begin{chunk}{package RATFACT RationalFactorize} )abbrev package RATFACT RationalFactorize ++ Author: P. Gianni -++ Date created: ?? ++ Date last updated: December 1993 -++ Keywords: factorization, hensel, rational number ++ Description: ++ Factorization of extended polynomials with rational coefficients. ++ This package implements factorization of extended polynomials @@ -138051,7 +137187,6 @@ RationalFactorize(RP) : public == private where ++ Author: Manuel Bronstein ++ Date Created: 1987 ++ Date Last Updated: 18 April 1991 -++ Keywords: polynomial, fraction ++ Description: ++ Utilities that provide the same top-level manipulations on ++ fractions than on polynomials. @@ -138242,8 +137377,6 @@ RationalFunctionDefiniteIntegration(R): Exports == Implementation where \begin{chunk}{package RFFACT RationalFunctionFactor} )abbrev package RFFACT RationalFunctionFactor ++ Author: Patrizia Gianni -++ Date Created: ??? -++ Date Last Updated: ??? ++ Description: ++ Factorization of univariate polynomials with coefficients which ++ are rational functions with integer coefficients. @@ -138293,14 +137426,7 @@ RationalFunctionFactor(UP): Exports == Implementation where \begin{chunk}{package RFFACTOR RationalFunctionFactorizer} )abbrev package RFFACTOR RationalFunctionFactorizer ++ Author: P. Gianni -++ Date Created: ++ Date Last Updated: March 1995 -++ Basic Functions: -++ Related Constructors: Fraction, Polynomial -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ \spadtype{RationalFunctionFactorizer} contains the factor function ++ (called factorFraction) which factors fractions of polynomials by factoring @@ -138360,7 +137486,6 @@ RationalFunctionFactorizer(R) : C == T ++ Author: Manuel Bronstein ++ Date Created: 1987 ++ Date Last Updated: 29 Mar 1990 -++ Keywords: polynomial, fraction, integration. ++ Description: ++ This package provides functions for the integration of rational functions. @@ -138437,7 +137562,6 @@ RationalFunctionIntegration(F): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 4 October 1989 ++ Date Last Updated: 26 November 1991 -++ Keywords: limit, rational function. ++ Description: ++ Computation of limits for rational functions. @@ -138835,6 +137959,8 @@ t1-t2 RationalFunctionSum examples ==================================================================== +Computes sums of rational functions. + There are 4 different forms of the sum operator in this domain. They are variations of the types of the input @@ -139858,10 +138984,6 @@ RationalRicDE(F, UP): Exports == Implementation where ++ Author: Marc Moreno Maza ++ Date Created: 01/1999 ++ Date Last Updated: 23/01/1999 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: ++ Description: ++ A package for computing the rational univariate representation ++ of a zero-dimensional algebraic variety given by a regular @@ -140052,13 +139174,6 @@ depending on several ``real roots''. )abbrev package POLUTIL RealPolynomialUtilitiesPackage ++ Author: Renaud Rioboo ++ Date Created: summer 1992 -++ Basic Functions: provides polynomial utilities -++ Related Constructors: RealClosure, -++ Date Last Updated: July 2004 -++ Also See: -++ AMS Classifications: -++ Keywords: Sturm sequences -++ References: ++ Description: ++ \axiomType{RealPolynomialUtilitiesPackage} provides common functions used ++ by interval coding. @@ -140293,6 +139408,9 @@ ans3 := realSolve(lp,lsv,0.01)$REALSOLV RealSolvePackage examples ==================================================================== +This package provides numerical solutions of systems of polynomial +equations for use in ACPLOT + p := 4*x^3 - 3*x^2 + 2*x - 4 ans1 := solve(p,0.01)$REALSOLV @@ -140414,14 +139532,6 @@ RealSolvePackage(): Exports == Implementation where \begin{chunk}{package REAL0 RealZeroPackage} )abbrev package REAL0 RealZeroPackage ++ Author: Andy Neff -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: UnivariatePolynomial, RealZeroPackageQ -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package provides functions for finding the real zeros ++ of univariate polynomials over the integers to arbitrary user-specified @@ -140723,14 +139833,7 @@ RealZeroPackage(Pol): T == C where \begin{chunk}{package REAL0Q RealZeroPackageQ} )abbrev package REAL0Q RealZeroPackageQ ++ Author: Andy Neff, Barry Trager -++ Date Created: ++ Date Last Updated: 7 April 1991 -++ Basic Functions: -++ Related Constructors: UnivariatePolynomial, RealZeroPackageQ -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package provides functions for finding the real zeros of univariate ++ polynomials over the rational numbers to arbitrary user-specified @@ -140816,15 +139919,6 @@ RealZeroPackageQ(Pol): T == C where ++ Author: Clifton J. Williamson ++ Date Created: 21 November 1989 ++ Date Last Updated: 12 June 1991 -++ Basic Operations: -++ Related Domains: IndexedMatrix(R,minRow,minCol), Matrix(R), -++ RectangularMatrix(n,m,R), SquareMatrix(n,R) -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Keywords: matrix, map, reduce -++ Examples: -++ References: ++ Description: ++ \spadtype{RectangularMatrixCategoryFunctions2} provides functions between ++ two matrix domains. The functions provided are \spadfun{map} and @@ -141537,7 +140631,6 @@ ReduceLODE(F, L, UP, A, LO): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 4 November 1991 ++ Date Last Updated: 3 February 1994 -++ Keywords: differential equation, ODE ++ Description: ++ \spadtype{ReductionOfOrder} provides ++ functions for reducing the order of linear ordinary differential equations @@ -141628,11 +140721,6 @@ ReductionOfOrder(F, L): Exports == Impl where ++ Author: Marc Moreno Maza ++ Date Created: 09/16/1998 ++ Date Last Updated: 12/16/1998 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: ++ References : ++ [1] M. MORENO MAZA "A new algorithm for computing triangular ++ decomposition of algebraic varieties" NAG Tech. Rep. 4/98. @@ -141930,11 +141018,6 @@ RegularSetDecompositionPackage(R,E,V,P,TS): Exports == Implementation where ++ Author: Marc Moreno Maza (marc@nag.co.uk) ++ Date Created: 08/30/1998 ++ Date Last Updated: 12/15/1998 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: ++ References : ++ [1] M. MORENO MAZA and R. RIOBOO "Computations of gcd over ++ algebraic towers of simple extensions" In proceedings of AAECC11 @@ -142410,12 +141493,6 @@ RepeatedSquaring(S): Exports == Implementation where ++ Authors: Holger Gollan, Johannes Grabmeier, Thorsten Werther ++ Date Created: 12 September 1987 ++ Date Last Updated: 24 May 1991 -++ Basic Operations: antisymmetricTensors,symmetricTensors, -++ tensorProduct, permutationRepresentation -++ Related Constructors: RepresentationPackage1, Permutation -++ Also See: IrrRepSymNatPackage -++ AMS Classifications: -++ Keywords: representation, symmetrization, tensor product ++ References: ++ G. James, A. Kerber: The Representation Theory of the Symmetric ++ Group. Encycl. of Math. and its Appl. Vol 16., Cambr. Univ Press 1981; @@ -142762,12 +141839,6 @@ RepresentationPackage1(R): public == private where ++ Authors: Holger Gollan, Johannes Grabmeier ++ Date Created: 10 September 1987 ++ Date Last Updated: 20 August 1990 -++ Basic Operations: areEquivalent?, isAbsolutelyIrreducible?, -++ split, meatAxe -++ Related Constructors: RepresentationTheoryPackage1 -++ Also See: IrrRepSymNatPackage -++ AMS Classifications: -++ Keywords: meat-axe, modular representation ++ Reference: ++ R. A. Parker: The Computer Calculation of Modular Characters ++ (The Meat-Axe), in M. D. Atkinson (Ed.), Computational Group Theory @@ -143549,13 +142620,6 @@ RepresentationPackage2(R): public == private where ++ Author: Stephen M. Watt ++ Date Created: 1986 ++ Date Last Updated: May 30, 1991 -++ Basic Operations: -++ Related Domains: ErrorFunctions, Exit, Void -++ Also See: -++ AMS Classifications: -++ Keywords: mode, resolve, type lattice -++ Examples: -++ References: ++ Description: ++ This package provides coercions for the special types \spadtype{Exit} ++ and \spadtype{Void}. @@ -143697,6 +142761,16 @@ RetractSolvePackage(Q, R): Exports == Implementation where RootsFindingPackage examples ==================================================================== +This pacackage finds all the roots of a polynomial. If the constant field is +not large enough then it returns the list of found zeros and the degree +of the extension need to find the other roots missing. If the return +degree is 1 then all the roots have been found. If 0 is return +for the extension degree then there are an infinite number of zeros, +that is you ask for the zeroes of 0. In the case of infinite field +a list of all found zeros is kept and for each other call of a function +that finds zeroes, a check is made on that list; this is to keep +a kind of "canonical" representation of the elements. + See Also: o )show RootsFindingPackage @@ -143846,9 +142920,6 @@ RootsFindingPackage(K):P==T where \begin{chunk}{package SAERFFC SAERationalFunctionAlgFactor} )abbrev package SAERFFC SAERationalFunctionAlgFactor ++ Author: Patrizia Gianni -++ Date Created: ??? -++ Date Last Updated: ??? -++ Keywords: factorization, algebraic extension, univariate polynomial ++ Description: ++ Factorization of univariate polynomials with coefficients in an ++ algebraic extension of \spadtype{Fraction Polynomial Integer}. @@ -143887,12 +142958,6 @@ SAERationalFunctionAlgFactor(UP, SAE, UPA): Exports == Implementation where )abbrev package FORMULA1 ScriptFormulaFormat1 ++ Author: Robert S. Sutor ++ Date Created: 1987 through 1990 -++ Change History: -++ Basic Operations: coerce -++ Related Constructors: ScriptFormulaFormat -++ Also See: TexFormat, TexFormat1 -++ AMS Classifications: -++ Keywords: output, format, SCRIPT, BookMaster, formula ++ References: ++ SCRIPT Mathematical Formula Formatter User's Guide, SH20-6453, ++ IBM Corporation, Publishing Systems Information