diff --git a/books/bookvol0.pamphlet b/books/bookvol0.pamphlet index 3487c16..c7c47e5 100644 --- a/books/bookvol0.pamphlet +++ b/books/bookvol0.pamphlet @@ -24701,6 +24701,262 @@ AXIOM domains and packages. The sections are organized by constructor names. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\domainhead{ApplicationProgramInterface} + +The ApplicationProgramInterface exposes Axiom internal functions +which might be useful for understanding, debugging, or creating tools. + +The getDomains function takes the name of a category and returns +a set of domains which inherit from that category: +\begin{verbatim} + getDomains 'Collection + + {AssociationList, Bits, CharacterClass, DataList, EqTable, FlexibleArray, + GeneralPolynomialSet, GeneralSparseTable, GeneralTriangularSet, HashTable, + IndexedBits, IndexedFlexibleArray, IndexedList, IndexedOneDimensionalArray, + IndexedString, IndexedVector, InnerTable, KeyedAccessFile, Library, List, + ListMultiDictionary, Multiset, OneDimensionalArray, Point, PrimitiveArray, + RegularChain, RegularTriangularSet, Result, RoutinesTable, Set, + SparseTable, SquareFreeRegularTriangularSet, Stream, String, StringTable, + Table, Vector, WuWenTsunTriangularSet} + Type: Set Symbol +\end{verbatim} +This can be used to form the set-difference of two categories: +\begin{verbatim} + difference(getDomains 'IndexedAggregate, getDomains 'Collection) + + {DirectProduct, DirectProductMatrixModule, DirectProductModule, + HomogeneousDirectProduct, OrderedDirectProduct, + SplitHomogeneousDirectProduct} + Type: Set Symbol +\end{verbatim} +The credits function prints a list of the people who have contributed +to the development of Axiom. This is equivalent to the )credits command. + +The summary function prints a short list of useful console commands. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\domainhead{ArrayStack} + +An ArrayStack object is represented as a list ordered by last-in, +first-out. It operates like a pile of books, where the ``next'' book is +the one on the top of the pile. + +Here we create an array stack of integers from a list. Notice that the +order in the list is the order in the stack. +\begin{verbatim} + a:ArrayStack INT:= arrayStack [1,2,3,4,5] + [1,2,3,4,5] +\end{verbatim} +We can remove the top of the stack using pop!: +\begin{verbatim} + pop! a + 1 +\end{verbatim} +Notice that the use of pop! is destructive (destructive operations +in Axiom usually end with ! to indicate that the underylying data +structure is changed). +\begin{verbatim} + a + [2,3,4,5] +\end{verbatim} +The extract! operation is another name for the pop! operation and +has the same effect. This operation treats the stack as a BagAggregate: +\begin{verbatim} + extract! a + 2 +\end{verbatim} +and you can see that it also has destructively modified the stack: +\begin{verbatim} + a + [3,4,5] +\end{verbatim} +Next we push a new element on top of the stack: +\begin{verbatim} + push!(9,a) + 9 +\end{verbatim} +Again, the push! operation is destructive so the stack is changed: +\begin{verbatim} + a + [9,3,4,5] +\end{verbatim} +Another name for push! is insert!, which treats the stack as a BagAggregate: +\begin{verbatim} + insert!(8,a) + [8,9,3,4,5] +\end{verbatim} +and it modifies the stack: +\begin{verbatim} + a + [8,9,3,4,5] +\end{verbatim} +The inspect function returns the top of the stack without modification, +viewed as a BagAggregate: +\begin{verbatim} + inspect a + 8 +\end{verbatim} +The empty? operation returns true only if there are no element on the +stack, otherwise it returns false: +\begin{verbatim} + empty? a + false +\end{verbatim} +The top operation returns the top of stack without modification, viewed +as a Stack: +\begin{verbatim} + top a + 8 +\end{verbatim} +The depth operation returns the number of elements on the stack: +\begin{verbatim} + depth a + 5 +\end{verbatim} +which is the same as the \# (length) operation: +\begin{verbatim} + #a + 5 +\end{verbatim} +The less? predicate will compare the stack length to an integer: +\begin{verbatim} + less?(a,9) + true +\end{verbatim} +The more? predicate will compare the stack length to an integer: +\begin{verbatim} + more?(a,9) + false +\end{verbatim} +The size? operation will compare the stack length to an integer: +\begin{verbatim} + size?(a,#a) + true +\end{verbatim} +and since the last computation must alwasy be true we try: +\begin{verbatim} + size?(a,9) + false +\end{verbatim} +The parts function will return the stack as a list of its elements: +\begin{verbatim} + parts a + [8,9,3,4,5] +\end{verbatim} +If we have a BagAggregate of elements we can use it to construct a stack. +Notice that the elements are pushed in reverse order: +\begin{verbatim} + bag([1,2,3,4,5])$ArrayStack(INT) + [5,4,3,2,1] +\end{verbatim} +The empty function will construct an empty stack of a given type: +\begin{verbatim} + b:=empty()$(ArrayStack INT) + [] +\end{verbatim} +and the empty? predicate allows us to find out if a stack is empty: +\begin{verbatim} + empty? b + true +\end{verbatim} +The sample function returns a sample, empty stack: +\begin{verbatim} + sample()$ArrayStack(INT) + [] +\end{verbatim} +We can copy a stack and it does not share storage so subsequent +modifications of the original stack will not affect the copy: +\begin{verbatim} + c:=copy a + [8,9,3,4,5] +\end{verbatim} +The eq? function is only true if the lists are the same reference, +so even though c is a copy of a, they are not the same: +\begin{verbatim} + eq?(a,c) + false +\end{verbatim} +However, a clearly shares a reference with itself: +\begin{verbatim} + eq?(a,a) + true +\end{verbatim} +But we can compare a and c for equality: +\begin{verbatim} + (a=c)@Boolean + true +\end{verbatim} +and clearly a is equal to itself: +\begin{verbatim} + (a=a)@Boolean + true +\end{verbatim} +and since a and c are equal, they are clearly NOT not-equal: +\begin{verbatim} + a~=c + false +\end{verbatim} +We can use the any? function to see if a predicate is true for any element: +\begin{verbatim} + any?(x+->(x=4),a) + true +\end{verbatim} +or false for every element: +\begin{verbatim} + any?(x+->(x=11),a) + false +\end{verbatim} +We can use the every? function to check every element satisfies a predicate: +\begin{verbatim} + every?(x+->(x=11),a) + false +\end{verbatim} +We can count the elements that are equal to an argument of this type: +\begin{verbatim} + count(4,a) + 1 +\end{verbatim} +or we can count against a boolean function: +\begin{verbatim} + count(x+->(x>2),a) + 5 +\end{verbatim} +You can also map a function over every element, returning a new stack: +\begin{verbatim} + map(x+->x+10,a) + [18,19,13,14,15] +\end{verbatim} +Notice that the orignal stack is unchanged: +\begin{verbatim} + a + [8,9,3,4,5] +\end{verbatim} +You can use map! to map a function over every element and change +the original stack since map! is destructive: +\begin{verbatim} + map!(x+->x+10,a) + [18,19,13,14,15] +\end{verbatim} +Notice that the orignal stack has been changed: +\begin{verbatim} + a + [18,19,13,14,15] +\end{verbatim} +The member function can also get the element of the stack as a list: +\begin{verbatim} + members a + [18,19,13,14,15] +\end{verbatim} +and using member? we can test if the stack holds a given element: +\begin{verbatim} + member?(14,a) + true +\end{verbatim} +Also see \domainref{Stack}, \domainref{Queue}, \domainref{Dequeue}, +\domainref{Heap}, and \domainref{BagAggregate} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \domainhead{AssociationList} The {\tt AssociationList} constructor provides a general structure for @@ -28825,6 +29081,320 @@ $$ \returnType{Type: Polynomial DecimalExpansion} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\domainhead{Dequeue} + +A Dequeue is a double-ended queue so elements can be added to either end. + +Here we create an dequeue of integers from a list. Notice that the +order in the list is the order in the dequeue. +\begin{verbatim} + a:Dequeue INT:= dequeue [1,2,3,4,5] + [1,2,3,4,5] +\end{verbatim} +We can remove the top of the dequeue using dequeue!: +\begin{verbatim} + dequeue! a + 1 +\end{verbatim} +Notice that the use of dequeue! is destructive (destructive operations +in Axiom usually end with ! to indicate that the underylying data +structure is changed). +\begin{verbatim} + a + [2,3,4,5] +\end{verbatim} +The extract! operation is another name for the dequeue! operation and +has the same effect. This operation treats the dequeue as a BagAggregate: +\begin{verbatim} + extract! a + 2 +\end{verbatim} +and you can see that it also has destructively modified the dequeue: +\begin{verbatim} + a + [3,4,5] +\end{verbatim} +Next we use enqueue! to add a new element to the end of the dequeue: +\begin{verbatim} + enqueue!(9,a) + 9 +\end{verbatim} +Again, the enqueue! operation is destructive so the dequeue is changed: +\begin{verbatim} + a + [3,4,5,9] +\end{verbatim} +Another name for enqueue! is insert!, which treats the dequeue as a +BagAggregate: +\begin{verbatim} + insert!(8,a) + [3,4,5,9,8] +\end{verbatim} +and it modifies the dequeue: +\begin{verbatim} + a + [3,4,5,9,8] +\end{verbatim} +The front operation returns the item at the front of the dequeue: +\begin{verbatim} + front a + 3 +\end{verbatim} +The back operation returns the item at the back of the dequeue: +\begin{verbatim} + back a + 8 +\end{verbatim} +The bottom! operation returns the item at the back of the dequeue: +\begin{verbatim} + bottom! a + 8 +\end{verbatim} +and it modifies the dequeue: +\begin{verbatim} + a + [3,4,5,9] +\end{verbatim} +The depth function returns the number of elements in the dequeue: +\begin{verbatim} + depth a + 4 +\end{verbatim} +The height function returns the number of elements in the dequeue: +\begin{verbatim} + height a + 4 +\end{verbatim} +The insertBottom! function adds the element at the end: +\begin{verbatim} + insertBottom!(6,a) + 6 +\end{verbatim} +and it modifies the dequeue: +\begin{verbatim} + a + [3,4,5,9,6] +\end{verbatim} +The extractBottom! function removes the element at the end: +\begin{verbatim} + extractBottom! a + 6 +\end{verbatim} +and it modifies the dequeue: +\begin{verbatim} + a + [3,4,5,9] +\end{verbatim} +The insertTop! function adds the element at the top: +\begin{verbatim} + insertTop!(7,a) + 7 +\end{verbatim} +and it modifies the dequeue: +\begin{verbatim} + a + [7,3,4,5,9] +\end{verbatim} +The extractTop! function adds the element at the top: +\begin{verbatim} + extractTop! a + 7 +\end{verbatim} +and it modifies the dequeue: +\begin{verbatim} + a + [3,4,5,9] +\end{verbatim} +The top function returns the top element: +\begin{verbatim} + top a + 3 +\end{verbatim} +and it does not modifies the dequeue: +\begin{verbatim} + a + [3,4,5,9] +\end{verbatim} +The top! function returns the top element: +\begin{verbatim} + top! a + 3 +\end{verbatim} +and it modifies the dequeue: +\begin{verbatim} + a + [4,5,9] +\end{verbatim} +The reverse! operation destructively reverses the elements of the dequeue: +\begin{verbatim} + reverse! a + [9,5,4] +\end{verbatim} +The rotate! operation moves the top element to the bottom: +\begin{verbatim} + rotate! a + [5,4,9] +\end{verbatim} +The inspect function returns the top of the dequeue without modification, +viewed as a BagAggregate: +\begin{verbatim} + inspect a + 5 +\end{verbatim} +The empty? operation returns true only if there are no element on the +dequeue, otherwise it returns false: +\begin{verbatim} + empty? a + false +\end{verbatim} +The \# (length) operation: +\begin{verbatim} + #a + 3 +\end{verbatim} +The length operation does the same thing: +\begin{verbatim} + length a + 3 +\end{verbatim} +The less? predicate will compare the dequeue length to an integer: +\begin{verbatim} + less?(a,9) + true +\end{verbatim} +The more? predicate will compare the dequeue length to an integer: +\begin{verbatim} + more?(a,9) + false +\end{verbatim} +The size? operation will compare the dequeue length to an integer: +\begin{verbatim} + size?(a,#a) + true +\end{verbatim} +and since the last computation must alwasy be true we try: +\begin{verbatim} + size?(a,9) + false +\end{verbatim} +The parts function will return the dequeue as a list of its elements: +\begin{verbatim} + parts a + [5,4,9] +\end{verbatim} +If we have a BagAggregate of elements we can use it to construct a dequeue: +\begin{verbatim} + bag([1,2,3,4,5])$Dequeue(INT) + [1,2,3,4,5] +\end{verbatim} +The empty function will construct an empty dequeue of a given type: +\begin{verbatim} + b:=empty()$(Dequeue INT) + [] +\end{verbatim} +and the empty? predicate allows us to find out if a dequeue is empty: +\begin{verbatim} + empty? b + true +\end{verbatim} +The sample function returns a sample, empty dequeue: +\begin{verbatim} + sample()$Dequeue(INT) + [] +\end{verbatim} +We can copy a dequeue and it does not share storage so subsequent +modifications of the original dequeue will not affect the copy: +\begin{verbatim} + c:=copy a + [5,4,9] +\end{verbatim} +The eq? function is only true if the lists are the same reference, +so even though c is a copy of a, they are not the same: +\begin{verbatim} + eq?(a,c) + false +\end{verbatim} +However, a clearly shares a reference with itself: +\begin{verbatim} + eq?(a,a) + true +\end{verbatim} +But we can compare a and c for equality: +\begin{verbatim} + (a=c)@Boolean + true +\end{verbatim} +and clearly a is equal to itself: +\begin{verbatim} + (a=a)@Boolean + true +\end{verbatim} +and since a and c are equal, they are clearly NOT not-equal: +\begin{verbatim} + a~=c + false +\end{verbatim} +We can use the any? function to see if a predicate is true for any element: +\begin{verbatim} + any?