\documentclass{article} \usepackage{open-axiom} \begin{document} \title{\$SPAD/src/algebra indexedp.spad} \author{James Davenport} \maketitle \begin{abstract} \end{abstract} \eject \tableofcontents \eject \section{category IDPC IndexedDirectProductCategory} <>= )abbrev category IDPC IndexedDirectProductCategory ++ Author: James Davenport, Gabriel Dos Reis ++ Date Created: ++ Date Last Updated: June 28, 2010 ++ Basic Functions: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ This category represents the direct product of some set with ++ respect to an ordered indexing set. IndexedDirectProductCategory(A:SetCategory,S:OrderedSet): Category == SetCategory with map: (A -> A, %) -> % ++ map(f,z) returns the new element created by applying the ++ function f to each component of the direct product element z. monomial: (A, S) -> % ++ monomial(a,s) constructs a direct product element with the s ++ component set to \spad{a} leadingCoefficient: % -> A ++ leadingCoefficient(z) returns the coefficient of the leading ++ (with respect to the ordering on the indexing set) ++ monomial of z. ++ Error: if z has no support. leadingSupport: % -> S ++ leadingSupport(z) returns the index of leading ++ (with respect to the ordering on the indexing set) monomial of z. ++ Error: if z has no support. reductum: % -> % ++ reductum(z) returns a new element created by removing the ++ leading coefficient/support pair from the element z. ++ Error: if z has no support. terms: % -> List Pair(S,A) ++ \spad{terms x} returns the list of terms in \spad{x}. ++ Each term is a pair of a support (the first component) ++ and the corresponding value (the second component). @ \section{domain IDPO IndexedDirectProductObject} <>= )abbrev domain IDPO IndexedDirectProductObject ++ Author: James Davenport, Gabriel Dos Reis ++ Date Created: ++ Date Last Updated: June 28, 2010 ++ Description: ++ Indexed direct products of objects over a set \spad{A} ++ of generators indexed by an ordered set S. All items have finite support. IndexedDirectProductObject(A,S): Public == Private where A: SetCategory S: OrderedSet Public == IndexedDirectProductCategory(A,S) Private == add Term == Pair(S,A) Rep == List Term -- Return the index of a term termIndex(t: Term): S == first t -- Return the value of a term termValue(t: Term): A == second t x = y == x' := rep x y' := rep y while not null x' and not null y' repeat termIndex first x' ~= termIndex first y' => return false termValue first x' ~= termValue first y' => return false x' := rest x' y' := rest y' null x' and null y' coerce(x:%):OutputForm == bracket [rarrow(termIndex(t)::OutputForm, termValue(t)::OutputForm) for t in rep x] -- sample():% == [[sample()$S,sample()$A]$Term]$Rep monomial(r,s) == per [[s,r]] map(f,x) == per [[termIndex tm,f termValue tm] for tm in rep x] reductum x == per rest rep x leadingCoefficient x == null rep x => error "Can't take leadingCoefficient of empty product element" termValue first rep x leadingSupport x == null rep x => error "Can't take leadingCoefficient of empty product element" termIndex first rep x terms x == rep x @ \section{domain IDPAM IndexedDirectProductAbelianMonoid} <>= )abbrev domain IDPAM IndexedDirectProductAbelianMonoid ++ Indexed direct products of abelian monoids over an abelian monoid \spad{A} of ++ generators indexed by the ordered set S. All items have finite support. ++ Only non-zero terms are stored. IndexedDirectProductAbelianMonoid(A:AbelianMonoid,S:OrderedSet): Join(AbelianMonoid,IndexedDirectProductCategory(A,S)) == IndexedDirectProductObject(A,S) add --representations Term == Pair(S,A) termIndex(t: Term): S == first t termValue(t: Term): A == second t r: A n: NonNegativeInteger f: A -> A s: S 0 == per indexedDirectProductObject [] zero? x == null terms x qsetrest!: (List Term, List Term) -> List Term qsetrest!(l, e) == RPLACD(l, e)$Lisp -- PERFORMANCE CRITICAL; Should build list up -- by merging 2 sorted lists. Doing this will -- avoid the recursive calls (very useful if there is a -- large number of vars in a polynomial. x + y == x' := terms x y' := terms y null x' => y null y' => x endcell: List Term := nil res: List Term := nil while not empty? x' and not empty? y' repeat newcell: List Term := nil if termIndex x'.first = termIndex y'.first then r := termValue x'.first + termValue y'.first if not zero? r then newcell := cons([termIndex x'.first, r], empty()) x' := rest x' y' := rest y' else if termIndex x'.first > termIndex y'.first then newcell := cons(x'.first, empty()) x' := rest x' else newcell := cons(y'.first, empty()) y' := rest y' if not empty? newcell then if not empty? endcell then qsetrest!