Development, @@ -143935,16 +143000,7 @@ ScriptFormulaFormat1(S : SetCategory): public == private where \begin{chunk}{package SEGBIND2 SegmentBindingFunctions2} )abbrev package SEGBIND2 SegmentBindingFunctions2 -++ Author: -++ Date Created: ++ Date Last Updated: June 4, 1991 -++ Basic Operations: -++ Related Domains: SegmentBinding, Segment, Equation -++ Also See: -++ AMS Classifications: -++ Keywords: equation -++ Examples: -++ References: ++ Description: ++ This package provides operations for mapping functions onto ++ \spadtype{SegmentBinding}s. @@ -143973,16 +143029,7 @@ SegmentBindingFunctions2(R:Type, S:Type): with \begin{chunk}{package SEG2 SegmentFunctions2} )abbrev package SEG2 SegmentFunctions2 -++ Author: -++ Date Created: ++ Date Last Updated: June 4, 1991 -++ Basic Operations: -++ Related Domains: Segment, UniversalSegment -++ Also See: -++ AMS Classifications: -++ Keywords: equation -++ Examples: -++ References: ++ Description: ++ This package provides operations for mapping functions onto segments. @@ -144037,9 +143084,6 @@ SegmentFunctions2(R:Type, S:Type): public == private where \begin{chunk}{package SAEFACT SimpleAlgebraicExtensionAlgFactor} )abbrev package SAEFACT SimpleAlgebraicExtensionAlgFactor ++ Author: Patrizia Gianni -++ Date Created: ??? -++ Date Last Updated: ??? -++ Keywords: factorization, algebraic extension, univariate polynomial ++ Description: ++ Factorization of univariate polynomials with coefficients in an ++ algebraic extension of the rational numbers (\spadtype{Fraction Integer}). @@ -144112,14 +143156,6 @@ SimplifyAlgebraicNumberConvertPackage(): with )abbrev package SMITH SmithNormalForm ++ Author: Patrizia Gianni ++ Date Created: October 1992 -++ Date Last Updated: -++ Basic Operations: -++ Related Domains: Matrix(R) -++ Also See: -++ AMS Classifications: -++ Keywords: matrix, canonical forms, linear algebra -++ Examples: -++ References: ++ Description: ++ \spadtype{SmithNormalForm} is a package ++ which provides some standard canonical forms for matrices. @@ -144523,15 +143559,6 @@ SortPackage(S,A) : Exports == Implementation where \begin{chunk}{package SUP2 SparseUnivariatePolynomialFunctions2} )abbrev package SUP2 SparseUnivariatePolynomialFunctions2 -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package lifts a mapping from coefficient rings R to S to ++ a mapping from sparse univariate polynomial over R to @@ -144574,12 +143601,6 @@ SparseUnivariatePolynomialFunctions2(R:Ring, S:Ring): with ++ Author: Stephen M. Watt ++ Date Created: September 1986 ++ Date Last Updated: May 23, 1991 -++ Basic Operations: outputAsFortran, outputAsScript, outputAsTex -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ SpecialOutputPackage allows FORTRAN, Tex and ++ Script Formula Formatter output from programs. @@ -144690,11 +143711,6 @@ SpecialOutputPackage: public == private where ++ Author: Marc Moreno Maza ++ Date Created: 09/23/1998 ++ Date Last Updated: 12/16/1998 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: ++ References : ++ [1] D. LAZARD "A new method for solving algebraic systems of ++ positive dimension" Discr. App. Math. 33:147-160,1991 @@ -145114,11 +144130,6 @@ SquareFreeQuasiComponentPackage(R,E,V,P,TS): Exports == Implementation where ++ Author: Marc Moreno Maza ++ Date Created: 09/23/1998 ++ Date Last Updated: 12/16/1998 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: ++ References : ++ [1] M. MORENO MAZA "A new algorithm for computing triangular ++ decomposition of algebraic varieties" NAG Tech. Rep. 4/98. @@ -145419,11 +144430,6 @@ SquareFreeRegularSetDecompositionPackage(R,E,V,P,TS): Exports == Implementation ++ Author: Marc Moreno Maza ++ Date Created: 09/23/1998 ++ Date Last Updated: 10/01/1998 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: ++ References : ++ [1] M. MORENO MAZA and R. RIOBOO "Computations of gcd over ++ algebraic towers of simple extensions" In proceedings of AAECC11 @@ -145890,13 +144896,6 @@ SquareFreeRegularTriangularSetGcdPackage(R,E,V,P,TS): Exports == Implementation ++ Author: Clifton J. Williamson ++ Date Created: 18 July 1990 ++ Date Last Updated: 18 July 1990 -++ Basic Operations: -++ Related Domains: Matrix(R) -++ Also See: -++ AMS Classifications: -++ Keywords: matrix, linear algebra -++ Examples: -++ References: ++ Description: ++ This package provides standard arithmetic operations on matrices. ++ The functions in this package store the results of computations @@ -146096,7 +145095,6 @@ StorageEfficientMatrixOperations(R): Exports == Implementation where ++ Authors: Burge, Watt; updated by Clifton J. Williamson ++ Date Created: July 1986 ++ Date Last Updated: 29 January 1990 -++ Keywords: stream, infinite list, infinite sequence ++ Description: ++ Functions defined on streams with entries in one set. @@ -146144,7 +145142,6 @@ StreamFunctions1(S:Type): Exports == Implementation where ++ Authors: Burge, Watt; updated by Clifton