(x+->(x=4),a) + true +\end{verbatim} +or false for every element: +\begin{verbatim} + any?(x+->(x=11),a) + false +\end{verbatim} +We can use the every? function to check every element satisfies a predicate: +\begin{verbatim} + every?(x+->(x=11),a) + false +\end{verbatim} +We can count the elements that are equal to an argument of this type: +\begin{verbatim} + count(4,a) + 1 +\end{verbatim} +or we can count against a boolean function: +\begin{verbatim} + count(x+->(x>2),a) + 3 +\end{verbatim} +You can also map a function over every element, returning a new dequeue: +\begin{verbatim} + map(x+->x+10,a) + [15,14,19] +\end{verbatim} +Notice that the orignal dequeue is unchanged: +\begin{verbatim} + a + [5,4,9] +\end{verbatim} +You can use map! to map a function over every element and change +the original dequeue since map! is destructive: +\begin{verbatim} + map!(x+->x+10,a) + [15,14,19] +\end{verbatim} +Notice that the orignal dequeue has been changed: +\begin{verbatim} + a + [15,14,19] +\end{verbatim} +The member function can also get the element of the dequeue as a list: +\begin{verbatim} + members a + [15,14,19] +\end{verbatim} +and using member? we can test if the dequeue holds a given element: +\begin{verbatim} + member?(14,a) + true +\end{verbatim} + +See \domainref{Stack}, \domainref{ArrayStack}, \domainref{Queue}, +\domainref{Dequeue}, \domainref{Heap} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \domainhead{DistributedMultivariatePolynomial} \hyphenation{Homo-gen-eous-Dis-tributed-Multi-var-i-ate-Pol-y-nomial} @@ -29259,6 +29829,62 @@ $$ \returnType{Type: Boolean} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\domainhead{EuclideanGroebnerBasisPackage} + +Example to call euclideanGroebner: +\begin{verbatim} + a1:DMP([y,x],INT):= (9*x**2 + 5*x - 3)+ y*(3*x**2 + 2*x + 1) + a2:DMP([y,x],INT):= (6*x**3 - 2*x**2 - 3*x +3) + y*(2*x**3 - x - 1) + a3:DMP([y,x],INT):= (3*x**3 + 2*x**2) + y*(x**3 + x**2) + an:=[a1,a2,a3] + euclideanGroebner(an) +\end{verbatim} +This will return the weak euclidean Groebner basis set. +All reductions are total reductions. + +You can get more information by providing a second argument. +To get the reduced critical pairs do: +\begin{verbatim} + euclideanGroebner(an,"redcrit") +\end{verbatim} +You can get other information by calling: +\begin{verbatim} + euclideanGroebner(an,"info") +\end{verbatim} +which returns: +\begin{verbatim} + ci => Leading monomial for critpair calculation + tci => Number of terms of polynomial i + cj => Leading monomial for critpair calculation + tcj => Number of terms of polynomial j + c => Leading monomial of critpair polynomial + tc => Number of terms of critpair polynomial + rc => Leading monomial of redcritpair polynomial + trc => Number of terms of redcritpair polynomial + tH => Number of polynomials in reduction list H + tD => Number of critpairs still to do +\end{verbatim} +The three argument form returns all of the information: +\begin{verbatim} + euclideanGroebner(an,"info","redcrit") +\end{verbatim} + +The term ordering is determined by the polynomial type used. +Suggested types include +\begin{verbatim} + DistributedMultivariatePolynomial + HomogeneousDistributedMultivariatePolynomial + GeneralDistributedMultivariatePolynomial +\end{verbatim} + +See +\domainref{EuclideanGroebnerBasisPackage}, +\domainref{DistributedMultivariatePolynomial}, +\domainref{HomogeneousDistributedMultivariatePolynomial}, +\domainref{GeneralDistributedMultivariatePolynomial}, and +\domainref{GroebnerPackage} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \domainhead{Exit} A function that does not return directly to its caller has {\tt Exit} @@ -31353,6 +31979,105 @@ For more information, see the paper: Bronstein, M and Salvy, B. \domainref{PartialFraction} for standard partial fraction decompositions. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\domainhead{GeneralDistributedMultivariatePolynomial} + +DistributedMultivariatePolynomial which is abbreviated as DMP and +HomogeneousDistributedMultivariatePolynomial, which is abbreviated +as HDMP, are very similar to MultivariatePolynomial except that +they are represented and displayed in a non-recursive manner. +\begin{verbatim} + (d1,d2,d3) : DMP([z,y,x],FRAC INT) + Type: Void +\end{verbatim} +The constructor DMP orders its monomials lexicographically while +HDMP orders them by total order refined by reverse lexicographic +order. +\begin{verbatim} + d1 := -4*z + 4*y**2*x + 16*x**2 + 1 + 2 2 + - 4z + 4y x + 16x + 1 + Type: DistributedMultivariatePolynomial([z,y,x],Fraction Integer) +\end{verbatim} +\begin{verbatim} + d2 := 2*z*y**2 + 4*x + 1 + 2 + 2z y + 4x + 1 + Type: DistributedMultivariatePolynomial([z,y,x],Fraction Integer) +\end{verbatim} +\begin{verbatim} + d3 := 2*z*x**2 - 2*y**2 - x + 2 2 + 2z x - 2y - x + Type: DistributedMultivariatePolynomial([z,y,x],Fraction Integer) +\end{verbatim} +These constructors are mostly used in Groebner basis calculations. +\begin{verbatim} + groebner [d1,d2,d3] + 1568 6 1264 5 6 4 182 3 2047 2 103 2857 + [z - ---- x - ---- x + --- x + --- x - ---- x - ---- x - -----, + 2745 305 305 549 610 2745 10980 + 2 112 6 84 5 1264 4 13 3 84 2 1772 2 + y + ---- x - --- x - ---- x - --- x + --- x + ---- x + ----, + 2745 305 305 549 305 2745 2745 + 7 29 6 17 4 11 3 1 2 15 1 + x + -- x - -- x - -- x + -- x + -- x + -] + 4 16 8 32 16 4 + Type: List DistributedMultivariatePolynomial([z,y,x],Fraction Integer) +\end{verbatim} +\begin{verbatim} + (n1,n2,n3) : HDMP([z,y,x],FRAC INT) + Type: Void +\end{verbatim} +\begin{verbatim} + n1 := d1 + 2 2 + 4y x + 16x - 4z + 1 + Type: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer) +\end{verbatim} +\begin{verbatim} + n2 := d2 + 2 + 2z y + 4x + 1 + Type: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer) +\end{verbatim} +\begin{verbatim} + n3 := d3 + 2 2 + 2z x - 2y - x + Type: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer) +\end{verbatim} +Note that we get a different Groebner basis when we use the HDMP +polynomials, as expected. +\begin{verbatim} + groebner [n1,n2,n3] + 4 3 3 2 1 1 4 29 3 1 2 7 9 1 + [y + 2x - - x + - z - -, x + -- x - - y - - z x - -- x - -, + 2 2 8 4 8 4 16 4 + 2 1 2 2 1 2 2 1 + z y + 2x + -, y x + 4x - z + -, z x - y - - x, + 2 4 2 + 2 2 2 1 3 + z - 4y + 2x - - z - - x] + 4 2 + Type: List HomogeneousDistributedMultivariatePolynomial([z,y,x], + Fraction Integer) +\end{verbatim} +GeneralDistributedMultivariatePolynomial is somewhat more flexible in +the sense that as well as accepting a list of variables to specify the +variable ordering, it also takes a predicate on exponent vectors to +specify the term ordering. With this polynomial type the user can +experiment with the effect of using completely arbitrary term orderings. +This flexibility is mostly important for algorithms such as Groebner +basis calculations which can be very sensitive to term ordering. + +See +\domainref{Polynomial}\\ +\domainref{UnivariatePolynomial}\\ +\domainref{MultivariatePolynomial}\\ +\domainref{HomogeneousDistributedMultivariatePolynomial}, and\\ +\domainref{DistributedMultivariatePolynomial} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \domainhead{GeneralSparseTable} Sometimes when working with tables there is a natural value to use as @@ -31613,6 +32338,53 @@ code for {\tt GroebnerFactorizationPackage} in {\bf groebf.input} for more details about the algorithms used. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\domainhead{GroebnerPackage} + +Example to call groebner: +\begin{verbatim} + s1:DMP[w,p,z,t,s,b]RN:= 45*p + 35*s - 165*b - 36 + s2:DMP[w,p,z,t,s,b]RN:= 35*p + 40*z + 25*t - 27*s + s3:DMP[w,p,z,t,s,b]RN:= 15*w + 25*p*s + 30*z - 18*t - 165*b**2 + s4:DMP[w,p,z,t,s,b]RN:= -9*w + 15*p*t + 20*z*s + s5:DMP[w,p,z,t,s,b]RN:= w*p + 2*z*t - 11*b**3 + s6:DMP[w,p,z,t,s,b]RN:= 99*w - 11*b*s + 3*b**2 + s7:DMP[w,p,z,t,s,b]RN:= b**2 + 33/50*b + 2673/10000 + + sn7:=[s1,s2,s3,s4,s5,s6,s7] + + groebner(sn7,info) +\end{verbatim} +groebner calculates a minimal Groebner Basis +all reductions are TOTAL reductions + +To get the reduced critical pairs do: +\begin{verbatim} + groebner(sn7,"redcrit") +\end{verbatim} +You can get other information by calling: +\begin{verbatim} + groebner(sn7,"info") +\end{verbatim} +which returns: +\begin{verbatim} + ci => Leading monomial for critpair calculation + tci => Number of terms of polynomial i + cj => Leading monomial for critpair calculation + tcj => Number of terms of polynomial j + c => Leading monomial of critpair polynomial + tc => Number of terms of critpair polynomial + rc => Leading monomial of redcritpair polynomial + trc => Number of terms of redcritpair polynomial + tF => Number of polynomials in reduction list F + tD => Number of critpairs still to do +\end{verbatim} +See +\domainref{GroebnerPackage}\\ +\domainref{DistributedMultivariatePolynomial}\\ +\domainref{HomogeneousDistributedMultivariatePolynomial}\\ +\domainref{EuclideanGroebnerBasisPackage} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \domainhead{Heap} The domain {\tt Heap(S)} implements a priority queue of objects of @@ -31786,6 +32558,108 @@ $$ \returnType{Type: Polynomial HexadecimalExpansion} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\domainhead{HomogeneousDistributedMultivariatePolynomial} + +DistributedMultivariatePolynomial which is abbreviated as DMP and +HomogeneousDistributedMultivariatePolynomial, which is abbreviated +as HDMP, are very similar to MultivariatePolynomial except that +they are represented and displayed in a non-recursive manner. +\begin{verbatim} + (d1,d2,d3) : DMP([z,y,x],FRAC INT) + Type: Void +\end{verbatim} +The constructor DMP orders its monomials lexicographically while +HDMP orders them by total order refined by reverse lexicographic +order. +\begin{verbatim} + d1 := -4*z + 4*y**2*x + 16*x**2 + 1 + 2 2 + - 4z + 4y x + 16x + 1 + Type: DistributedMultivariatePolynomial([z,y,x],Fraction Integer) +\end{verbatim} +\begin{verbatim} + d2 := 2*z*y**2 + 4*x + 1 + 2 + 2z y + 4x + 1 + Type: DistributedMultivariatePolynomial([z,y,x],Fraction Integer) +\end{verbatim} +\begin{verbatim} + d3 := 2*z*x**2 - 2*y**2 - x + 2 2 + 2z x - 2y - x + Type: DistributedMultivariatePolynomial([z,y,x],Fraction Integer) +\end{verbatim} +These constructors are mostly used in Groebner basis calculations. +\begin{verbatim} + groebner [d1,d2,d3] + 1568 6 1264 5 6 4 182 3 2047 2 103 2857 + [z - ---- x - ---- x + --- x + --- x - ---- x - ---- x - -----, + 2745 305 305 549 610 2745 10980 + 2 112 6 84 5 1264 4 13 3 84 2 1772 2 + y + ---- x - --- x - ---- x - --- x + --- x + ---- x + ----, + 2745 305 305 549 305 2745 2745 + 7 29 6 17 4 11 3 1 2 15 1 + x + -- x - -- x - -- x + -- x + -- x + -] + 4 16 8 32 16 4 + Type: List DistributedMultivariatePolynomial([z,y,x],Fraction Integer) +\end{verbatim} +\begin{verbatim} + (n1,n2,n3) : HDMP([z,y,x],FRAC INT) + Type: Void +\end{verbatim} +\begin{verbatim} + n1 := d1 + 2 2 + 4y x + 16x - 4z + 1 + Type: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer) +\end{verbatim} +\begin{verbatim} + n2 := d2 + 2 + 2z y + 4x + 1 + Type: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer) +\end{verbatim} +\begin{verbatim} + n3 := d3 + 2 2 + 2z x - 2y - x + Type: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer) +\end{verbatim} +\begin{verbatim} + +Note that we get a different Groebner basis when we use the HDMP +polynomials, as expected. +\begin{verbatim} + groebner [n1,n2,n3] + 4 3 3 2 1 1 4 29 3 1 2 7 9 1 + [y + 2x - - x + - z - -, x + -- x - - y - - z x - -- x - -, + 2 2 8 4 8 4 16 4 + 2 1 2 2 1 2 2 1 + z y + 2x + -, y x + 4x - z + -, z x - y - - x, + 2 4 2 + 2 2 2 1 3 + z - 4y + 2x - - z - - x] + 4 2 + Type: List HomogeneousDistributedMultivariatePolynomial([z,y,x], + Fraction Integer) +\end{verbatim} + +GeneralDistributedMultivariatePolynomial is somewhat more flexible in +the sense that as well as accepting a list of variables to specify the +variable ordering, it also takes a predicate on exponent vectors to +specify the term ordering. With this polynomial type the user can +experiment with the effect of using completely arbitrary term orderings. +This flexibility is mostly important for algorithms such as Groebner +basis calculations which can be very sensitive to term ordering. + +See +\domainref{Polynomial},\\ +\domainref{UnivariatePolynomial},\\ +\domainref{MultivariatePolynomial},\\ +\domainref{DistributedMultivariatePolynomial}, and\\ +\domainref{GeneralDistributedMultivariatePolynomial} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \domainhead{Integer} Axiom provides many operations for manipulating arbitrary precision @@ -41738,6 +42612,42 @@ $$ \returnType{Type: List NonNegativeInteger} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\domainhead{NottinghamGroup} + +If F is a finite field with $p^n$ elements, then we may form the group +of all formal power series $\{t(1+a_1 t+a_2 t+...)