J. Williamson ++ Date Created: July 1986 ++ Date Last Updated: 29 January 1990 -++ Keywords: stream, infinite list, infinite sequence ++ Description: ++ Functions defined on streams with entries in two sets. @@ -146233,7 +145230,6 @@ StreamFunctions2(A:Type,B:Type): Exports == Implementation where ++ Authors: Burge, Watt; updated by Clifton J. Williamson ++ Date Created: July 1986 ++ Date Last Updated: 29 January 1990 -++ Keywords: stream, infinite list, infinite sequence ++ Description: ++ Functions defined on streams with entries in three sets. @@ -146294,14 +145290,6 @@ StreamFunctions3(A,B,C): Exports == Implementation where ++ Author: Clifton J. Williamson ++ Date Created: 23 February 1990 ++ Date Last Updated: 23 February 1990 -++ Basic Operations: infiniteProduct, evenInfiniteProduct, oddInfiniteProduct, -++ generalInfiniteProduct -++ Related Domains: UnivariateTaylorSeriesCategory -++ Also See: -++ AMS Classifications: -++ Keywords: Taylor series, infinite product -++ Examples: -++ References: ++ Description: ++ This package computes infinite products of Taylor series over an ++ integral domain of characteristic 0. Here Taylor series are @@ -146420,15 +145408,6 @@ just take the definition there. ++ Author: William Burge, Stephen Watt, Clifton J. Williamson ++ Date Created: 1986 ++ Date Last Updated: 26 May 1994 -++ Basic Operations: -++ Related Domains: Stream(A), ParadoxicalCombinatorsForStreams(A), -++ StreamTranscendentalFunctions(A), -++ StreamTranscendentalFunctionsNonCommutative(A) -++ Also See: -++ AMS Classifications: -++ Keywords: stream, Taylor series -++ Examples: -++ References: ++ Description: ++ StreamTaylorSeriesOperations implements Taylor series arithmetic, ++ where a Taylor series is represented by a stream of its coefficients. @@ -146906,6 +145885,8 @@ StreamTaylorSeriesOperations(A): Exports == Implementation where StreamTensor examples ==================================================================== +This package has no description + See Also: o )show StreamTensor @@ -146918,6 +145899,8 @@ o )show StreamTensor \begin{chunk}{package STNSR StreamTensor} )abbrev package STNSR StreamTensor +++ Description: +++ This package has no description StreamTensor(R: Type): with tensorMap: (Stream R, R -> List R) -> Stream R @@ -146967,13 +145950,6 @@ StreamTensor(R: Type): with ++ Author: William Burge, Clifton J. Williamson ++ Date Created: 1986 ++ Date Last Updated: 6 March 1995 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: Taylor series, elementary function, transcendental function -++ Examples: -++ References: ++ Description: ++ StreamTranscendentalFunctions implements transcendental functions on ++ Taylor series, where a Taylor series is represented by a stream of @@ -147446,13 +146422,6 @@ StreamTranscendentalFunctions(Coef): Exports == Implementation where ++ Author: Clifton J. Williamson ++ Date Created: 26 May 1994 ++ Date Last Updated: 26 May 1994 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: Taylor series, transcendental function, non-commutative -++ Examples: -++ References: ++ Description: ++ StreamTranscendentalFunctionsNonCommutative implements transcendental ++ functions on Taylor series over a non-commutative ring, where a Taylor @@ -147701,12 +146670,6 @@ StreamTranscendentalFunctionsNonCommutative(Coef): _ ++ Authors: J. Grabmeier ++ Date Created: 02 April 1992 ++ Date Last Updated: 14 April 1992 -++ Basic Operations: -++ Related Constructors: AlgebraPackage, AlgebraGivenByStructuralConstants -++ Also See: -++ AMS Classifications: -++ Keywords: structural constants -++ Reference: ++ Description: ++ StructuralConstantsPackage provides functions creating ++ structural constants from a multiplication tables or a basis @@ -147861,12 +146824,6 @@ must be linear polynomials in the generators" ++ Author: Lalo Gonzalez-Vega ++ Date Created: 1994? ++ Date Last Updated: 30 January 1996 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: localization -++ References: ++ Description: ++ This package produces functions for counting etc. real roots of univariate ++ polynomials in x over R, which must be an OrderedIntegralDomain @@ -148391,12 +147348,6 @@ SubResultantPackage(R, UP): Exports == Implementation where ++ Author: P. Gianni ++ Date Created: October 1993 ++ Date Last Updated: March 1995 -++ Basic Functions: -++ Related Constructors: MultivariateFactorize -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ SupFractionFactorize contains the factor function for univariate ++ polynomials over the quotient field of a ring S such that the package @@ -148475,7 +147426,6 @@ SupFractionFactorizer(E,OV,R,P) : C == T ++ Author: Manuel Bronstein ++ Date Created: 11 June 1991 ++ Date Last Updated: 13 April 1994 -++ Keywords: differential equation, ODE, system ++ Description: ++ SystemODESolver provides tools for triangulating ++ and solving some systems of linear ordinary differential equations. @@ -148729,13 +147679,6 @@ SystemODESolver(F, LO): Exports == Implementation where ++ Author: P. Gianni ++ Date Created: summer 1988 ++ Date Last Updated: summer 1990 -++ Basic Functions: -++ Related Constructors: Fraction, Polynomial, FloatingRealPackage, -++ FloatingComplexPackage, RadicalSolvePackage -++ Also See: LinearSystemMatrixPackage, GroebnerSolve -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Symbolic solver for systems of rational functions with coefficients ++ in an integral domain R. @@ -148997,13 +147940,6 @@ SystemSolvePackage(R): Cat == Cap where ++ Authors: Johannes Grabmeier, Thorsten Werther ++ Date Created: 03 September 1988 ++ Date Last Updated: 07 June 1990 -++ Basic Operations: nextPartition, numberOfImproperPartitions, -++ listYoungTableaus, subSet, unrankImproperPartitions0 -++ Related Constructors: IntegerCombinatoricFunctions -++ Also See: RepresentationTheoryPackage1, RepresentationTheoryPackage2, -++ IrrRepSymNatPackage -++ AMS Classifications: -++ Keywords: improper partition, partition, subset, Coleman ++ References: ++ G. James/ A. Kerber: The Representation Theory of the Symmetric ++ Group. Encycl. of Math. and its Appl., Vol. 16., Cambridge @@ -149546,12 +148482,6 @@ SymmetricFunctions(R:Ring): Exports == Implementation where ++ Author: William H. Burge ++ Date Created: 1987 ++ Date Last Updated: 23 Sept 1991 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: Young tableau -++ References: ++ Description: ++ TableauBumpers implements the Schenstead-Knuth ++ correspondence between sequences and pairs of Young tableaux. @@ -149699,12 +148629,6 @@ TableauxBumpers(S:OrderedSet):T==C where ++ Author: Marc Moreno Maza ++ Date Created: 09/09/1998 ++ Date Last Updated: 12/16/1998 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ \axiom{TabulatedComputationPackage(Key ,Entry)} provides some modest support ++ for dealing with operations with type \axiom{Key -> Entry}. The result of @@ -150075,14 +148999,6 @@ should be unique. )abbrev package TEMUTL TemplateUtilities ++ Author: Mike Dewar ++ Date Created: October 1992 -++ Date Last Updated: -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ This package provides functions for template manipulation @@ -150133,12 +149049,6 @@ TemplateUtilities(): Exports == Implementation where )abbrev package TEX1 TexFormat1 ++ Author: Robert S. Sutor ++ Date Created: 1987 through 1990 -++ Change History: -++ Basic Operations: coerce -++ Related Constructors: TexFormat -++ Also See: ScriptFormulaFormat, ScriptFormulaFormat1 -++ AMS Classifications: -++ Keywords: TeX, output, format ++ Description: ++ \spadtype{TexFormat1} provides a utility coercion for changing ++ to TeX format anything that has a coercion to the standard output format. @@ -150251,12 +149161,6 @@ ToolsForSign(R:Ring): with ++ Author: Clifton J. Williamson ++ Date Created: 23 January 1990 ++ Date Last Updated: October 1991 by Jon Steinbach -++ Basic Operations: draw -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ TopLevelDrawFunctions provides top level functions for ++ drawing graphics of expressions. @@ -150590,15 +149494,9 @@ TopLevelDrawFunctions(Ex:Join(ConvertibleTo InputForm,SetCategory)): \begin{chunk}{package DRAWCURV TopLevelDrawFunctionsForAlgebraicCurves} )abbrev package DRAWCURV TopLevelDrawFunctionsForAlgebraicCurves -++ Author: Clifton J. Williamson +++ Author: Clifton J. Williamson, Jon Steinbach ++ Date Created: 26 June 1990 -++ Date Last Updated: October 1991 by Jon Steinbach -++ Basic Operations: draw -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: +++ Date Last Updated: October 1991 ++ Description: ++ TopLevelDrawFunctionsForAlgebraicCurves provides top level ++ functions for drawing non-singular algebraic curves. @@ -150731,15 +149629,9 @@ TopLevelDrawFunctionsForAlgebraicCurves(R,Ex): Exports == Implementation where \begin{chunk}{package DRAWCFUN TopLevelDrawFunctionsForCompiledFunctions} )abbrev package DRAWCFUN TopLevelDrawFunctionsForCompiledFunctions -++ Author: Clifton J. Williamson +++ Author: Clifton J. Williamson, Scott Morrison ++ Date Created: 22 June 1990 -++ Date Last Updated: January 1992 by Scott Morrison -++ Basic Operations: draw, recolor -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: +++ Date Last Updated: January 1992 ++ Description: ++ TopLevelDrawFunctionsForCompiledFunctions provides top level ++ functions for drawing graphics of expressions. @@ -151388,12 +150280,6 @@ SingleFloat value. ++ Author: Mike Dewar ++ Date Created: 24 May 1995 ++ Date Last Updated: 25 November 1996 -++ Basic Operations: draw -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ TopLevelDrawFunctionsForPoints provides top level functions ++ for drawing curves and surfaces described by sets of points. @@ -152103,7 +150989,6 @@ TranscendentalIntegration(F, UP): Exports == Implementation where ++ Author: Bob Sutor, Manuel Bronstein ++ Date Created: Way back ++ Date Last