\}$ where $u(0)=0$ and +$u^{'}(0)=1$ and $a_i \in F$. +The group operation is substitution (composition). +This is called the Nottingham Group. + +The Nottingham Group is the projective limit of finite p-groups. +Every finite p-group can be embedded in the Nottingham Group. +\begin{verbatim} + x:=monomial(1,1)$UFPS PF 1783 + x +\end{verbatim} +\begin{verbatim} + s:=retract(sin x)$NOTTING PF 1783 + 3 5 7 9 11 + x + 297x + 1679x + 427x + 316x + O(x ) + + s^2 + 3 5 7 9 11 + x + 594x + 535x + 1166x + 1379x + O(x ) + + s^-1 + 3 5 7 9 11 + x + 1486x + 847x + 207x + 1701x + O(x ) + + s^-1*s + 11 + x + O(x ) + + s*s^-1 + 11 + x + O(x ) +\end{verbatim} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \domainhead{Octonion} The Octonions, also called the Cayley-Dixon algebra, defined over a @@ -42964,6 +43874,55 @@ $$ \returnType{Type: List NonNegativeInteger} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\domainhead{Permuation} + + +We represent a permutation as two lists of equal length representing preimages +and images of moved points. Fixed points do not occur in either of these +lists. This enables us to compute the set of fixed points and the set of moved +points easily. +\begin{verbatim} + p := coercePreimagesImages([[1,2,3],[1,2,3]]) + 1 + Type: Permutation PositiveInteger +\end{verbatim} +\begin{verbatim} + movedPoints p + {} + Type: Set PositiveInteger +\end{verbatim} +\begin{verbatim} + even? p + true + Type: Boolean +\end{verbatim} +\begin{verbatim} + p := coercePreimagesImages([[0,1,2,3],[3,0,2,1]])$PERM ZMOD 4 + (1 0 3) + Type: Permutation IntegerMod 4 +\end{verbatim} +\begin{verbatim} + fixedPoints p + {2} + Type: Set IntegerMod 4 +\end{verbatim} +\begin{verbatim} + q := coercePreimagesImages([[0,1,2,3],[1,0]])$PERM ZMOD 4 + (1 0) + Type: Permutation IntegerMod 4 +\end{verbatim} +\begin{verbatim} + fixedPoints(p*q) + {2,0} + Type: Set IntegerMod 4 +\end{verbatim} +\begin{verbatim} + even?(p*q) + false + Type: Boolean +\end{verbatim} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \domainhead{Polynomial} The domain constructor {\tt Polynomial} (abbreviation: {\tt POLY}) @@ -44519,6 +45478,105 @@ $$ \returnType{Type: Boolean} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\domainhead{RealSolvePackage} +\begin{verbatim} +p := 4*x^3 - 3*x^2 + 2*x - 4 + + + 3 2 + (1) 4x - 3x + 2x - 4 + Type: Polynomial Integer +\end{verbatim} +\begin{verbatim} +ans1 := solve(p,0.01)$REALSOLV + + + (2) [1.11328125] + Type: List Float +\end{verbatim} +\begin{verbatim} +ans2 := solve(p::POLY(FRAC(INT)),0.01)$REALSOLV + + + (3) [1.11328125] + Type: List Float +\end{verbatim} +\begin{verbatim} +R := Integer + + + (4) Integer + Type: Domain +\end{verbatim} +\begin{verbatim} +ls : List Symbol := [x,y,z,t] + + + (5) [x,y,z,t] + Type: List Symbol +\end{verbatim} +\begin{verbatim} +ls2 : List Symbol := [x,y,z,t,new()$Symbol] + + + (6) [x,y,z,t,%A] + Type: List Symbol +\end{verbatim} +\begin{verbatim} +pack := ZDSOLVE(R,ls,ls2) + + + (7) ZeroDimensionalSolvePackage(Integer,[x,y,z,t],[x,y,z,t,%A]) + Type: Domain +\end{verbatim} +\begin{verbatim} +p1 := x**2*y*z + y*z + + + 2 + (8) (x + 1)y z + Type: Polynomial Integer +\end{verbatim} +\begin{verbatim} +p2 := x**2*y**2*z + x + z + + + 2 2 + (9) (x y + 1)z + x + Type: Polynomial Integer +\end{verbatim} +\begin{verbatim} +p3 := x**2*y**2*z**2 + z + 1 + + + 2 2 2 + (10) x y z + z + 1 + Type: Polynomial Integer +\end{verbatim} +\begin{verbatim} +lp := [p1, p2, p3] + + + 2 2 2 2 2 2 + (11) [(x + 1)y z,(x y + 1)z + x,x y z + z + 1] + Type: List Polynomial Integer +\end{verbatim} +\begin{verbatim} +lsv:List(Symbol):=[x,y,z] + + + (12) [x,y,z] + Type: List Symbol +\end{verbatim} +\begin{verbatim} +ans3 := realSolve(lp,lsv,0.01)$REALSOLV + + + (13) [ [1.0,0.0,- 1.0] ] + Type: List List Float +\end{verbatim} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \domainhead{RegularTriangularSet} The {\tt RegularTriangularSet} domain constructor implements regular @@ -48300,6 +49358,261 @@ information on related topics, see \domainref{Matrix} and \domainref{OneDimensionalArray}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\domainhead{TwoDimensionalViewport} + +We want to graph $x^3 * (a+b*x)$ on the interval $x=-1..1$ +so we clear out the workspace. + +We assign values to the constants +\begin{verbatim} + a:=0.5 + 0.5 + Type: Float +\end{verbatim} +\begin{verbatim} + b:=0.5 + 0.5 + Type: Float +\end{verbatim} + +We draw the first case of the graph +\begin{verbatim} + y1:=draw(x^3*(a+b*x),x=-1..1,title=="2.2.10 explicit") + TwoDimensionalViewport: "2.2.10 explicit" + Type: TwoDimensionalViewport +\end{verbatim} +which results in the image:\\ +\includegraphics[scale=2.00]{ps/v0plot1.eps} + +We fetch the graph of the first object +\begin{verbatim} + g1:=getGraph(y1,1) + Graph with 1 point list + Type: GraphImage +\end{verbatim} + +We extract its points +\begin{verbatim} + pointLists g1 + [ + [ [-1.,0.,1.,3.], [-0.95833333333333337,-1.8336166570216028E-2,1.,3.], + [-0.91666666666666674,-3.2093942901234518E-2,1.,3.], + [-0.87500000000000011,-4.18701171875E-2,1.,3.], + [-0.83333333333333348,-4.8225308641975301E-2,1.,3.], + [-0.79166666666666685,-5.1683967496141986E-2,1.,3.], + [-0.75000000000000022,-5.2734375E-2,1.,3.], + [-0.70833333333333359,-5.1828643422067916E-2,1.,3.], + [-0.66666666666666696,-4.9382716049382741E-2,1.,3.], + [-0.62500000000000033,-4.5776367187500042E-2,1.,3.], + [-0.5833333333333337,-4.1353202160493867E-2,1.,3.], + [-0.54166666666666707,-3.6420657310956832E-2,1.,3.], + [-0.50000000000000044,-3.1250000000000056E-2,1.,3.], + [-0.45833333333333376,-2.6076328607253136E-2,1.,3.], + [-0.41666666666666707,-2.1098572530864244E-2,1.,3.], + [-0.37500000000000039,-1.6479492187500042E-2,1.,3.], + [-0.3333333333333337,-1.2345679012345713E-2,1.,3.], + [-0.29166666666666702,-8.7875554591049648E-3,1.,3.], + [-0.25000000000000033,-5.8593750000000208E-3,1.,3.], + [-0.20833333333333368,-3.5792221257716214E-3,1.,3.], + [-0.16666666666666702,-1.9290123456790237E-3,1.,3.], + [-0.12500000000000036,-8.5449218750000705E-4,1.,3.], + [-8.3333333333333703E-2,-2.6523919753086765E-4,1.,3.], + [-4.1666666666667039E-2,-3.4661940586420673E-5,1.,3.], + [-3.7470027081099033E-16,-2.6304013894372334E-47,1.,3.], + [4.166666666666629E-2,3.7676022376542178E-5,1.,3.], + [8.3333333333332954E-2,3.1346450617283515E-4,1.,3.], + [0.12499999999999961,1.0986328124999894E-3,1.,3.], + [0.16666666666666627,2.7006172839505972E-3,1.,3.], + [0.20833333333333293,5.463023244598731E-3,1.,3.], + [0.24999999999999958,9.765624999999948E-3,1.,3.], + [0.29166666666666624,1.6024365837191284E-2,1.,3.], + [0.33333333333333293,2.469135802469126E-2,1.,3.], + [0.37499999999999961,3.6254882812499882E-2,1.,3.], + [0.4166666666666663,5.1239390432098617E-2,1.,3.], + [0.45833333333333298,7.0205500096450435E-2,1.,3.], + [0.49999999999999967,9.3749999999999792E-2,1.,3.], + [0.5416666666666663,0.12250584731867258,1.,3.], + [0.58333333333333293,0.15714216820987617,1.,3.], + [0.62499999999999956,0.1983642578124995,1.,3.], + [0.66666666666666619,0.24691358024691298,1.,3.], + [0.70833333333333282,0.30356776861496837,1.,3.], + [0.74999999999999944,0.369140624999999,1.,3.], + [0.79166666666666607,0.44448212046681984,1.,3.], + [0.8333333333333327,0.530478395061727,1.,3.], + [0.87499999999999933,0.62805175781249845,1.,3.], + [0.91666666666666596,0.73816068672839308,1.,3.], + [0.95833333333333259,0.86179982880015205,1.,3.], [1.,1.,1.,3.]] + ] + Type: List List Point DoubleFloat +\end{verbatim} +Now we create a second graph with a changed parameter +\begin{verbatim} + b:=1.0 + 1.0 + Type: Float +\end{verbatim} +We draw it +\begin{verbatim} + y2:=draw(x^3*(a+b*x),x=-1..1) + TwoDimensionalViewport: "AXIOM2D" + Type: TwoDimensionalViewport +\end{verbatim} +which results in the image:\\ +\includegraphics[scale=2.00]{ps/v0plot2.eps} + +We fetch this new graph +\begin{verbatim} + g2:=getGraph(y2,1) + Graph with 1 point list + Type: GraphImage +\end{verbatim} +We get the points from this graph +\begin{verbatim} + pointLists g2 + [ + [ [-1.,0.5,1.,3.], [-0.95833333333333337,0.40339566454475323,1.,3.], + [-0.91666666666666674,0.32093942901234584,1.,3.], + [-0.87500000000000011,0.25122070312500017,1.,3.], + [-0.83333333333333348,0.19290123456790137,1.,3.], + [-0.79166666666666685,0.14471510898919768,1.,3.], + [-0.75000000000000022,0.10546875000000019,1.,3.], + [-0.70833333333333359,7.404091917438288E-2,1.,3.], + [-0.66666666666666696,4.938271604938288E-2,1.,3.], + [-0.62500000000000033,3.0517578125000125E-2,1.,3.], + [-0.5833333333333337,1.6541280864197649E-2,1.,3.], + [-0.54166666666666707,6.6219376929013279E-3,1.,3.], + [-0.50000000000000044,5.5511151231257827E-17,1.,3.], + [-0.45833333333333376,-4.011742862654287E-3,1.,3.], + [-0.41666666666666707,-6.0281635802469057E-3,1.,3.], + [-0.37500000000000039,-6.5917968750000035E-3,1.,3.], + [-0.3333333333333337,-6.1728395061728461E-3,1.,3.], + [-0.29166666666666702,-5.1691502700617377E-3,1.,3.], + [-0.25000000000000033,-3.9062500000000104E-3,1.,3.], + [-0.20833333333333368,-2.6373215663580349E-3,1.,3.], + [-0.16666666666666702,-1.543209876543218E-3,1.,3.], + [-0.12500000000000036,-7.3242187500000564E-4,1.,3.], + [-8.3333333333333703E-2,-2.4112654320987957E-4,1.,3.], + [-4.1666666666667039E-2,-3.315489969135889E-5,1.,3.], + [-3.7470027081099033E-16,-2.6304013894372324E-47,1.,3.], + [4.166666666666629E-2,3.9183063271603852E-5,1.,3.], + [8.3333333333332954E-2,3.3757716049382237E-4,1.,3.], + [0.12499999999999961,1.2207031249999879E-3,1.,3.], + [0.16666666666666627,3.0864197530863957E-3,1.,3.], + [0.20833333333333293,6.4049238040123045E-3,1.,3.], + [0.24999999999999958,1.1718749999999934E-2,1.,3.], + [0.29166666666666624,1.9642771026234473E-2,1.,3.], + [0.33333333333333293,3.0864197530864071E-2,1.,3.], + [0.37499999999999961,4.6142578124999847E-2,1.,3.], + [0.4166666666666663,6.6309799382715848E-2,1.,3.], + [0.45833333333333298,9.2270085841049135E-2,1.,3.], + [0.49999999999999967,0.12499999999999971,1.,3.], + [0.5416666666666663,0.16554844232253049,1.,3.], + [0.58333333333333293,0.21503665123456736,1.,3.], + [0.62499999999999956,0.27465820312499928,1.,3.], + [0.66666666666666619,0.3456790123456781,1.,3.], + [0.70833333333333282,0.42943733121141858,1.,3.], + [0.74999999999999944,0.52734374999999845,1.,3.], + [0.79166666666666607,0.64088119695215873,1.,3.], + [0.8333333333333327,0.77160493827160281,1.,3.], + [0.87499999999999933,0.92114257812499756,1.,3.], + [0.91666666666666596,1.0911940586419722,1.,3.], + [0.95833333333333259,1.2835316599151199,1.,3.], [1.,1.5,1.,3.]] + ] + Type: List List Point DoubleFloat +\end{verbatim} +and we put these points, g2 onto the first graph y1 as graph 2 +\begin{verbatim} + putGraph(y1,g2,2) + Type: Void +\end{verbatim} +And now we do the whole sequence again +\begin{verbatim} + b:=2.0 + 2.0 + Type: Float +\end{verbatim} +\begin{verbatim} + y3:=draw(x^3*(a+b*x),x=-1..1) + TwoDimensionalViewport: "AXIOM2D" + Type: TwoDimensionalViewport +\end{verbatim} +which results in the image:\\ +\includegraphics[scale=2.00]{ps/v0plot3.eps} + +\begin{verbatim} + g3:=getGraph(y3,1) + Graph with 1 point list + Type: GraphImage +\end{verbatim} +\begin{verbatim} + pointLists g3 + [ + [ [-1.,1.5,1.,3.], [-0.95833333333333337,1.2468593267746917,1.,3.], + [-0.91666666666666674,1.0270061728395066,1.,3.], + [-0.87500000000000011,0.83740234375000044,1.,3.], + [-0.83333333333333348,0.67515432098765471,1.,3.], + [-0.79166666666666685,0.53751326195987703,1.,3.], + [-0.75000000000000022,0.42187500000000056,1.,3.], + [-0.70833333333333359,0.32578004436728447,1.,3.], + [-0.66666666666666696,0.24691358024691412,1.,3.], + [-0.62500000000000033,0.18310546875000044,1.,3.], + [-0.5833333333333337,0.1323302469135807,1.,3.], + [-0.54166666666666707,9.2707127700617648E-2,1.,3.], + [-0.50000000000000044,6.2500000000000278E-2,1.,3.], + [-0.45833333333333376,4.0117428626543411E-2,1.,3.], + [-0.41666666666666707,2.4112654320987775E-2,1.,3.], + [-0.37500000000000039,1.3183593750000073E-2,1.,3.], + [-0.3333333333333337,6.1728395061728877E-3,1.,3.], + [-0.29166666666666702,2.0676601080247183E-3,1.,3.], + [-0.25000000000000033,1.0408340855860843E-17,1.,3.], + [-0.20833333333333368,-7.5352044753086191E-4,1.,3.], + [-0.16666666666666702,-7.7160493827160663E-4,1.,3.], + [-0.12500000000000036,-4.8828125000000282E-4,1.,3.], + [-8.3333333333333703E-2,-1.9290123456790339E-4,1.,3.], + [-4.1666666666667039E-2,-3.0140817901235325E-5,1.,3.], + [-3.7470027081099033E-16,-2.6304013894372305E-47,1.,3.], + [4.166666666666629E-2,4.21971450617272E-5,1.,3.], + [8.3333333333332954E-2,3.8580246913579681E-4,1.,3.], + [0.12499999999999961,1.4648437499999848E-3,1.,3.], + [0.16666666666666627,3.8580246913579933E-3,1.,3.], + [0.20833333333333293,8.2887249228394497E-3,1.,3.], + [0.24999999999999958,1.562499999999991E-2,1.,3.], + [0.29166666666666624,2.6879581404320851E-2,1.,3.], + [0.33333333333333293,4.3209876543209694E-2,1.,3.], + [0.37499999999999961,6.5917968749999764E-2,1.,3.], + [0.4166666666666663,9.6450617283950296E-2,1.,3.], + [0.45833333333333298,0.13639925733024652,1.,3.], + [0.49999999999999967,0.18749999999999956,1.,3.], + [0.5416666666666663,0.25163363233024633,1.,3.], + [0.58333333333333293,0.33082561728394977,1.,3.], + [0.62499999999999956,0.42724609374999883,1.,3.], + [0.66666666666666619,0.5432098765432084,1.,3.], + [0.70833333333333282,0.68117645640431912,1.,3.], + [0.74999999999999944,0.84374999999999756,1.,3.], + [0.79166666666666607,1.0336793499228365,1.,3.], + [0.8333333333333327,1.2538580246913544,1.,3.], + [0.87499999999999933,1.507324218749996,1.,3.], + [0.91666666666666596,1.7972608024691306,1.,3.], + [0.95833333333333259,2.1269953221450555,1.,3.], [1.,2.5,1.,3.]] + ] + Type: List List Point DoubleFloat +\end{verbatim} +and put the third graphs points g3 onto the first graph y1 as graph 3 +\begin{verbatim} + putGraph(y1,g3,3) + Type: Void +\end{verbatim} +Finally we show the combined result +\begin{verbatim} + vp:=makeViewport2D(y1) + TwoDimensionalViewport: "2.2.10 explicit" + Type: TwoDimensionalViewport +\end{verbatim} +which results in the image:\\ +\includegraphics[scale=2.00]{ps/v0plot4.eps}\\ +which shows all of the graphs in a single image. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \domainhead{UnivariatePolynomial} The domain constructor {\tt UnivariatePolynomial} (abbreviated {\tt diff --git a/books/bookvol10.3.pamphlet b/books/bookvol10.3.pamphlet index 2a4d9e4..2299641 100644 --- a/books/bookvol10.3.pamphlet +++ b/books/bookvol10.3.pamphlet @@ -2196,7 +2196,7 @@ the original stack since map! is destructive: map!(x+->x+10,a) [18,19,13,14,15] -o + Notice that the orignal stack has been changed: a @@ -63017,6 +63017,17 @@ sample()$NOTTING(PF(1783)) ==================================================================== NottinghamGroup examples ==================================================================== + n +If F is a finite field with p elements, then we may form the group +of all formal power series {t(1+a t+a t+...)} where u(0)=0 and + , 1 2 +u (0)=1 and $a is an element of F. + i +The group operation is substitution (composition). +This is called the Nottingham Group. + +The Nottingham Group is the projective limit of finite p-groups. +Every finite p-group can be embedded in the Nottingham Group. x:=monomial(1,1)$UFPS PF 1783 x @@ -63049,14 +63060,6 @@ o )show UnivariateFormalPowerSeries \pagehead{NottinghamGroup}{NOTTING} \pagepic{ps/v103nottinghamgroup.ps}{NOTTING}{1.00} -If F is a finite field with $p^n$ elements, then we may form the group -of all formal power series $\{t(1+a_1t+a_2t+\ldots)\}$ where $u(0)=0$ and -$u^{'}(0)=1$ and $a_i \in F$. -The group operation is substitution (composition). -This is called the Nottingham Group. - -The Nottingham Group is the projective limit of finite p-groups. -Every finite p-group can be embedded in the Nottingham Group. {\bf Exports:}\\ \begin{tabular}{lllll} \cross{NOTTING}{1} & @@ -70096,6 +70099,124 @@ PendantTree(S: SetCategory): T == C where @ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{domain PERM Permutation} +<>= +)set break resume +)sys rm -f Permutation.output +)spool Permutation.output +)set message test on +)set message auto off +)clear all +--S 1 of 8 +p := coercePreimagesImages([[1,2,3],[1,2,3]]) +--R +--R +--R (1) 1 +--R Type: Permutation PositiveInteger +--E 1 + +--S 2 of 8 +movedPoints p -- should return {} +--R +--R +--R (2) {} +--R Type: Set PositiveInteger +--E 2 + +--S 3 of 8 +even? p -- should return true +--R +--R +--R (3) true +--R Type: Boolean +--E 3 + +--S 4 of 8 +p := coercePreimagesImages([[0,1,2,3],[3,0,2,1]])$PERM ZMOD 4 +--R +--R +--R (4) (1 0 3) +--R Type: Permutation IntegerMod 4 +--E 4 + +--S 5 of 8 +fixedPoints p -- should return {2} +--R +--R +--R (5) {2} +--R Type: Set IntegerMod 4 +--E 5 + +--S 6 of 8 +q := coercePreimagesImages([[0,1,2,3],[1,0]])$PERM ZMOD 4 +--R +--R +--R (6) (1 0) +--R Type: Permutation IntegerMod 4 +--E 6 + +--S 7 of 8 +fixedPoints(p*q) -- should return {2,0} +--R +--R +--R (7) {2,0} +--R Type: Set IntegerMod 4 +--E 7 + +--S 8 of 8 +even?(p*q) -- should return false +--R +--R +--R (8) false +--R Type: Boolean +--E 8 +)spool +)lisp (bye) +@ +<>= +==================================================================== +Permutation Examples +==================================================================== + +We represent a permutation as two lists of equal length representing preimages +and images of moved points. I.e., fixed points do not occur in either of these +lists. This enables us to compute the set of fixed points and the set of moved +points easily. + + p := coercePreimagesImages([[1,2,3],[1,2,3]]) + 1 + Type: Permutation PositiveInteger + + movedPoints p + {} + Type: Set PositiveInteger + + even? p + true + Type: Boolean + + p := coercePreimagesImages([[0,1,2,3],[3,0,2,1]])$PERM ZMOD 4 + (1 0 3) + Type: Permutation IntegerMod 4 + + fixedPoints p + {2} + Type: Set IntegerMod 4 + + q := coercePreimagesImages([[0,1,2,3],[1,0]])$PERM ZMOD 4 + (1 0) + Type: Permutation IntegerMod 4 + + fixedPoints(p*q) + {2,0} + Type: Set IntegerMod 4 + even?(p*q) + false + Type: Boolean + +See Also: +o )show Permutation + +@ \pagehead{Permutation}{PERM} \pagepic{ps/v103permutation.ps}{PERM}{1.00} @@ -70191,6 +70312,9 @@ Permutation(S:SetCategory): public == private where ++ coercePreimagesImages(lls) coerces the representation {\em lls} ++ of a permutation as a list of preimages and images to a permutation. ++ We assume that both preimage and image do not contain repetitions. + ++ + ++X p := coercePreimagesImages([[1,2,3],[1,2,3]]) + ++X q := coercePreimagesImages([[0,1,2,3],[3,0,2,1]])$PERM ZMOD 4 coerce : List List S -> % ++ coerce(lls) coerces a list of cycles {\em lls} to a ++ permutation, each cycle being a list with no @@ -70216,6 +70340,9 @@ Permutation(S:SetCategory): public == private where ++ permutation p. movedPoints : % -> Set S ++ movedPoints(p) returns the set of points moved by the permutation p. + ++ + ++X p := coercePreimagesImages([[1,2,3],[1,2,3]]) + ++X movedPoints p cyclePartition : % -> Partition ++ cyclePartition(p) returns the cycle structure of a permutation ++ p including cycles of length 1 only if S is finite. @@ -70229,6 +70356,9 @@ Permutation(S:SetCategory): public == private where even? : % -> Boolean ++ even?(p) returns true if and only if p is an even permutation, ++ i.e. {\em sign(p)} is 1. + ++ + ++X p := coercePreimagesImages([[1,2,3],[1,2,3]]) + ++X even? p odd? : % -> Boolean ++ odd?(p) returns true if and only if p is an odd permutation ++ i.e. {\em sign(p)} is {\em -1}. @@ -70241,6 +70371,8 @@ Permutation(S:SetCategory): public == private where if S has Finite then fixedPoints : % -> Set S ++ fixedPoints(p) returns the points fixed by the permutation p. + ++X p := coercePreimagesImages([[0,1,2,3],[3,0,2,1]])$PERM ZMOD 4 + ++X fixedPoints p if S has IntegerNumberSystem or S has Finite then coerceImages : L S -> % ++ coerceImages(ls) coerces the list {\em ls} to a permutation @@ -70254,130 +70386,7 @@ Permutation(S:SetCategory): public == private where -- representation of the object: Rep := V L S -@ - -We represent a permutation as two lists of equal length representing preimages -and images of moved points. I.e., fixed points do not occur in either of these -lists. This enables us to compute the set of fixed points and the set of moved -points easily. - -Note that this was not respected in versions before [[patch--50]] of this -domain. - -<>= -)set break resume -)sys rm -f Permutation.output -)spool Permutation.output -)set message test on -)set message auto off -)clear all ---S 1 of 8 -p := coercePreimagesImages([[1,2,3],[1,2,3]]) ---R ---R ---R (1) 1 ---R Type: Permutation PositiveInteger ---E 1 - ---S 2 of 8 -movedPoints p -- should return {} ---R ---R ---R (2) {} ---R Type: Set PositiveInteger ---E 2 - ---S 3 of 8 -even? p -- should return true ---R ---R ---R (3) true ---R Type: Boolean ---E 3 - ---S 4 of 8 -p := coercePreimagesImages([[0,1,2,3],[3,0,2,1]])$PERM ZMOD 4 ---R ---R ---R (4) (1 0 3) ---R Type: Permutation IntegerMod 4 ---E 4 - ---S 5 of 8 -fixedPoints p -- should return {2} ---R ---R ---R (5) {2} ---R Type: Set IntegerMod 4 ---E 5 - ---S 6 of 8 -q := coercePreimagesImages([[0,1,2,3],[1,0]])$PERM ZMOD 4 ---R ---R ---R (6) (1 0) ---R Type: Permutation IntegerMod 4 ---E 6 - ---S 7 of 8 -fixedPoints(p*q) -- should return {2,0} ---R ---R ---R (7) {2,0} ---R Type: Set IntegerMod 4 ---E 7 - ---S 8 of 8 -even?(p*q) -- should return false ---R ---R ---R (8) false ---R Type: Boolean ---E 8 -)spool -)lisp (bye) -@ -<>= -==================================================================== -Permutation Examples -==================================================================== - - p := coercePreimagesImages([[1,2,3],[1,2,3]]) - 1 - Type: Permutation PositiveInteger - - movedPoints p - {} - Type: Set PositiveInteger - - even? p - true - Type: Boolean - - p := coercePreimagesImages([[0,1,2,3],[3,0,2,1]])$PERM ZMOD 4 - (1 0 3) - Type: Permutation IntegerMod 4 - - fixedPoints p - {2} - Type: Set IntegerMod 4 - - q := coercePreimagesImages([[0,1,2,3],[1,0]])$PERM ZMOD 4 - (1 0) - Type: Permutation IntegerMod 4 - - fixedPoints(p*q) - {2,0} - Type: Set IntegerMod 4 - even?(p*q) - false - Type: Boolean - -See Also: -o )show Permutation -@ -<>= -- import of domains and packages import OutputForm @@ -70491,26 +70500,6 @@ o )show Permutation image := cons(pi, image) [preImage, image] -@ - -This operation transforms a pair of preimages and images into an element of the -domain. Since we assume that fixed points do not occur in the representation, -we have to sort them out here. - -Note that before [[patch--50]] this read -\begin{verbatim} - coercePreimagesImages preImageAndImage == - p : % := [preImageAndImage.1,preImageAndImage.2] -\end{verbatim} -causing bugs when computing [[movedPoints]], [[fixedPoints]], [[even?]], -[[odd?]], etc., as reported in Issue~\#295. - -The other coercion facilities check for fixed points. It also seems that [[*]] -removes fixed points from its result. - -Up to [[patch--50]] we did not check for duplicates. - -<>= movedPoints p == construct p.1 @@ -70546,7 +70535,6 @@ Up to [[patch--50]] we did not check for duplicates. even? p => 1 - 1 - even?(p) == even?(#(p.1) - numberOfCycles p) -- see the book of James and Kerber on symmetric groups -- for this formula. @@ -71613,8 +71601,10 @@ Pi(): Exports == Implementation where )set message test on )set message auto off )clear all ---S 1 of 1 -makeSketch(x+y,x,y,-1/2..1/2,-1/2..1/2)$ACPLOT + +--S 1 of 5 +sketch:=makeSketch(x+y,x,y,-1/2..1/2,-1/2..1/2)$ACPLOT +--R --R (1) ACPLOT --R 1 1 1 1 --R y + x = 0, - - <= x <= -, - - <= y <= - @@ -71623,6 +71613,548 @@ makeSketch(x+y,x,y,-1/2..1/2,-1/2..1/2)$ACPLOT --R [- 0.5,0.5] --R Type: PlaneAlgebraicCurvePlot --E 1 + +--S 2 of 5 +refined:=refine(sketch,0.1) +--R +--R (2) ACPLOT +--R 1 1 1 1 +--R y + x = 0, - - <= x <= -, - - <= y <= - +--R 2 2 2 2 +--R [0.5,- 0.5] +--R [0.49600000000000083,- 0.49600000000000083] +--R [0.49200000000000083,- 0.49200000000000083] +--R [0.48800000000000082,- 0.48800000000000082] +--R [0.48400000000000082,- 0.48400000000000082] +--R [0.48000000000000081,- 0.48000000000000081] +--R [0.47600000000000081,- 0.47600000000000081] +--R [0.47200000000000081,- 0.47200000000000081] +--R [0.4680000000000008,- 0.4680000000000008] +--R [0.4640000000000008,- 0.4640000000000008] +--R [0.4600000000000008,- 0.4600000000000008] +--R [0.45600000000000079,- 0.45600000000000079] +--R [0.45200000000000079,- 0.45200000000000079] +--R [0.44800000000000079,- 0.44800000000000079] +--R [0.44400000000000078,- 0.44400000000000078] +--R [0.44000000000000078,- 0.44000000000000078] +--R [0.43600000000000078,- 0.43600000000000078] +--R [0.43200000000000077,- 0.43200000000000077] +--R [0.42800000000000077,- 0.42800000000000077] +--R [0.42400000000000077,- 0.42400000000000077] +--R [0.42000000000000076,- 0.42000000000000076] +--R [0.41600000000000076,- 0.41600000000000076] +--R [0.41200000000000075,- 0.41200000000000075] +--R [0.40800000000000075,- 0.40800000000000075] +--R [0.40400000000000075,- 0.40400000000000075] +--R [0.40000000000000074,- 0.40000000000000074] +--R [0.39600000000000074,- 0.39600000000000074] +--R [0.39200000000000074,- 0.39200000000000074] +--R [0.38800000000000073,- 0.38800000000000073] +--R [0.38400000000000073,- 0.38400000000000073] +--R [0.38000000000000073,- 0.38000000000000073] +--R [0.37600000000000072,- 0.37600000000000072] +--R [0.37200000000000072,- 0.37200000000000072] +--R [0.36800000000000072,- 0.36800000000000072] +--R [0.36400000000000071,- 0.36400000000000071] +--R [0.36000000000000071,- 0.36000000000000071] +--R [0.3560000000000007,- 0.3560000000000007] +--R [0.3520000000000007,- 0.3520000000000007] +--R [0.3480000000000007,- 0.3480000000000007] +--R [0.34400000000000069,- 0.34400000000000069] +--R [0.34000000000000069,- 0.34000000000000069] +--R [0.33600000000000069,- 0.33600000000000069] +--R [0.33200000000000068,- 0.33200000000000068] +--R [0.32800000000000068,- 0.32800000000000068] +--R [0.32400000000000068,- 0.32400000000000068] +--R [0.32000000000000067,- 0.32000000000000067] +--R [0.31600000000000067,- 0.31600000000000067] +--R [0.31200000000000067,- 0.31200000000000067] +--R [0.30800000000000066,- 0.30800000000000066] +--R [0.30400000000000066,- 0.30400000000000066] +--R [0.30000000000000066,- 0.30000000000000066] +--R [0.29600000000000065,- 0.29600000000000065] +--R [0.29200000000000065,- 0.29200000000000065] +--R [0.28800000000000064,- 0.28800000000000064] +--R [0.28400000000000064,- 0.28400000000000064] +--R [0.28000000000000064,- 0.28000000000000064] +--R [0.27600000000000063,- 0.27600000000000063] +--R [0.27200000000000063,- 0.27200000000000063] +--R [0.26800000000000063,- 0.26800000000000063] +--R [0.26400000000000062,- 0.26400000000000062] +--R [0.26000000000000062,- 0.26000000000000062] +--R [0.25600000000000062,- 0.25600000000000062] +--R [0.25200000000000061,- 0.25200000000000061] +--R [0.24800000000000061,- 0.24800000000000061] +--R [0.24400000000000061,- 0.24400000000000061] +--R [0.2400000000000006,- 0.2400000000000006] +--R [0.2360000000000006,- 