Updated: 22 January 1996, added simplifyLog MCD. -++ Keywords: transcendental, manipulation. ++ Description: ++ TranscendentalManipulations provides functions to simplify and ++ expand expressions involving transcendental operators. @@ -153425,6 +152310,28 @@ definingPolynomial %phi1 TransSolvePackage examples ==================================================================== +This package tries to find solutions of equations of type Expression(R). + +This means expressions involving transcendental, exponential, logarithmic +and nthRoot functions. + +After trying to transform different kernels to one kernel by applying +several rules, it calls zerosOf for the SparseUnivariatePolynomial in +the remaining kernel. + +For example the expression + sin(x)*cos(x)-2 +will be transformed to + -2 tan(x/2)**4 -2 tan(x/2)**3 -4 tan(x/2)**2 +2 tan(x/2) -2 +by using the function normalize and then to + -2 tan(x)**2 + tan(x) -2 +with help of subsTan. This function tries to express the given function +in terms of tan(x/2) to express in terms of tan(x). + +Other examples are the expressions + sqrt(x+1)+sqrt(x+7)+1 + sqrt(sin(x))+1 + solve(1/2*v*v*cos(theta+phi)*cos(theta+phi)+g*l*cos(phi)=g*l,phi) definingPolynomial %phi0 @@ -153445,10 +152352,7 @@ o )show TransSolvePackage )abbrev package SOLVETRA TransSolvePackage ++ Author: W. Wiwianka, Martin Rubey ++ Date Created: Summer 1991 -++ Change History: 9/91 -++ Basic Operations: solve -++ Related Constructors: RadicalSolvePackage, FloatingRealPackage -++ Keywords: +++ Date Last Changed: 9/91 ++ Description: ++ This package tries to find solutions of equations of type Expression(R). ++ This means expressions involving transcendental, exponential, logarithmic @@ -153998,10 +152902,7 @@ generates the error (reported as bug \# 102): )abbrev package SOLVESER TransSolvePackageService ++ Author: W. Wiwianka ++ Date Created: Summer 1991 -++ Change History: 9/91 -++ Basic Operations: decomposeFunc, unvectorise -++ Related Constructors: -++ Keywords: +++ Date Last Changed: 9/91 ++ Description: ++ This package finds the function func3 where func1 and func2 ++ are given and func1 = func3(func2) . If there is no solution then @@ -154121,11 +153022,7 @@ TransSolvePackageService(R) : Exports == Implementation where \begin{chunk}{package TRIMAT TriangularMatrixOperations} )abbrev package TRIMAT TriangularMatrixOperations ++ Author: Victor Miller -++ Date Created: ++ Date Last Updated: 24 Jul 1990 -++ Keywords: -++ Examples: -++ References: ++ Description: ++ This package provides functions that compute "fraction-free" ++ inverses of upper and lower triangular matrices over a integral @@ -154212,7 +153109,6 @@ TriangularMatrixOperations(R,Row,Col,M): Exports == Implementation where ++ Author: Manuel Bronstein ++ Date Created: 4 April 1988 ++ Date Last Updated: 14 February 1994 -++ Keywords: trigonometric, function, manipulation. ++ Description: ++ \spadtype{TrigonometricManipulations} provides transformations from ++ trigonometric functions to complex exponentials and logarithms, and back. @@ -154381,8 +153277,6 @@ TrigonometricManipulations(R, F): Exports == Implementation where ++ Author: Clifton J. Williamson ++ Date Created: Bastille Day 1989 ++ Date Last Updated: 5 June 1990 -++ Keywords: -++ Examples: ++ Description: ++ Tools for constructing tubes around 3-dimensional parametric curves. @@ -154518,9 +153412,6 @@ TubePlotTools(): Exports == Implementation where ++ Author: Clifton J. Williamson ++ Date Created: 22 December 1989 ++ Date Last Updated: 10 July 1990 -++ Keywords: plot, singularity -++ Examples: -++ References: ++ Description: ++ Automatic clipping for 2-dimensional plots ++ The purpose of this package is to provide reasonable plots of @@ -155431,13 +154322,6 @@ UnivariateFormalPowerSeriesFunctions(Coef: Ring): Exports == Implementation ++ Author: Clifton J. Williamson ++ Date Created: 5 March 1990 ++ Date Last Updated: 5 March 1990 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: Laurent series, map -++ Examples: -++ References: ++ Description: ++ Mapping package for univariate Laurent series ++ This package allows one to apply a function to the coefficients of @@ -155485,15 +154369,6 @@ UnivariateLaurentSeriesFunctions2(Coef1,Coef2,var1,var2,cen1,cen2):_ \begin{chunk}{package UPOLYC2 UnivariatePolynomialCategoryFunctions2} )abbrev package UPOLYC2 UnivariatePolynomialCategoryFunctions2 -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ Mapping from polynomials over R to polynomials over S ++ given a map from R to S assumed to send zero to zero. @@ -155543,7 +154418,6 @@ UnivariatePolynomialCategoryFunctions2(R,PR,S,PS): Exports == Impl where ++ Author: Manuel Bronstein ++ Date Created: 2 May 1988 ++ Date Last Updated: 22 Feb 1990 -++ Keywords: gcd, quotient, common, denominator, polynomial. ++ Description: ++ UnivariatePolynomialCommonDenominator provides ++ functions to compute the common denominator