0.2360000000000006] +--R [0.23200000000000059,- 0.23200000000000059] +--R [0.22800000000000059,- 0.22800000000000059] +--R [0.22400000000000059,- 0.22400000000000059] +--R [0.22000000000000058,- 0.22000000000000058] +--R [0.21600000000000058,- 0.21600000000000058] +--R [0.21200000000000058,- 0.21200000000000058] +--R [0.20800000000000057,- 0.20800000000000057] +--R [0.20400000000000057,- 0.20400000000000057] +--R [0.20000000000000057,- 0.20000000000000057] +--R [0.19600000000000056,- 0.19600000000000056] +--R [0.19200000000000056,- 0.19200000000000056] +--R [0.18800000000000056,- 0.18800000000000056] +--R [0.18400000000000055,- 0.18400000000000055] +--R [0.18000000000000055,- 0.18000000000000055] +--R [0.17600000000000054,- 0.17600000000000054] +--R [0.17200000000000054,- 0.17200000000000054] +--R [0.16800000000000054,- 0.16800000000000054] +--R [0.16400000000000053,- 0.16400000000000053] +--R [0.16000000000000053,- 0.16000000000000053] +--R [0.15600000000000053,- 0.15600000000000053] +--R [0.15200000000000052,- 0.15200000000000052] +--R [0.14800000000000052,- 0.14800000000000052] +--R [0.14400000000000052,- 0.14400000000000052] +--R [0.14000000000000051,- 0.14000000000000051] +--R [0.13600000000000051,- 0.13600000000000051] +--R [0.13200000000000051,- 0.13200000000000051] +--R [0.1280000000000005,- 0.1280000000000005] +--R [0.1240000000000005,- 0.1240000000000005] +--R [0.1200000000000005,- 0.1200000000000005] +--R [0.11600000000000049,- 0.11600000000000049] +--R [0.11200000000000049,- 0.11200000000000049] +--R [0.10800000000000048,- 0.10800000000000048] +--R [0.10400000000000048,- 0.10400000000000048] +--R [0.10000000000000048,- 0.10000000000000048] +--R [9.6000000000000474E-2,- 9.6000000000000474E-2] +--R [9.200000000000047E-2,- 9.200000000000047E-2] +--R [8.8000000000000467E-2,- 8.8000000000000467E-2] +--R [8.4000000000000463E-2,- 8.4000000000000463E-2] +--R [8.000000000000046E-2,- 8.000000000000046E-2] +--R [7.6000000000000456E-2,- 7.6000000000000456E-2] +--R [7.2000000000000453E-2,- 7.2000000000000453E-2] +--R [6.8000000000000449E-2,- 6.8000000000000449E-2] +--R [6.4000000000000445E-2,- 6.4000000000000445E-2] +--R [6.0000000000000442E-2,- 6.0000000000000442E-2] +--R [5.6000000000000438E-2,- 5.6000000000000438E-2] +--R [5.2000000000000435E-2,- 5.2000000000000435E-2] +--R [4.8000000000000431E-2,- 4.8000000000000431E-2] +--R [4.4000000000000428E-2,- 4.4000000000000428E-2] +--R [4.0000000000000424E-2,- 4.0000000000000424E-2] +--R [3.6000000000000421E-2,- 3.6000000000000421E-2] +--R [3.2000000000000417E-2,- 3.2000000000000417E-2] +--R [2.8000000000000417E-2,- 2.8000000000000417E-2] +--R [2.4000000000000417E-2,- 2.4000000000000417E-2] +--R [2.0000000000000417E-2,- 2.0000000000000417E-2] +--R [1.6000000000000417E-2,- 1.6000000000000417E-2] +--R [1.2000000000000417E-2,- 1.2000000000000417E-2] +--R [8.0000000000004165E-3,- 8.0000000000004165E-3] +--R [4.0000000000004164E-3,- 4.0000000000004164E-3] +--R [4.163336342344337E-16,- 4.163336342344337E-16] +--R [- 3.9999999999995837E-3,3.9999999999995837E-3] +--R [- 7.9999999999995838E-3,7.9999999999995838E-3] +--R [- 1.1999999999999584E-2,1.1999999999999584E-2] +--R [- 1.5999999999999584E-2,1.5999999999999584E-2] +--R [- 1.9999999999999584E-2,1.9999999999999584E-2] +--R [- 2.3999999999999584E-2,2.3999999999999584E-2] +--R [- 2.7999999999999584E-2,2.7999999999999584E-2] +--R [- 3.1999999999999584E-2,3.1999999999999584E-2] +--R [- 3.5999999999999588E-2,3.5999999999999588E-2] +--R [- 3.9999999999999591E-2,3.9999999999999591E-2] +--R [- 4.3999999999999595E-2,4.3999999999999595E-2] +--R [- 4.7999999999999599E-2,4.7999999999999599E-2] +--R [- 5.1999999999999602E-2,5.1999999999999602E-2] +--R [- 5.5999999999999606E-2,5.5999999999999606E-2] +--R [- 5.9999999999999609E-2,5.9999999999999609E-2] +--R [- 6.3999999999999613E-2,6.3999999999999613E-2] +--R [- 6.7999999999999616E-2,6.7999999999999616E-2] +--R [- 7.199999999999962E-2,7.199999999999962E-2] +--R [- 7.5999999999999623E-2,7.5999999999999623E-2] +--R [- 7.9999999999999627E-2,7.9999999999999627E-2] +--R [- 8.3999999999999631E-2,8.3999999999999631E-2] +--R [- 8.7999999999999634E-2,8.7999999999999634E-2] +--R [- 9.1999999999999638E-2,9.1999999999999638E-2] +--R [- 9.5999999999999641E-2,9.5999999999999641E-2] +--R [- 9.9999999999999645E-2,9.9999999999999645E-2] +--R [- 0.10399999999999965,0.10399999999999965] +--R [- 0.10799999999999965,0.10799999999999965] +--R [- 0.11199999999999966,0.11199999999999966] +--R [- 0.11599999999999966,0.11599999999999966] +--R [- 0.11999999999999966,0.11999999999999966] +--R [- 0.12399999999999967,0.12399999999999967] +--R [- 0.12799999999999967,0.12799999999999967] +--R [- 0.13199999999999967,0.13199999999999967] +--R [- 0.13599999999999968,0.13599999999999968] +--R [- 0.13999999999999968,0.13999999999999968] +--R [- 0.14399999999999968,0.14399999999999968] +--R [- 0.14799999999999969,0.14799999999999969] +--R [- 0.15199999999999969,0.15199999999999969] +--R [- 0.15599999999999969,0.15599999999999969] +--R [- 0.1599999999999997,0.1599999999999997] +--R [- 0.1639999999999997,0.1639999999999997] +--R [- 0.16799999999999971,0.16799999999999971] +--R [- 0.17199999999999971,0.17199999999999971] +--R [- 0.17599999999999971,0.17599999999999971] +--R [- 0.17999999999999972,0.17999999999999972] +--R [- 0.18399999999999972,0.18399999999999972] +--R [- 0.18799999999999972,0.18799999999999972] +--R [- 0.19199999999999973,0.19199999999999973] +--R [- 0.19599999999999973,0.19599999999999973] +--R [- 0.19999999999999973,0.19999999999999973] +--R [- 0.20399999999999974,0.20399999999999974] +--R [- 0.20799999999999974,0.20799999999999974] +--R [- 0.21199999999999974,0.21199999999999974] +--R [- 0.21599999999999975,0.21599999999999975] +--R [- 0.21999999999999975,0.21999999999999975] +--R [- 0.22399999999999975,0.22399999999999975] +--R [- 0.22799999999999976,0.22799999999999976] +--R [- 0.23199999999999976,0.23199999999999976] +--R [- 0.23599999999999977,0.23599999999999977] +--R [- 0.23999999999999977,0.23999999999999977] +--R [- 0.24399999999999977,0.24399999999999977] +--R [- 0.24799999999999978,0.24799999999999978] +--R [- 0.25199999999999978,0.25199999999999978] +--R [- 0.25599999999999978,0.25599999999999978] +--R [- 0.25999999999999979,0.25999999999999979] +--R [- 0.26399999999999979,0.26399999999999979] +--R [- 0.26799999999999979,0.26799999999999979] +--R [- 0.2719999999999998,0.2719999999999998] +--R [- 0.2759999999999998,0.2759999999999998] +--R [- 0.2799999999999998,0.2799999999999998] +--R [- 0.28399999999999981,0.28399999999999981] +--R [- 0.28799999999999981,0.28799999999999981] +--R [- 0.29199999999999982,0.29199999999999982] +--R [- 0.29599999999999982,0.29599999999999982] +--R [- 0.29999999999999982,0.29999999999999982] +--R [- 0.30399999999999983,0.30399999999999983] +--R [- 0.30799999999999983,0.30799999999999983] +--R [- 0.31199999999999983,0.31199999999999983] +--R [- 0.31599999999999984,0.31599999999999984] +--R [- 0.31999999999999984,0.31999999999999984] +--R [- 0.32399999999999984,0.32399999999999984] +--R [- 0.32799999999999985,0.32799999999999985] +--R [- 0.33199999999999985,0.33199999999999985] +--R [- 0.33599999999999985,0.33599999999999985] +--R [- 0.33999999999999986,0.33999999999999986] +--R [- 0.34399999999999986,0.34399999999999986] +--R [- 0.34799999999999986,0.34799999999999986] +--R [- 0.35199999999999987,0.35199999999999987] +--R [- 0.35599999999999987,0.35599999999999987] +--R [- 0.35999999999999988,0.35999999999999988] +--R [- 0.36399999999999988,0.36399999999999988] +--R [- 0.36799999999999988,0.36799999999999988] +--R [- 0.37199999999999989,0.37199999999999989] +--R [- 0.37599999999999989,0.37599999999999989] +--R [- 0.37999999999999989,0.37999999999999989] +--R [- 0.3839999999999999,0.3839999999999999] +--R [- 0.3879999999999999,0.3879999999999999] +--R [- 0.3919999999999999,0.3919999999999999] +--R [- 0.39599999999999991,0.39599999999999991] +--R [- 0.39999999999999991,0.39999999999999991] +--R [- 0.40399999999999991,0.40399999999999991] +--R [- 0.40799999999999992,0.40799999999999992] +--R [- 0.41199999999999992,0.41199999999999992] +--R [- 0.41599999999999993,0.41599999999999993] +--R [- 0.41999999999999993,0.41999999999999993] +--R [- 0.42399999999999993,0.42399999999999993] +--R [- 0.42799999999999994,0.42799999999999994] +--R [- 0.43199999999999994,0.43199999999999994] +--R [- 0.43599999999999994,0.43599999999999994] +--R [- 0.43999999999999995,0.43999999999999995] +--R [- 0.44399999999999995,0.44399999999999995] +--R [- 0.44799999999999995,0.44799999999999995] +--R [- 0.45199999999999996,0.45199999999999996] +--R [- 0.45599999999999996,0.45599999999999996] +--R [- 0.45999999999999996,0.45999999999999996] +--R [- 0.46399999999999997,0.46399999999999997] +--R [- 0.46799999999999997,0.46799999999999997] +--R [- 0.47199999999999998,0.47199999999999998] +--R [- 0.47599999999999998,0.47599999999999998] +--R [- 0.47999999999999998,0.47999999999999998] +--R [- 0.48399999999999999,0.48399999999999999] +--R [- 0.48799999999999999,0.48799999999999999] +--R [- 0.49199999999999999,0.49199999999999999] +--R [- 0.496,0.496] +--R [- 0.5,0.5] +--R Type: PlaneAlgebraicCurvePlot +--E 2 + +--S 3 of 5 +listBranches(sketch) +--R +--R (3) [[[0.5,- 0.5],[- 0.5,0.5]]] +--R Type: List List Point DoubleFloat +--E 3 + +--S 4 of 5 +listBranches(refined) +--R +--R (4) +--R [ +--R [[0.5,- 0.5], [0.49600000000000083,- 0.49600000000000083], +--R [0.49200000000000083,- 0.49200000000000083], +--R [0.48800000000000082,- 0.48800000000000082], +--R [0.48400000000000082,- 0.48400000000000082], +--R [0.48000000000000081,- 0.48000000000000081], +--R [0.47600000000000081,- 0.47600000000000081], +--R [0.47200000000000081,- 0.47200000000000081], +--R [0.4680000000000008,- 0.4680000000000008], +--R [0.4640000000000008,- 0.4640000000000008], +--R [0.4600000000000008,- 0.4600000000000008], +--R [0.45600000000000079,- 0.45600000000000079], +--R [0.45200000000000079,- 0.45200000000000079], +--R [0.44800000000000079,- 0.44800000000000079], +--R [0.44400000000000078,- 0.44400000000000078], +--R [0.44000000000000078,- 0.44000000000000078], +--R [0.43600000000000078,- 0.43600000000000078], +--R [0.43200000000000077,- 0.43200000000000077], +--R [0.42800000000000077,- 0.42800000000000077], +--R [0.42400000000000077,- 0.42400000000000077], +--R [0.42000000000000076,- 0.42000000000000076], +--R [0.41600000000000076,- 0.41600000000000076], +--R [0.41200000000000075,- 0.41200000000000075], +--R [0.40800000000000075,- 0.40800000000000075], +--R [0.40400000000000075,- 0.40400000000000075], +--R [0.40000000000000074,- 0.40000000000000074], +--R [0.39600000000000074,- 0.39600000000000074], +--R [0.39200000000000074,- 0.39200000000000074], +--R [0.38800000000000073,- 0.38800000000000073], +--R [0.38400000000000073,- 0.38400000000000073], +--R [0.38000000000000073,- 0.38000000000000073], +--R [0.37600000000000072,- 0.37600000000000072], +--R [0.37200000000000072,- 0.37200000000000072], +--R [0.36800000000000072,- 0.36800000000000072], +--R [0.36400000000000071,- 0.36400000000000071], +--R [0.36000000000000071,- 0.36000000000000071], +--R [0.3560000000000007,- 0.3560000000000007], +--R [0.3520000000000007,- 0.3520000000000007], +--R [0.3480000000000007,- 0.3480000000000007], +--R [0.34400000000000069,- 0.34400000000000069], +--R [0.34000000000000069,- 0.34000000000000069], +--R [0.33600000000000069,- 0.33600000000000069], +--R [0.33200000000000068,- 0.33200000000000068], +--R [0.32800000000000068,- 0.32800000000000068], +--R [0.32400000000000068,- 0.32400000000000068], +--R [0.32000000000000067,- 0.32000000000000067], +--R [0.31600000000000067,- 0.31600000000000067], +--R [0.31200000000000067,- 0.31200000000000067], +--R [0.30800000000000066,- 0.30800000000000066], +--R [0.30400000000000066,- 0.30400000000000066], +--R [0.30000000000000066,- 0.30000000000000066], +--R [0.29600000000000065,- 0.29600000000000065], +--R [0.29200000000000065,- 0.29200000000000065], +--R [0.28800000000000064,- 0.28800000000000064], +--R [0.28400000000000064,- 0.28400000000000064], +--R [0.28000000000000064,- 0.28000000000000064], +--R [0.27600000000000063,- 0.27600000000000063], +--R [0.27200000000000063,- 0.27200000000000063], +--R [0.26800000000000063,- 0.26800000000000063], +--R [0.26400000000000062,- 0.26400000000000062], +--R [0.26000000000000062,- 0.26000000000000062], +--R [0.25600000000000062,- 0.25600000000000062], +--R [0.25200000000000061,- 0.25200000000000061], +--R [0.24800000000000061,- 0.24800000000000061], +--R [0.24400000000000061,- 0.24400000000000061], +--R [0.2400000000000006,- 0.2400000000000006], +--R [0.2360000000000006,- 0.2360000000000006], +--R [0.23200000000000059,- 0.23200000000000059], +--R [0.22800000000000059,- 0.22800000000000059], +--R [0.22400000000000059,- 0.22400000000000059], +--R [0.22000000000000058,- 0.22000000000000058], +--R [0.21600000000000058,- 0.21600000000000058], +--R [0.21200000000000058,- 0.21200000000000058], +--R [0.20800000000000057,- 0.20800000000000057], +--R [0.20400000000000057,- 0.20400000000000057], +--R [0.20000000000000057,- 0.20000000000000057], +--R [0.19600000000000056,- 0.19600000000000056], +--R [0.19200000000000056,- 0.19200000000000056], +--R [0.18800000000000056,- 0.18800000000000056], +--R [0.18400000000000055,- 0.18400000000000055], +--R [0.18000000000000055,- 0.18000000000000055], +--R [0.17600000000000054,- 0.17600000000000054], +--R [0.17200000000000054,- 0.17200000000000054], +--R [0.16800000000000054,- 0.16800000000000054], +--R [0.16400000000000053,- 0.16400000000000053], +--R [0.16000000000000053,- 0.16000000000000053], +--R [0.15600000000000053,- 0.15600000000000053], +--R [0.15200000000000052,- 0.15200000000000052], +--R [0.14800000000000052,- 0.14800000000000052], +--R [0.14400000000000052,- 0.14400000000000052], +--R [0.14000000000000051,- 0.14000000000000051], +--R [0.13600000000000051,- 0.13600000000000051], +--R [0.13200000000000051,- 0.13200000000000051], +--R [0.1280000000000005,- 0.1280000000000005], +--R [0.1240000000000005,- 0.1240000000000005], +--R [0.1200000000000005,- 0.1200000000000005], +--R [0.11600000000000049,- 0.11600000000000049], +--R [0.11200000000000049,- 0.11200000000000049], +--R [0.10800000000000048,- 0.10800000000000048], +--R [0.10400000000000048,- 0.10400000000000048], +--R [0.10000000000000048,- 0.10000000000000048], +--R [9.6000000000000474E-2,- 9.6000000000000474E-2], +--R [9.200000000000047E-2,- 9.200000000000047E-2], +--R [8.8000000000000467E-2,- 8.8000000000000467E-2], +--R [8.4000000000000463E-2,- 8.4000000000000463E-2], +--R [8.000000000000046E-2,- 8.000000000000046E-2], +--R [7.6000000000000456E-2,- 7.6000000000000456E-2], +--R [7.2000000000000453E-2,- 7.2000000000000453E-2], +--R [6.8000000000000449E-2,- 6.8000000000000449E-2], +--R [6.4000000000000445E-2,- 6.4000000000000445E-2], +--R [6.0000000000000442E-2,- 6.0000000000000442E-2], +--R [5.6000000000000438E-2,- 5.6000000000000438E-2], +--R [5.2000000000000435E-2,- 5.2000000000000435E-2], +--R [4.8000000000000431E-2,- 4.8000000000000431E-2], +--R [4.4000000000000428E-2,- 4.4000000000000428E-2], +--R [4.0000000000000424E-2,- 4.0000000000000424E-2], +--R [3.6000000000000421E-2,- 3.6000000000000421E-2], +--R [3.2000000000000417E-2,- 3.2000000000000417E-2], +--R [2.8000000000000417E-2,- 2.8000000000000417E-2], +--R [2.4000000000000417E-2,- 2.4000000000000417E-2], +--R [2.0000000000000417E-2,- 2.0000000000000417E-2], +--R [1.6000000000000417E-2,- 1.6000000000000417E-2], +--R [1.2000000000000417E-2,- 1.2000000000000417E-2], +--R [8.0000000000004165E-3,- 8.0000000000004165E-3], +--R [4.0000000000004164E-3,- 4.0000000000004164E-3], +--R [4.163336342344337E-16,- 4.163336342344337E-16], +--R [- 3.9999999999995837E-3,3.9999999999995837E-3], +--R [- 7.9999999999995838E-3,7.9999999999995838E-3], +--R [- 1.1999999999999584E-2,1.1999999999999584E-2], +--R [- 1.5999999999999584E-2,1.5999999999999584E-2], +--R [- 1.9999999999999584E-2,1.9999999999999584E-2], +--R [- 2.3999999999999584E-2,2.3999999999999584E-2], +--R [- 2.7999999999999584E-2,2.7999999999999584E-2], +--R [- 3.1999999999999584E-2,3.1999999999999584E-2], +--R [- 3.5999999999999588E-2,3.5999999999999588E-2], +--R [- 3.9999999999999591E-2,3.9999999999999591E-2], +--R [- 4.3999999999999595E-2,4.3999999999999595E-2], +--R [- 4.7999999999999599E-2,4.7999999999999599E-2], +--R [- 5.1999999999999602E-2,5.1999999999999602E-2], +--R [- 5.5999999999999606E-2,5.5999999999999606E-2], +--R [- 5.9999999999999609E-2,5.9999999999999609E-2], +--R [- 6.3999999999999613E-2,6.3999999999999613E-2], +--R [- 6.7999999999999616E-2,6.7999999999999616E-2], +--R [- 7.199999999999962E-2,7.199999999999962E-2], +--R [- 7.5999999999999623E-2,7.5999999999999623E-2], +--R [- 7.9999999999999627E-2,7.9999999999999627E-2], +--R [- 8.3999999999999631E-2,8.3999999999999631E-2], +--R [- 8.7999999999999634E-2,8.7999999999999634E-2], +--R [- 9.1999999999999638E-2,9.1999999999999638E-2], +--R [- 9.5999999999999641E-2,9.5999999999999641E-2], +--R [- 9.9999999999999645E-2,9.9999999999999645E-2], +--R [- 0.10399999999999965,0.10399999999999965], +--R [- 0.10799999999999965,0.10799999999999965], +--R [- 0.11199999999999966,0.11199999999999966], +--R [- 0.11599999999999966,0.11599999999999966], +--R [- 0.11999999999999966,0.11999999999999966], +--R [- 0.12399999999999967,0.12399999999999967], +--R [- 0.12799999999999967,0.12799999999999967], +--R [- 0.13199999999999967,0.13199999999999967], +--R [- 0.13599999999999968,0.13599999999999968], +--R [- 0.13999999999999968,0.13999999999999968], +--R [- 0.14399999999999968,0.14399999999999968], +--R [- 0.14799999999999969,0.14799999999999969], +--R [- 0.15199999999999969,0.15199999999999969], +--R [- 0.15599999999999969,0.15599999999999969], +--R [- 0.1599999999999997,0.1599999999999997], +--R [- 0.1639999999999997,0.1639999999999997], +--R [- 0.16799999999999971,0.16799999999999971], +--R [- 0.17199999999999971,0.17199999999999971], +--R [- 0.17599999999999971,0.17599999999999971], +--R [- 0.17999999999999972,0.17999999999999972], +--R [- 0.18399999999999972,0.18399999999999972], +--R [- 0.18799999999999972,0.18799999999999972], +--R [- 0.19199999999999973,0.19199999999999973], +--R [- 0.19599999999999973,0.19599999999999973], +--R [- 0.19999999999999973,0.19999999999999973], +--R [- 0.20399999999999974,0.20399999999999974], +--R [- 0.20799999999999974,0.20799999999999974], +--R [- 0.21199999999999974,0.21199999999999974], +--R [- 0.21599999999999975,0.21599999999999975], +--R [- 0.21999999999999975,0.21999999999999975], +--R [- 0.22399999999999975,0.22399999999999975], +--R [- 0.22799999999999976,0.22799999999999976], +--R [- 0.23199999999999976,0.23199999999999976], +--R [- 0.23599999999999977,0.23599999999999977], +--R [- 0.23999999999999977,0.23999999999999977], +--R [- 0.24399999999999977,0.24399999999999977], +--R [- 0.24799999999999978,0.24799999999999978], +--R [- 0.25199999999999978,0.25199999999999978], +--R [- 0.25599999999999978,0.25599999999999978], +--R [- 0.25999999999999979,0.25999999999999979], +--R [- 0.26399999999999979,0.26399999999999979], +--R [- 0.26799999999999979,0.26799999999999979], +--R [- 0.2719999999999998,0.2719999999999998], +--R [- 0.2759999999999998,0.2759999999999998], +--R [- 0.2799999999999998,0.2799999999999998], +--R [- 0.28399999999999981,0.28399999999999981], +--R [- 0.28799999999999981,0.28799999999999981], +--R [- 0.29199999999999982,0.29199999999999982], +--R [- 0.29599999999999982,0.29599999999999982], +--R [- 0.29999999999999982,0.29999999999999982], +--R [- 0.30399999999999983,0.30399999999999983], +--R [- 0.30799999999999983,0.30799999999999983], +--R [- 0.31199999999999983,0.31199999999999983], +--R [- 0.31599999999999984,0.31599999999999984], +--R [- 0.31999999999999984,0.31999999999999984], +--R [- 0.32399999999999984,0.32399999999999984], +--R [- 0.32799999999999985,0.32799999999999985], +--R [- 0.33199999999999985,0.33199999999999985], +--R [- 0.33599999999999985,0.33599999999999985], +--R [- 0.33999999999999986,0.33999999999999986], +--R [- 0.34399999999999986,0.34399999999999986], +--R [- 0.34799999999999986,0.34799999999999986], +--R [- 0.35199999999999987,0.35199999999999987], +--R [- 0.35599999999999987,0.35599999999999987], +--R [- 0.35999999999999988,0.35999999999999988], +--R [- 0.36399999999999988,0.36399999999999988], +--R [- 0.36799999999999988,0.36799999999999988], +--R [- 0.37199999999999989,0.37199999999999989], +--R [- 0.37599999999999989,0.37599999999999989], +--R [- 0.37999999999999989,0.37999999999999989], +--R [- 0.3839999999999999,0.3839999999999999], +--R [- 0.3879999999999999,0.3879999999999999], +--R [- 0.3919999999999999,0.3919999999999999], +--R [- 0.39599999999999991,0.39599999999999991], +--R [- 0.39999999999999991,0.39999999999999991], +--R [- 0.40399999999999991,0.40399999999999991], +--R [- 0.40799999999999992,0.40799999999999992], +--R [- 0.41199999999999992,0.41199999999999992], +--R [- 0.41599999999999993,0.41599999999999993], +--R [- 0.41999999999999993,0.41999999999999993], +--R [- 0.42399999999999993,0.42399999999999993], +--R [- 0.42799999999999994,0.42799999999999994], +--R [- 0.43199999999999994,0.43199999999999994], +--R [- 0.43599999999999994,0.43599999999999994], +--R [- 0.43999999999999995,0.43999999999999995], +--R [- 0.44399999999999995,0.44399999999999995], +--R [- 0.44799999999999995,0.44799999999999995], +--R [- 0.45199999999999996,0.45199999999999996], +--R [- 0.45599999999999996,0.45599999999999996], +--R [- 0.45999999999999996,0.45999999999999996], +--R [- 0.46399999999999997,0.46399999999999997], +--R [- 0.46799999999999997,0.46799999999999997], +--R [- 0.47199999999999998,0.47199999999999998], +--R [- 0.47599999999999998,0.47599999999999998], +--R [- 0.47999999999999998,0.47999999999999998], +--R [- 0.48399999999999999,0.48399999999999999], +--R [- 0.48799999999999999,0.48799999999999999], +--R [- 0.49199999999999999,0.49199999999999999], [- 0.496,0.496], +--R [- 0.5,0.5]] +--R ] +--R Type: List List Point DoubleFloat +--E 4 + +--S 5 of 5 +)show ACPLOT +--R PlaneAlgebraicCurvePlot is a domain constructor +--R Abbreviation for PlaneAlgebraicCurvePlot is ACPLOT +--R This constructor is not exposed in this frame. +--R Issue )edit bookvol10.3.spad.pamphlet to see algebra source code for ACPLOT +--R +--R------------------------------- Operations -------------------------------- +--R coerce : % -> OutputForm refine : (%,DoubleFloat) -> % +--R xRange : % -> Segment DoubleFloat yRange : % -> Segment DoubleFloat +--R listBranches : % -> List List Point DoubleFloat +--R makeSketch : (Polynomial Integer,Symbol,Symbol,Segment Fraction Integer,Segment Fraction Integer) -> % +--R +--E 5 + )spool )lisp (bye) @ @@ -71631,10 +72163,50 @@ makeSketch(x+y,x,y,-1/2..1/2,-1/2..1/2)$ACPLOT PlaneAlgebraicCurvePlot examples ==================================================================== - makeSketch(x+y,x,y,-1/2..1/2,-1/2..1/2)$ACPLOT +sketch:=makeSketch(x+y,x,y,-1/2..1/2,-1/2..1/2)$ACPLOT -See Also: -o )show PlaneAlgebraicCurvePlot + ACPLOT + 1 1 1 1 + y + x = 0, - - <= x <= -, - - <= y <= - + 2 2 2 2 + [0.5,- 0.5] + [- 0.5,0.5] + + + +refined:=refine(sketch,0.1) + + ACPLOT + 1 1 1 1 + y + x = 0, - - <= x <= -, - - <= y <= - + 2 2 2 2 + [0.5,- 0.5] + [0.49600000000000083,- 0.49600000000000083] + [0.49200000000000083,- 0.49200000000000083] + [0.48800000000000082,- 0.48800000000000082] + [0.48400000000000082,- 0.48400000000000082] + ... + [- 0.48399999999999999,0.48399999999999999] + [- 0.48799999999999999,0.48799999999999999] + [- 0.49199999999999999,0.49199999999999999] + [- 0.496,0.496] + [- 0.5,0.5] + +listBranches(sketch) + + [[[0.5,- 0.5],[- 0.5,0.5]]] + + +listBranches(refined) + + [ + [[0.5,- 0.5], [0.49600000000000083,- 0.49600000000000083], + [0.49200000000000083,- 0.49200000000000083], + [0.48800000000000082,- 0.48800000000000082], + ... + [- 0.48399999999999999,0.48399999999999999], + [- 0.48799999999999999,0.48799999999999999], + [- 0.49199999999999999,0.49199999999999999], [- 0.496,0.496], @ \pagehead{PlaneAlgebraicCurvePlot}{ACPLOT} @@ -71681,8 +72253,14 @@ PlaneAlgebraicCurvePlot(): PlottablePlaneCurveCategory _ ++ the vertical axis is the y-axis. The rational numbers ++ xMin,...,yMax specify the boundaries of the region in ++ which the curve is to be plotted. + ++ + ++X makeSketch(x+y,x,y,-1/2..1/2,-1/2..1/2)$ACPLOT + refine:(%,DoubleFloat) -> % ++ refine(p,x) \undocumented{} + ++ + ++X sketch:=makeSketch(x+y,x,y,-1/2..1/2,-1/2..1/2)$ACPLOT + ++X refined:=refine(sketch,0.1) == add diff --git a/books/bookvol10.4.pamphlet b/books/bookvol10.4.pamphlet index 0e014b7..0760a9d 100644 --- a/books/bookvol10.4.pamphlet +++ b/books/bookvol10.4.pamphlet @@ -43692,7 +43692,8 @@ factor t6 )set message test on )set message auto off )clear all ---S 1 + +--S 1 of 36 M := SQMATRIX(2,INT) --R --R @@ -43700,7 +43701,7 @@ M := SQMATRIX(2,INT) --R Type: Domain --E 1 ---S 2 +--S 2 of 36 m1: M := squareMatrix matrix [ [1, 2], [0, -1] ] --R --R @@ -43710,7 +43711,7 @@ m1: M := squareMatrix matrix [ [1, 2], [0, -1] ] --R Type: SquareMatrix(2,Integer) --E 2 ---S 3 +--S 3 of 36 m2: M := squareMatrix matrix [ [2, 3], [1, -2] ] --R --R @@ -43720,7 +43721,7 @@ m2: M := squareMatrix matrix [ [2, 3], [1, -2] ] --R Type: SquareMatrix(2,Integer) --E 3 ---S 4 +--S 4 of 36 m3: M := squareMatrix matrix [ [3, 4], [2, -3] ] --R --R @@ -43730,7 +43731,7 @@ m3: M := squareMatrix matrix [ [3, 4], [2, -3] ] --R Type: SquareMatrix(2,Integer) --E 4 ---S 5 +--S 5 of 36 linearlyDependentOverZ? vector [m1, m2, m3] --R --R @@ -43738,7 +43739,7 @@ linearlyDependentOverZ? vector [m1, m2, m3] --R Type: Boolean --E 5 ---S 6 +--S 6 of 36 c := linearDependenceOverZ vector [m1, m2, m3] --R --R @@ -43746,7 +43747,7 @@ c := linearDependenceOverZ vector [m1, m2, m3] --R Type: Union(Vector Integer,...) --E 6 ---S 7 +--S 7 of 36 c.1 * m1 + c.2 * m2 + c.3 * m3 --R --R @@ -43756,7 +43757,7 @@ c.1 * m1 + c.2 * m2 + c.3 * m3 --R Type: SquareMatrix(2,Integer) --E 7 ---S 8 +--S 8 of 36 solveLinearlyOverQ(vector [m1, m3], m2) --R --R @@ -47406,6 +47407,970 @@ LaplaceTransform(R, F): Exports == Implementation where @ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{package LAZM3PK LazardSetSolvingPackage} +<>= +)set break resume +)sys rm -f LazardSetSolvingPackage.output +)spool LazardSetSolvingPackage.output +)set message test on +)set message auto off +)clear all + +--S 1 of 36 +R := Integer +--R +--R (1) Integer +--R Type: Domain +--E 1 + +--S 2 of 36 +ls : List Symbol := [b1,x,y,z,t,v,u,w] +--R +--R (2) [b1,x,y,z,t,v,u,w] +--R Type: List Symbol +--E 2 + +--S 3 of 36 +V := OVAR(ls) +--R +--R (3) OrderedVariableList [b1,x,y,z,t,v,u,w] +--R Type: Domain +--E 3 + +--S 4 of 36 +E := IndexedExponents V +--R +--R (4) IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w] +--R Type: Domain +--E 4 + +--S 5 of 36 +P := NSMP(R, V) +--R +--R (5) +--R NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w +--R ]) +--R Type: Domain +--E 5 + +--S 6 of 36 +b1: P := 'b1 +--R +--R (6) b1 +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 6 + +--S 7 of 36 +x: P := 'x +--R +--R (7) x +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 7 + +--S 8 of 36 +y: P := 'y +--R +--R (8) y +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 8 + +--S 9 of 36 +z: P := 'z +--R +--R (9) z +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 9 + +--S 10 of 36 +t: P := 't +--R +--R (10) t +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 10 + +--S 11 of 36 +u: P := 'u +--R +--R (11) u +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 11 + +--S 12 of 36 +v: P := 'v +--R +--R (12) v +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 12 + +--S 13 of 36 +w: P := 'w +--R +--R (13) w +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 13 + +--S 14 of 36 +T := REGSET(R,E,V,P) +--R +--R (14) +--R RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [b1,x,y,z,t +--R ,v,u,w],OrderedVariableList [b1,x,y,z,t,v,u,w],NewSparseMultivariatePolynomia +--R l(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])) +--R Type: Domain +--E 14 + +--S 15 of 36 +p0 := b1 + y + z - t - w +--R +--R (15) b1 + y + z - t - w +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 15 + +--S 16 of 36 +p1 := 2*z*u + 2*y*v + 2*t*w - 2*w**2 - w - 1 +--R +--R 2 +--R (16) 2v y + 2u z + 2w t - 2w - w - 1 +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 16 + +--S 17 of 36 +p2 := 3*z*u**2 + 3*y*v**2 - 3*t*w**2 + 3*w**3 + 3*w**2 - t + 4*w +--R +--R 2 2 2 3 2 +--R (17) 3v y + 3u z + (- 3w - 1)t + 3w + 3w + 4w +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 17 + +--S 18 of 36 +p3 := 6*x*z*v - 6*t*w**2 + 6*w**3 - 3*t*w + 6*w**2 - t + 4*w +--R +--R 2 3 2 +--R (18) 6v z x + (- 6w - 3w - 1)t + 6w + 6w + 4w +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 18 + +--S 19 of 36 +p4 := 4*z*u**3+ 4*y*v**3+ 4*t*w**3- 4*w**4 - 6*w**3+ 4*t*w- 10*w**2- w- 1 +--R +--R 3 3 3 4 3 2 +--R (19) 4v y + 4u z + (4w + 4w)t - 4w - 6w - 10w - w - 1 +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 19 + +--S 20 of 36 +p5 := 8*x*z*u*v +8*t*w**3 -8*w**4 +4*t*w**2 -12*w**3 +4*t*w -14*w**2 -3*w -1 +--R +--R 3 2 4 3 2 +--R (20) 8u v z x + (8w + 4w + 4w)t - 8w - 12w - 14w - 3w - 1 +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 20 + +--S 21 of 36 +p6 := 12*x*z*v**2+12*t*w**3 -12*w**4 +12*t*w**2 -18*w**3 +8*t*w -14*w**2 -w -1 +--R +--R 2 3 2 4 3 2 +--R (21) 12v z x + (12w + 12w + 8w)t - 12w - 18w - 14w - w - 1 +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 21 + +--S 22 of 36 +p7 := -24*t*w**3 + 24*w**4 - 24*t*w**2 + 36*w**3 - 8*t*w + 26*w**2 + 7*w + 1 +--R +--R 3 2 4 3 2 +--R (22) (- 24w - 24w - 8w)t + 24w + 36w + 26w + 7w + 1 +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 22 + +--S 23 of 36 +lp := [p0, p1, p2, p3, p4, p5, p6, p7] +--R +--R (23) +--R 2 +--R [b1 + y + z - t - w, 2v y + 2u z + 2w t - 2w - w - 1, +--R 2 2 2 3 2 +--R 3v y + 3u z + (- 3w - 1)t + 3w + 3w + 4w, +--R 2 3 2 +--R 6v z x + (- 6w - 3w - 1)t + 6w + 6w + 4w, +--R 3 3 3 4 3 2 +--R 4v y + 4u z + (4w + 4w)t - 4w - 6w - 10w - w - 1, +--R 3 2 4 3 2 +--R 8u v z x + (8w + 4w + 4w)t - 8w - 12w - 14w - 3w - 1, +--R 2 3 2 4 3 2 +--R 12v z x + (12w + 12w + 8w)t - 12w - 18w - 14w - w - 1, +--R 3 2 4 3 2 +--R (- 24w - 24w - 8w)t + 24w + 36w + 26w + 7w + 1] +--RType: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 23 + +--S 24 of 36 +lts := zeroSetSplit(lp,false)$T +--R +--R (24) +--R [{w + 1,u,v,t + 1,b1 + y + z + 2}, {w + 1,v,t + 1,z,b1 + y + 2}, +--R {w + 1,t + 1,z,y,b1 + 2}, {w + 1,v - u,t + 1,y + z,x,b1 + 2}, +--R {w + 1,u,t + 1,y,x,b1 + z + 2}, +--R +--R 5 4 3 2 +--R {144w + 216w + 96w + 6w - 11w - 1, +--R 2 5 4 3 2 +--R (12w + 9w + 1)u - 72w - 108w - 42w - 9w - 3w, +--R 2 4 3 2 +--R (12w + 9w + 1)v + 36w + 54w + 18w , +--R 3 2 4 3 2 +--R (24w + 24w + 8w)t - 24w - 36w - 26w - 7w - 1, +--R +--R 2 2 4 3 2 +--R (12u v - 12u )z + (12w v + 12w + 4)t + (3w - 5)v + 36w + 42w + 6w +--R + +--R - 16w +--R , +--R 2 +--R 2v y + 2u z + 2w t - 2w - w - 1, +--R 2 3 2 +--R 6v z x + (- 6w - 3w - 1)t + 6w + 6w + 4w, b1 + y + z - t - w} +--R ] +--RType: List RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w],OrderedVariableList [b1,x,y,z,t,v,u,w],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])) +--E 24 + +--S 25 of 36 +[coHeight(ts) for ts in lts] +--R +--R (25) [3,3,3,2,2,0] +--R Type: List NonNegativeInteger +--E 25 + +--S 26 of 36 +ST := SREGSET(R,E,V,P) +--R +--R (26) +--R SquareFreeRegularTriangularSet(Integer,IndexedExponents OrderedVariableList [ +--R b1,x,y,z,t,v,u,w],OrderedVariableList [b1,x,y,z,t,v,u,w],NewSparseMultivariat +--R ePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])) +--R Type: Domain +--E 26 + +--S 27 of 36 +pack := LAZM3PK(R,E,V,P,T,ST) +--R +--R (27) +--R LazardSetSolvingPackage(Integer,IndexedExponents OrderedVariableList [b1,x,y, +--R z,t,v,u,w],OrderedVariableList [b1,x,y,z,t,v,u,w],NewSparseMultivariatePolyno +--R mial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]),RegularTriangularSet(Int +--R eger,IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w],OrderedVariableL +--R ist [b1,x,y,z,t,v,u,w],NewSparseMultivariatePolynomial(Integer,OrderedVariabl +--R eList [b1,x,y,z,t,v,u,w])),SquareFreeRegularTriangularSet(Integer,IndexedExpo +--R nents OrderedVariableList [b1,x,y,z,t,v,u,w],OrderedVariableList [b1,x,y,z,t, +--R v,u,w],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z, +--R t,v,u,w]))) +--R Type: Domain +--E 27 + +--S 28 of 36 +zeroSetSplit(lp,false)$pack +--R +--R (28) +--R [{w + 1,t + 1,z,y,b1 + 2}, {w + 1,v,t + 1,z,b1 + y + 2}, +--R {w + 1,u,v,t + 1,b1 + y + z + 2}, {w + 1,v - u,t + 1,y + z,x,b1 + 2}, +--R {w + 1,u,t + 1,y,x,b1 + z + 2}, +--R +--R 5 4 3 2 4 3 2 +--R {144w + 216w + 96w + 6w - 11w - 1, u - 24w - 36w - 14w + w + 1, +--R 4 3 2 4 3 2 +--R 3v - 48w - 60w - 10w + 8w + 2, t - 24w - 36w - 14w - w + 1, +--R 4 3 2 +--R 486z - 2772w - 4662w - 2055w + 30w + 127, +--R 4 3 2 +--R 2916y - 22752w - 30312w - 8220w + 2064w + 1561, +--R 4 3 2 +--R 356x - 3696w - 4536w - 968w + 822w + 371, +--R 4 3 2 +--R 2916b1 - 30600w - 46692w - 20274w - 8076w + 593} +--R ] +--RType: List SquareFreeRegularTriangularSet(Integer,IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w],OrderedVariableList [b1,x,y,z,t,v,u,w],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])) +--E 28 + +--S 29 of 36 +f0 := (w - v) ** 2 + (u - t) ** 2 - 1 +--R +--R 2 2 2 2 +--R (29) t - 2u t + v - 2w v + u + w - 1 +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 29 + +--S 30 of 36 +f1 := t ** 2 - v ** 3 +--R +--R 2 3 +--R (30) t - v +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 30 + +--S 31 of 36 +f2 := 2 * t * (w - v) + 3 * v ** 2 * (u - t) +--R +--R 2 2 +--R (31) (- 3v - 2v + 2w)t + 3u v +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 31 + +--S 32 of 36 +f3 := (3 * z * v ** 2 - 1) * (2 * z * t - 1) +--R +--R 2 2 2 +--R (32) 6v t z + (- 2t - 3v )z + 1 +--RType: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 32 + +--S 33 of 36 +lf := [f0, f1, f2, f3] +--R +--R (33) +--R 2 2 2 2 2 3 2 2 +--R [t - 2u t + v - 2w v + u + w - 1, t - v , (- 3v - 2v + 2w)t + 3u v , +--R 2 2 2 +--R 6v t z + (- 2t - 3v )z + 1] +--RType: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]) +--E 33 + +--S 34 of 36 +zeroSetSplit(lf,true)$T +--R +--R (34) +--R [ +--R { +--R 6 3 2 4 +--R 729u + (- 1458w + 729w - 4158w - 1685)u +--R + +--R 6 5 4 3 2 2 8 +--R (729w - 1458w - 2619w - 4892w - 297w + 5814w + 427)u + 729w +--R + +--R 7 6 5 4 3 2 +--R 216w - 2900w - 2376w + 3870w + 4072w - 1188w - 1656w + 529 +--R , +--R +--R 4 3 2 2 6 5 +--R 2187u + (- 4374w - 972w - 12474w - 2868)u + 2187w - 1944w +--R + +--R 4 3 2 +--R - 10125w - 4800w + 2501w + 4968w - 1587 +--R * +--R v +--R + +--R 3 2 2 6 5 4 3 2 +--R (1944w - 108w )u + 972w + 3024w - 1080w + 496w + 1116w +--R , +--R 2 2 2 2 2 +--R (3v + 2v - 2w)t - 3u v , ((4v - 4w)t - 6u v )z + (2t + 3v )z - 1} +--R ] +--RType: List RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w],OrderedVariableList [b1,x,y,z,t,v,u,w],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])) +--E 34 + +--S 35 of 36 +zeroSetSplit(lf,false)$T +--R +--R (35) +--R [ +--R { +--R 6 3 2 4 +--R 729u + (- 1458w + 729w - 4158w - 1685)u +--R + +--R 6 5 4 3 2 2 8 +--R (729w - 1458w - 2619w - 4892w - 297w + 5814w + 427)u + 729w +--R + +--R 7 6 5 4 3 2 +--R 216w - 2900w - 2376w + 3870w + 4072w - 1188w - 1656w + 529 +--R , +--R +--R 4 3 2 2 6 5 +--R 2187u + (- 4374w - 972w - 12474w - 2868)u + 2187w - 1944w +--R + +--R 4 3 2 +--R - 10125w - 4800w + 2501w + 4968w - 1587 +--R * +--R v +--R + +--R 3 2 2 6 5 4 3 2 +--R (1944w - 108w )u + 972w + 3024w - 1080w + 496w + 1116w +--R , +--R 2 2 2 2 2 +--R (3v + 2v - 2w)t - 3u v , ((4v - 4w)t - 6u v )z + (2t + 3v )z - 1} +--R , +--R +--R 4 3 2 2 +--R {27w + 4w - 54w - 36w + 23, u, (12w + 2)v - 9w - 2w + 9, +--R 2 2 +--R 6t - 2v - 3w + 2w + 3, 2t z - 1} +--R , +--R +--R 6 5 4 3 2 +--R {59049w + 91854w - 45198w + 145152w + 63549w + 60922w + 21420, +--R +--R 5 4 3 +--R 31484448266904w - 18316865522574w + 23676995746098w +--R + +--R 2 +--R 6657857188965w + 8904703998546w + 3890631403260 +--R * +--R 2 +--R u +--R + +--R 5 4 3 +--R 94262810316408w - 82887296576616w + 89801831438784w +--R + +--R 2 +--R 28141734167208w + 38070359425432w + 16003865949120 +--R , +--R 2 2 2 3 2 3 2 +--R (243w + 36w + 85)v + (- 81u - 162w + 36w + 154w + 72)v - 72w + 4w , +--R 2 2 2 2 2 +--R (3v + 2v - 2w)t - 3u v , ((4v - 4w)t - 6u v )z + (2t + 3v )z - 1} +--R , +--R +--R 4 3 2 2 +--R {27w + 4w - 54w - 36w + 23, u, (12w + 2)v - 9w - 2w + 9, +--R 2 2 2 +--R 6t - 2v - 3w + 2w + 3, 3v z - 1} +--R ] +--RType: List RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w],OrderedVariableList [b1,x,y,z,t,v,u,w],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])) +--E 35 + +--S 36 of 36 +zeroSetSplit(lf,false)$pack +--R +--R (36) +--R [ +--R { +--R 6 3 2 4 +--R 729u + (- 1458w + 729w - 4158w - 1685)u +--R + +--R 6 5 4 3 2 2 8 +--R (729w - 1458w - 2619w - 4892w - 297w + 5814w + 427)u + 729w +--R + +--R 7 6 5 4 3 2 +--R 216w - 2900w - 2376w + 3870w + 4072w - 1188w - 1656w + 529 +--R , +--R +--R 4 3 2 2 6 5 +--R 2187u + (- 4374w - 972w - 12474w - 2868)u + 2187w - 1944w +--R + +--R 4 3 2 +--R - 10125w - 4800w + 2501w + 4968w - 1587 +--R * +--R v +--R + +--R 3 2 2 6 5 4 3 2 +--R (1944w - 108w )u + 972w + 3024w - 1080w + 496w + 1116w +--R , +--R 2 2 2 2 2 +--R (3v + 2v - 2w)t - 3u v , ((4v - 4w)t - 6u v )z + (2t + 3v )z - 1} +--R , +--R +--R 2 2 2 +--R {81w + 18w + 28, 729u - 1890w - 533, 81v + (- 162w + 27)v - 72w - 112, +--R 11881t + (972w + 2997)u v + (- 11448w - 11536)u, +--R +--R 2 +--R 641237934604288z +--R + +--R (78614584763904w + 26785578742272)u + 236143618655616w +--R + +--R 70221988585728 +--R * +--R v +--R + +--R (358520253138432w + 101922133759488)u + 142598803536000w +--R + +--R 54166419595008 +--R * +--R z +--R + +--R (32655103844499w - 44224572465882)u v +--R + +--R (43213900115457w - 32432039102070)u +--R } +--R , +--R +--R 4 3 2 3 2 +--R {27w + 4w - 54w - 36w + 23, u, 218v - 162w + 3w + 160w + 153, +--R 2 3 2 3 2 +--R 109t - 27w - 54w + 63w + 80, 1744z + (- 1458w + 27w + 1440w + 505)t} +--R , +--R +--R 4 3 2 3 2 +--R {27w + 4w - 54w - 36w + 23, u, 218v - 162w + 3w + 160w + 153, +--R 2 3 2 3 2 +--R 109t - 27w - 54w + 63w + 80, 1308z + 162w - 3w - 814w - 153} +--R , +--R +--R 4 3 2 2 2 +--R {729w + 972w - 1026w + 1684w + 765, 81u + 72w + 16w - 72, +--R 3 2 +--R 702v - 162w - 225w + 40w - 99, +--R 3 2 +--R 11336t + (324w - 603w - 1718w - 1557)u, +--R +--R 2 +--R 595003968z +--R + +--R 3 2 +--R (- 963325386w - 898607682w + 1516286466w - 3239166186)u +--R + +--R 3 2 +--R - 1579048992w - 1796454288w + 2428328160w - 4368495024 +--R * +--R z +--R + +--R 3 2 +--R (9713133306w + 9678670317w - 16726834476w + 28144233593)u +--R } +--R ] +--RType: List SquareFreeRegularTriangularSet(Integer,IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w],OrderedVariableList [b1,x,y,z,t,v,u,w],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])) +--E 36 + +)spool +)lisp (bye) +@ +<>= +==================================================================== +LazardSetSolvingPackage examples +==================================================================== + +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 +dimension are not necessarily normalized. + +The decompositions are computed in two steps. First the algorithm of +Moreno Maza (implemented in the RegularTriangularSet domain +constructor) is called. Then the resulting decompositions are +converted into lists of square-free regular triangular sets and the +redundant components are removed. Moreover, zero-dimensional regular +triangular sets are normalized. + +Note that the way of understanding triangular decompositions is +detailed in the example of the RegularTriangularSet constructor. + +The LazardSetSolvingPackage constructor takes six arguments. The +first one, R, is the coefficient ring of the polynomials; it must +belong to the category GcdDomain. The second one, E, is the exponent +monoid of the polynomials; it must belong to the category +OrderedAbelianMonoidSup. the third one, V, is the ordered set of +variables; it must belong to the category OrderedSet. The fourth one +is the polynomial ring; it must belong to the category +RecursivePolynomialCategory(R,E,V). The fifth one is a domain of the +category RegularTriangularSetCategory(R,E,V,P) and the last one is a +domain of the category SquareFreeRegularTriangularSetCategory(R,E,V,P). +The abbreviation for LazardSetSolvingPackage is LAZM3PK. + +For the purpose of solving zero-dimensional algebraic systems, see +also LexTriangularPackage and ZeroDimensionalSolvePackage. These +packages are easier to call than LAZM3PK. Moreover, the +ZeroDimensionalSolvePackage package provides operations to compute +either the complex roots or the real roots. + +We illustrate now the use of the LazardSetSolvingPackage package +constructor with two examples (Butcher and Vermeer). + +Define the coefficient ring. + +R := Integer + Integer + +Define the list of variables, + +ls : List Symbol := [b1,x,y,z,t,v,u,w] + [b1,x,y,z,t,v,u,w] + +and make it an ordered set: + +V := OVAR(ls) + OrderedVariableList [b1,x,y,z,t,v,u,w] + +then define the exponent monoid. + +E := IndexedExponents V + +Define the polynomial ring. + +P := NSMP(R, V) + +Let the variables be polynomial. + +b1: P := 'b1 + +x: P := 'x + +y: P := 'y + +z: P := 'z + +t: P := 't + +u: P := 'u + +v: P := 'v + +w: P := 'w + +Now call the {\tt RegularTriangularSet} domain constructor. + +T := REGSET(R,E,V,P) + +Define a polynomial system (the Butcher example). + +p0 := b1 + y + z - t - w + +p1 := 2*z*u + 2*y*v + 2*t*w - 2*w**2 - w - 1 + +p2 := 3*z*u**2 + 3*y*v**2 - 3*t*w**2 + 3*w**3 + 3*w**2 - t + 4*w + +p3 := 6*x*z*v - 6*t*w**2 + 6*w**3 - 3*t*w + 6*w**2 - t + 4*w + +p4 := 4*z*u**3+ 4*y*v**3+ 4*t*w**3- 4*w**4 - 6*w**3+ 4*t*w- 10*w**2- w- 1 + +p5 := 8*x*z*u*v +8*t*w**3 -8*w**4 +4*t*w**2 -12*w**3 +4*t*w -14*w**2 -3*w -1 + +p6 := 12*x*z*v**2+12*t*w**3 -12*w**4 +12*t*w**2 -18*w**3 +8*t*w -14*w**2 -w -1 + +p7 := -24*t*w**3 + 24*w**4 - 24*t*w**2 + 36*w**3 - 8*t*w + 26*w**2 + 7*w + 1 + +lp := [p0, p1, p2, p3, p4, p5, p6, p7] + [b1 + y + z - t - w, 2v y + 2u z + 2w t - 2w - w - 1, + 2 2 2 3 2 + 3v y + 3u z + (- 3w - 1)t + 3w + 3w + 4w, + 2 3 2 + 6v z x + (- 6w - 3w - 1)t + 6w + 6w + 4w, + 3 3 3 4 3 2 + 4v y + 4u z + (4w + 4w)t - 4w - 6w - 10w - w - 1, + 3 2 4 3 2 + 8u v z x + (8w + 4w + 4w)t - 8w - 12w - 14w - 3w - 1, + 2 3 2 4 3 2 + 12v z x + (12w + 12w + 8w)t - 12w - 18w - 14w - w - 1, + 3 2 4 3 2 + (- 24w - 24w - 8w)t + 24w + 36w + 26w + 7w + 1] + +First of all, let us solve this system in the sense of Lazard by means +of the REGSET constructor: + +lts := zeroSetSplit(lp,false)$T + [{w + 1,u,v,t + 1,b1 + y + z + 2}, {w + 1,v,t + 1,z,b1 + y + 2}, + {w + 1,t + 1,z,y,b1 + 2}, {w + 1,v - u,t + 1,y + z,x,b1 + 2}, + {w + 1,u,t + 1,y,x,b1 + z + 2}, + + 5 4 3 2 + {144w + 216w + 96w + 6w - 11w - 1, + 2 5 4 3 2 + (12w + 9w + 1)u - 72w - 108w - 42w - 9w - 3w, + 2 4 3 2 + (12w + 9w + 1)v + 36w + 54w + 18w , + 3 2 4 3 2 + (24w + 24w + 8w)t - 24w - 36w - 26w - 7w - 1, + + 2 2 4 3 2 + (12u v - 12u )z + (12w v + 12w + 4)t + (3w - 5)v + 36w + 42w + 6w + + + - 16w + , + 2 + 2v y + 2u z + 2w t - 2w - w - 1, + 2 3 2 + 6v z x + (- 6w - 3w - 1)t + 6w + 6w + 4w, b1 + y + z - t - w} + ] + +We can get the dimensions of each component +of a decomposition as follows. + +[coHeight(ts) for ts in lts] + [3,3,3,2,2,0] + +The first five sets have a simple shape. However, the last one, which +has dimension zero, can be simplified by using Lazard triangular sets. + +Thus we call the SquareFreeRegularTriangularSet domain constructor, + +ST := SREGSET(R,E,V,P) + +and set the LAZM3PK package constructor to our situation. + +pack := LAZM3PK(R,E,V,P,T,ST) + +We are ready to solve the system by means of Lazard triangular sets: + +zeroSetSplit(lp,false)$pack + [{w + 1,t + 1,z,y,b1 + 2}, {w + 1,v,t + 1,z,b1 + y + 2}, + {w + 1,u,v,t + 1,b1 + y + z + 2}, {w + 1,v - u,t + 1,y + z,x,b1 + 2}, + {w + 1,u,t + 1,y,x,b1 + z + 2}, + + 5 4 3 2 4 3 2 + {144w + 216w + 96w + 6w - 11w - 1, u - 24w - 36w - 14w + w + 1, + 4 3 2 4 3 2 + 3v - 48w - 60w - 10w + 8w + 2, t - 24w - 36w - 14w - w + 1, + 4 3 2 + 486z - 2772w - 4662w - 2055w + 30w + 127, + 4 3 2 + 2916y - 22752w - 30312w - 8220w + 2064w + 1561, + 4 3 2 + 356x - 3696w - 4536w - 968w + 822w + 371, + 4 3 2 + 2916b1 - 30600w - 46692w - 20274w - 8076w + 593} + ] + +We see the sixth triangular set is {\em nicer} now: each one of its +polynomials has a constant initial. + +We follow with the Vermeer example. The ordering is the usual one +for this system. + +Define the polynomial system. + +f0 := (w - v) ** 2 + (u - t) ** 2 - 1 + + + 2 2 2 2 + t - 2u t + v - 2w v + u + w - 1 + + +f1 := t ** 2 - v ** 3 + + 2 3 + t - v + +f2 := 2 * t * (w - v) + 3 * v ** 2 * (u - t) + + 2 2 + (- 3v - 2v + 2w)t + 3u v + +f3 := (3 * z * v ** 2 - 1) * (2 * z * t - 1) + + 2 2 2 + 6v t z + (- 2t - 3v )z + 1 + +lf := [f0, f1, f2, f3] + + 2 2 2 2 2 3 2 2 + [t - 2u t + v - 2w v + u + w - 1, t - v , (- 3v - 2v + 2w)t + 3u v , + 2 2 2 + 6v t z + (- 2t - 3v )z + 1] + +First of all, let us solve this system in the sense of Kalkbrener by +means of the REGSET constructor: + +zeroSetSplit(lf,true)$T + [ + { + 6 3 2 4 + 729u + (- 1458w + 729w - 4158w - 1685)u + + + 6 5 4 3 2 2 8 + (729w - 1458w - 2619w - 4892w - 297w + 5814w + 427)u + 729w + + + 7 6 5 4 3 2 + 216w - 2900w - 2376w + 3870w + 4072w - 1188w - 1656w + 529 + , + + 4 3 2 2 6 5 + 2187u + (- 4374w - 972w - 12474w - 2868)u + 2187w - 1944w + + + 4 3 2 + - 10125w - 4800w + 2501w + 4968w - 1587 + * + v + + + 3 2 2 6 5 4 3 2 + (1944w - 108w )u + 972w + 3024w - 1080w + 496w + 1116w + , + 2 2 2 2 2 + (3v + 2v - 2w)t - 3u v , ((4v - 4w)t - 6u v )z + (2t + 3v )z - 1} + ] + +We have obtained one regular chain (i.e. regular triangular set) with +dimension 1. This set is in fact a characterist set of the (radical +of) of the ideal generated by the input system lf. Thus we have +only the generic points of the variety associated with lf +(for the elimination ordering given by ls). + +So let us get now a full description of this variety. + +Hence, we solve this system in the sense of Lazard by means of the REGSET +constructor: + +zeroSetSplit(lf,false)$T + [ + { + 6 3 2 4 + 729u + (- 1458w + 729w - 4158w - 1685)u + + + 6 5 4 3 2 2 8 + (729w - 1458w - 2619w - 4892w - 297w + 5814w + 427)u + 729w + + + 7 6 5 4 3 2 + 216w - 2900w - 2376w + 3870w + 4072w - 1188w - 1656w + 529 + , + + 4 3 2 2 6 5 + 2187u + (- 4374w - 972w - 12474w - 2868)u + 2187w - 1944w + + + 4 3 2 + - 10125w - 4800w + 2501w + 4968w - 1587 + * + v + + + 3 2 2 6 5 4 3 2 + (1944w - 108w )u + 972w + 3024w - 1080w + 496w + 1116w + , + 2 2 2 2 2 + (3v + 2v - 2w)t - 3u v , ((4v - 4w)t - 6u v )z + (2t + 3v )z - 1} + , + + 4 3 2 2 + {27w + 4w - 54w - 36w + 23, u, (12w + 2)v - 9w - 2w + 9, + 2 2 + 6t - 2v - 3w + 2w + 3, 2t z - 1} + , + + 6 5 4 3 2 + {59049w + 91854w - 45198w + 145152w + 63549w + 60922w + 21420, + + 5 4 3 + 31484448266904w - 18316865522574w + 23676995746098w + + + 2 + 6657857188965w + 8904703998546w + 3890631403260 + * + 2 + u + + + 5 4 3 + 94262810316408w - 82887296576616w + 89801831438784w + + + 2 + 28141734167208w + 38070359425432w + 16003865949120 + , + 2 2 2 3 2 3 2 + (243w + 36w + 85)v + (- 81u - 162w + 36w + 154w + 72)v - 72w + 4w , + 2 2 2 2 2 + (3v + 2v - 2w)t - 3u v , ((4v - 4w)t - 6u v )z + (2t + 3v )z - 1} + , + + 4 3 2 2 + {27w + 4w - 54w - 36w + 23, u, (12w + 2)v - 9w - 2w + 9, + 2 2 2 + 6t - 2v - 3w + 2w + 3, 3v z - 1} + ] + + +We retrieve our regular chain of dimension 1 and we get three regular +chains of dimension 0 corresponding to the degenerated cases. +We want now to simplify these zero-dimensional regular chains by using +Lazard triangular sets. Moreover, this will allow us to prove that +the above decomposition has no redundant component. + +Generally, decompositions computed by the REGSET constructor do +not have redundant components. However, to be sure that no redundant +component occurs one needs to use the SREGSET or LAZM3PK constructors. + +So let us solve the input system in the sense of Lazard by means of +the LAZM3PK constructor: + +zeroSetSplit(lf,false)$pack + [ + { + 6 3 2 4 + 729u + (- 1458w + 729w - 4158w - 1685)u + + + 6 5 4 3 2 2 8 + (729w - 1458w - 2619w - 4892w - 297w + 5814w + 427)u + 729w + + + 7 6 5 4 3 2 + 216w - 2900w - 2376w + 3870w + 4072w - 1188w - 1656w + 529 + , + + 4 3 2 2 6 5 + 2187u + (- 4374w - 972w - 12474w - 2868)u + 2187w - 1944w + + + 4 3 2 + - 10125w - 4800w + 2501w + 4968w - 1587 + * + v + + + 3 2 2 6 5 4 3 2 + (1944w - 108w )u + 972w + 3024w - 1080w + 496w + 1116w + , + 2 2 2 2 2 + (3v + 2v - 2w)t - 3u v , ((4v - 4w)t - 6u v )z + (2t + 3v )z - 1} + , + + 2 2 2 + {81w + 18w + 28, 729u - 1890w - 533, 81v + (- 162w + 27)v - 72w - 112, + 11881t + (972w + 2997)u v + (- 11448w - 11536)u, + + 2 + 641237934604288z + + + (78614584763904w + 26785578742272)u + 236143618655616w + + + 70221988585728 + * + v + + + (358520253138432w + 101922133759488)u + 142598803536000w + + + 54166419595008 + * + z + + + (32655103844499w - 44224572465882)u v + + + (43213900115457w - 32432039102070)u + } + , + + 4 3 2 3 2 + {27w + 4w - 54w - 36w + 23, u, 218v - 162w + 3w + 160w + 153, + 2 3 2 3 2 + 109t - 27w - 54w + 63w + 80, 1744z + (- 1458w + 27w + 1440w + 505)t} + , + + 4 3 2 3 2 + {27w + 4w - 54w - 36w + 23, u, 218v - 162w + 3w + 160w + 153, + 2 3 2 3 2 + 109t - 27w - 54w + 63w + 80, 1308z + 162w - 3w - 814w - 153} + , + + 4 3 2 2 2 + {729w + 972w - 1026w + 1684w + 765, 81u + 72w + 16w - 72, + 3 2 + 702v - 162w - 225w + 40w - 99, + 3 2 + 11336t + (324w - 603w - 1718w - 1557)u, + + 2 + 595003968z + + + 3 2 + (- 963325386w - 898607682w + 1516286466w - 3239166186)u + + + 3 2 + - 1579048992w - 1796454288w + 2428328160w - 4368495024 + * + z + + + 3 2 + (9713133306w + 9678670317w - 16726834476w + 28144233593)u + } + ] + +Due to square-free factorization, we obtained now four +zero-dimensional regular chains. Moreover, each of them is normalized +(the initials are constant). Note that these zero-dimensional +components may be investigated further with the +ZeroDimensionalSolvePackage package constructor. + +See also: +o )show LazardSetSolvingPackage +o )show ZeroDimensionalSolvePackage + +@ \pagehead{LazardSetSolvingPackage}{LAZM3PK} \pagepic{ps/v104lazardsetsolvingpackage.ps}{LAZM3PK}{1.00} diff --git a/books/ps/v0plot1.eps b/books/ps/v0plot1.eps new file mode 100644 index 0000000..6490f81 --- /dev/null +++ b/books/ps/v0plot1.eps @@ -0,0 +1,1340 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: GIMP PostScript file plugin V 1.17 by Peter Kirchgessner +%%Title: v0plot1.eps +%%CreationDate: Thu Mar 19 15:34:01 2009 +%%DocumentData: Clean7Bit +%%LanguageLevel: 2 +%%Pages: 1 +%%BoundingBox: 14 14 142 146 +%%EndComments +%%BeginProlog +% Use own dictionary to avoid conflicts +10 dict begin +%%EndProlog +%%Page: 1 1 +% Translate for offset +14.173228346456694 14.173228346456694 translate +% Translate to begin of first scanline +0 131.17539273108105 translate 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src/algebra/Makefile add LazardSetSolvingPackage.help +20090404 tpd books/ps/v0plot4.eps TwoDimensionalViewport example image 3 +20090404 tpd books/ps/v0plot3.eps TwoDimensionalViewport example image 3 +20090404 tpd books/ps/v0plot2.eps TwoDimensionalViewport example image 2 +20090404 tpd books/ps/v0plot1.eps TwoDimensionalViewport example image 1 20090404 tpd src/axiom-website/patches.html 20090404.01.tpd.patch 20090404 tpd books/bookvol0 fix cross references 20090403 tpd src/axiom-website/patches.html 20090403.02.tpd.patch diff --git a/src/algebra/Makefile.pamphlet b/src/algebra/Makefile.pamphlet index d58506c..bde991e 100644 --- a/src/algebra/Makefile.pamphlet +++ b/src/algebra/Makefile.pamphlet @@ -16460,6 +16460,7 @@ SPADHELP=\ ${HELP}/Integer.help ${HELP}/IntegerLinearDependence.help \ ${HELP}/IntegerNumberTheoryFunctions.help \ ${HELP}/Kernel.help ${HELP}/KeyedAccessFile.help \ + ${HELP}/LazardSetSolvingPackage.help \ ${HELP}/LexTriangularPackage.help ${HELP}/Library.help \ ${HELP}/LieExponentials.help ${HELP}/LiePolynomial.help \ ${HELP}/LinearOrdinaryDifferentialOperator.help \ @@ -16548,6 +16549,7 @@ REGRESS=\ Integer.regress IntegerLinearDependence.regress \ IntegerNumberTheoryFunctions.regress \ Kernel.regress KeyedAccessFile.regress \ + LazardSetSolvingPackage.regress \ LexTriangularPackage.regress Library.regress \ LieExponentials.regress LiePolynomial.regress \ LinearOrdinaryDifferentialOperator.regress \ @@ -17054,6 +17056,18 @@ ${HELP}/KeyedAccessFile.help: ${BOOKS}/bookvol10.3.pamphlet >${INPUT}/KeyedAccessFile.input @echo "KeyedAccessFile (KAFILE)" >>${HELPFILE} +${HELP}/LazardSetSolvingPackage.help: ${BOOKS}/bookvol10.3.pamphlet + @echo 7042 create LazardSetSolvingPackage.help from \ + ${BOOKS}/bookvol10.3.pamphlet + @${TANGLE} -R"LazardSetSolvingPackage.help" \ + ${BOOKS}/bookvol10.3.pamphlet \ + >${HELP}/LazardSetSolvingPackage.help + @cp ${HELP}/LazardSetSolvingPackage.help ${HELP}/LAZM3PK.help + @${TANGLE} -R"LazardSetSolvingPackage.input" \ + ${BOOKS}/bookvol10.3.pamphlet \ + >${INPUT}/LazardSetSolvingPackage.input + @echo "LazardSetSolvingPackage (LAZM3PK)" >>${HELPFILE} + ${HELP}/LexTriangularPackage.help: ${BOOKS}/bookvol10.4.pamphlet @echo 7043 create LexTriangularPackage.help from \ ${BOOKS}/bookvol10.4.pamphlet diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index 782bc59..cd448e8 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -1044,5 +1044,7 @@ remove gcl-2.6.7, gcl-2.6.8pre
bookvol5 collect top level loop code
20090404.01.tpd.patch bookvol0 fix cross references
+20090404.02.tpd.patch +bookvol0 document additional domains