of the coefficients of @@ -155604,12 +154478,6 @@ UnivariatePolynomialCommonDenominator(R, Q, UP): Exports == Impl where ++ Author: Frederic Lehobey ++ Date Created: 17 June 1996 ++ Date Last Updated: 4 June 1997 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keyword: -++ Exemples: ++ References: ++ [1] Peter Henrici, Automatic Computations with Power Series, ++ Journal of the Association for Computing Machinery, Volume 3, No. 1, @@ -155743,13 +154611,6 @@ UnivariatePolynomialDecompositionPackage(R,UP): Exports == Implementation where ++ Author: Frederic Lehobey ++ Date Created: 3 June 1997 ++ Date Last Updated: 3 June 1997 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keyword: -++ Exemples: -++ References: ++ Description: ++ UnivariatePolynomialDivisionPackage provides a ++ division for non monic univarite polynomials with coefficients in @@ -155800,15 +154661,6 @@ UnivariatePolynomialDivisionPackage(R,UP): Exports == Implementation where \begin{chunk}{package UP2 UnivariatePolynomialFunctions2} )abbrev package UP2 UnivariatePolynomialFunctions2 -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package lifts a mapping from coefficient rings R to S to ++ a mapping from \spadtype{UnivariatePolynomial}(x,R) to @@ -155955,14 +154807,6 @@ UnivariatePolynomialMultiplicationPackage(R: Ring, U: UnivariatePolynomialCatego \begin{chunk}{package UPSQFREE UnivariatePolynomialSquareFree} )abbrev package UPSQFREE UnivariatePolynomialSquareFree ++ Author: Dave Barton, Barry Trager -++ Date Created: -++ Date Last Updated: -++ Basic Functions: squareFree, squareFreePart -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package provides for square-free decomposition of ++ univariate polynomials over arbitrary rings, i.e. @@ -156091,9 +154935,6 @@ UnivariatePolynomialSquareFree(RC:IntegralDomain,P):C == T ++ Author: Scott C. Morrison ++ Date Created: 5 April 1991 ++ Date Last Updated: 5 April 1991 -++ Keywords: Puiseux series, map -++ Examples: -++ References: ++ Description: ++ Mapping package for univariate Puiseux series. ++ This package allows one to apply a function to the coefficients of @@ -156294,13 +155135,6 @@ UnivariateSkewPolynomialCategoryOps(R, C): Exports == Implementation where ++ Author: Clifton J. Williamson ++ Date Created: 9 February 1990 ++ Date Last Updated: 9 February 1990 -++ Basic Operations: -++ Related Domains: UnivariateTaylorSeries(Coef1,var,cen) -++ Also See: -++ AMS Classifications: -++ Keywords: Taylor series, map -++ Examples: -++ References: ++ Description: ++ Mapping package for univariate Taylor series. ++ This package allows one to apply a function to the coefficients of @@ -156352,9 +155186,6 @@ UnivariateTaylorSeriesFunctions2(Coef1,Coef2,UTS1,UTS2):_ ++ Author: Stephen Watt (revised by Clifton J. Williamson) ++ Date Created: February 1988 ++ Date Last Updated: 30 September 1993 -++ Keywords: differential equation, ODE, Taylor series -++ Examples: -++ References: ++ Description: ++ Taylor series solutions of explicit ODE's. ++ This package provides Taylor series solutions to regular @@ -156491,16 +155322,7 @@ UnivariateTaylorSeriesODESolver(Coef,UTS):_ \begin{chunk}{package UNISEG2 UniversalSegmentFunctions2} )abbrev package UNISEG2 UniversalSegmentFunctions2 -++ Author: -++ Date Created: ++ Date Last Updated: June 4, 1991 -++ Basic Operations: -++ Related Domains: Segment, UniversalSegment -++ Also See: -++ AMS Classifications: -++ Keywords: equation -++ Examples: -++ References: ++ Description: ++ This package provides operations for mapping functions onto segments. @@ -156841,11 +155663,12 @@ o )show U32VectorPolynomialOperations \begin{chunk}{package POLYVEC U32VectorPolynomialOperations} )abbrev package POLYVEC U32VectorPolynomialOperations -++ Description: This is a low-level package which implements operations -++ on vectors treated as univariate modular polynomials. Most -++ operations takes modulus as parameter. Modulus is machine -++ sized prime which should be small enough to avoid overflow -++ in intermediate calculations. +++ Description: +++ This is a low-level package which implements operations +++ on vectors treated as univariate modular polynomials. Most +++ operations takes modulus as parameter. Modulus is machine +++ sized prime which should be small enough to avoid overflow +++ in intermediate calculations. U32VectorPolynomialOperations() : Export == Implementation where PA ==> U32Vector Export ==> with @@ -157372,15 +156195,6 @@ U32VectorPolynomialOperations() : Export == Implementation where \begin{chunk}{package VECTOR2 VectorFunctions2} )abbrev package VECTOR2 VectorFunctions2 -++ Author: -++ Date Created: -++ Date Last Updated: -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package provides operations which all take as arguments ++ vectors of elements of some type \spad{A} and functions from \spad{A} to @@ -157460,15 +156274,6 @@ VectorFunctions2(A, B): Exports == Implementation where ++ Author: Jim Wen ++ Date Created: 15 January 1990 ++ Date Last Updated: -++ Basic Operations: pointColorDefault, lineColorDefault, axesColorDefault, -++ unitsColorDefault, pointSizeDefault, viewPosDefault, viewSizeDefault, -++ viewDefaults, viewWriteDefault, viewWriteAvailable, var1StepsDefault, -++ var2StepsDefault, tubePointsDefault, tubeRadiusDefault -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ ViewportDefaultsPackage describes default and user definable ++ values for graphics @@ -157688,12 +156493,6 @@ ViewDefaultsPackage():Exports == Implementation where ++ Author: Jim Wen ++ Date Created: 30 April 1989 ++ Date Last Updated: 15 June 1990 -++ Basic Operations: graphCurves, drawCurves, coerce -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ ViewportPackage provides functions for creating GraphImages ++ and TwoDimensionalViewports from lists of lists of points. @@ -157817,13 +156616,6 @@ ViewportPackage():Exports == Implementation where ++ Author:William H. Burge ++ Date Created:Sept 1988 ++ Date Last Updated:Feb 15 1992 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ Examples: -++ References: ++ Description: ++ This package implements the Weierstrass preparation ++ theorem f or multivariate power series. @@ -157982,14 +156774,6 @@ WeierstrassPreparation(R): Defn == Impl where ++ Authors: Victor Miller, Clifton Williamson ++ Date Created: 24 July 1991 ++ Date Last Updated: 20 September 1994 -++ Basic Operations: integralBasis, localIntegralBasis -++ Related Domains: IntegralBasisTools(R,UP,F), -++ TriangularMatrixOperations(R,Vector R,Vector R,Matrix R) -++ Also See: FunctionFieldIntegralBasis, NumberFieldIntegralBasis -++ AMS Classifications: -++ Keywords: function field, integral basis -++ Examples: -++ References: ++ Description: ++ In this package K is a finite field, R is a ring of univariate ++ polynomials over K, and F is a framed algebra over R. The package @@ -158203,13 +156987,6 @@ WildFunctionFieldIntegralBasis(K,R,UP,F): Exports == Implementation where ++ Author: Michel Petitot (petitot@lifl.fr). ++ Date Created: 91 ++ Date Last Updated: 7 Juillet 92 -++ Fix History: compilation v 2.1 le 13 dec 98 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ This package provides computations of logarithms and exponentials ++ for polynomials in non-commutative variables. @@ -159878,6 +158655,29 @@ lpr2 := positiveSolve(lf)$pack ZeroDimensionalSolvePackage examples ==================================================================== +A package for computing symbolically the complex and real roots of +zero-dimensional algebraic systems over the integer or rational +numbers. Complex roots are given by means of univariate representations +of irreducible regular chains. Real roots are given by means of tuples +of coordinates lying in the RealClosure of the coefficient ring. + +This constructor takes three arguments. The first one R is the +coefficient ring. The second one ls is the list of variables +involved in the systems to solve. The third one must be concat(ls,s) +where s is an additional symbol used for the univariate representations. + +WARNING. The third argument is not checked. + +All operations are based on triangular decompositions. +The default is to compute these decompositions directly from the input +system by using the RegularChain domain constructor. + +The lexTriangular algorithm can also be used for computing these +decompositions (see LexTriangularPackage package constructor). + +For that purpose, the operations univariateSolve, realSolve and +positiveSolve admit an optional argument. + The ZeroDimensionalSolvePackage package constructor provides operations for computing symbolically the complex or real roots of zero-dimensional algebraic systems. @@ -161420,12 +160220,6 @@ o )show ZeroDimensionalSolvePackage ++ Author: Marc Moreno Maza ++ Date Created: 23/01/1999 ++ Date Last Updated: 08/02/1999 -++ Basic Functions: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: ++ Description: ++ A package for computing symbolically the complex and real roots of ++ zero-dimensional algebraic systems over the integer or rational diff --git a/changelog b/changelog index b5c70a9..28cfdc4 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,5 @@ +20130305 tpd src/axiom-website/patches.html 20130305.02.tpd.patch +20120305 tpd books/bookvol10.4 write help documentation 20130305 tpd src/axiom-website/patches.html 20130305.01.tpd.patch 20120305 tpd books/bookvolbib add references 20130301 tpd src/axiom-website/patches.html 20130301.01.tpd.patch diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index 86b2e2c..043eaa6 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -4005,5 +4005,7 @@ books/bookvolbib add references books/bookvol10.3 write help documentation for all domains 20130305.01.tpd.patch books/bookvolbib add references +20130305.02.tpd.patch +books/bookvol10.4 write help documentation