(endcell, newcell) endcell := newcell else res := newcell; endcell := res end := empty? x' => y' x' if empty? res then res := end else qsetrest!(endcell, end) res pretend % n * x == n = 0 => 0 n = 1 => x [[termIndex u,a] for u in terms x | not zero?(a:=n * termValue u)] pretend % monomial(r,s) == zero? r => 0 per indexedDirectProductObject [[s,r]] map(f,x) == [[termIndex tm,a] for tm in terms x | not zero?(a:=f termValue tm)] pretend % reductum x == null terms x => 0 rest(terms x) pretend % leadingCoefficient x == null terms x => 0 termValue terms(x).first pair2Term(t: Pair(A,S)): Term == [second t, first t] @ \section{domain IDPOAM IndexedDirectProductOrderedAbelianMonoid} <>= )abbrev domain IDPOAM IndexedDirectProductOrderedAbelianMonoid ++ Indexed direct products of ordered abelian monoids \spad{A} of ++ generators indexed by the ordered set S. ++ The inherited order is lexicographical. ++ All items have finite support: only non-zero terms are stored. IndexedDirectProductOrderedAbelianMonoid(A:OrderedAbelianMonoid,S:OrderedSet): Join(OrderedAbelianMonoid,IndexedDirectProductCategory(A,S)) == IndexedDirectProductAbelianMonoid(A,S) add --representations Term:= Record(k:S,c:A) Rep:= List Term x,y: % x false empty? x => true -- note careful order of these two lines y.first.k > x.first.k => true y.first.k < x.first.k => false y.first.c > x.first.c => true y.first.c < x.first.c => false x.rest < y.rest @ \section{domain IDPOAMS IndexedDirectProductOrderedAbelianMonoidSup} <>= )abbrev domain IDPOAMS IndexedDirectProductOrderedAbelianMonoidSup ++ Indexed direct products of ordered abelian monoid sups \spad{A}, ++ generators indexed by the ordered set S. ++ All items have finite support: only non-zero terms are stored. IndexedDirectProductOrderedAbelianMonoidSup(A:OrderedAbelianMonoidSup,S:OrderedSet): Join(OrderedAbelianMonoidSup,IndexedDirectProductCategory(A,S)) == IndexedDirectProductOrderedAbelianMonoid(A,S) add --representations Term:= Record(k:S,c:A) Rep:= List Term x,y: % r: A s: S subtractIfCan(x,y) == empty? y => x empty? x => "failed" x.first.k < y.first.k => "failed" x.first.k > y.first.k => t:= subtractIfCan(x.rest, y) t case "failed" => "failed" cons( x.first, t) u:=subtractIfCan(x.first.c, y.first.c) u case "failed" => "failed" zero? u => subtractIfCan(x.rest, y.rest) t:= subtractIfCan(x.rest, y.rest) t case "failed" => "failed" cons([x.first.k,u],t) sup(x,y) == empty? y => x empty? x => y x.first.k < y.first.k => cons(y.first,sup(x,y.rest)) x.first.k > y.first.k => cons(x.first,sup(x.rest,y)) u:=sup(x.first.c, y.first.c) cons([x.first.k,u],sup(x.rest,y.rest)) @ \section{domain IDPAG IndexedDirectProductAbelianGroup} <>= )abbrev domain IDPAG IndexedDirectProductAbelianGroup ++ Indexed direct products of abelian groups over an abelian group \spad{A} of ++ generators indexed by the ordered set S. ++ All items have finite support: only non-zero terms are stored. IndexedDirectProductAbelianGroup(A:AbelianGroup,S:OrderedSet): Join(AbelianGroup,IndexedDirectProductCategory(A,S)) with construct: List Pair(A,S) -> % ++ \spad{construct l} returns an object that is a linear ++ combination with support in \spad{A} and coefficients ++ in \spad{A}. == IndexedDirectProductAbelianMonoid(A,S) add --representations Term == Pair(S,A) termIndex(t: Term): S == first t termValue(t: Term): A == second t -x == [[termIndex u,-termValue u] for u in terms x] pretend % n:Integer * x:% == n = 0 => 0 n = 1 => x [[termIndex u,a] for u in terms x | not zero?(a := n * termValue u)] pretend % qsetrest!: (List Term, List Term) -> List Term qsetrest!(l, e) == RPLACD(l, e)$Lisp x - y == x' := terms x y' := terms y null x' => -y null y' => x endcell: List Term := nil res: List Term := nil while not empty? x' and not empty? y' repeat newcell: List Term := nil if termIndex x'.first = termIndex y'.first then r := termValue x'.first - termValue y'.first if not zero? r then newcell := cons([termIndex x'.first, r], empty()) x' := rest x' y' := rest y' else if termIndex x'.first > termIndex y'.first then newcell := cons(x'.first, empty()) x' := rest x' else newcell := cons([termIndex y'.first,-termValue y'.first], empty()) y' := rest y' if not empty? newcell then if not empty? endcell then qsetrest!(endcell, newcell) endcell := newcell else res := newcell; endcell := res end := empty? x' => terms(-(y' pretend %)) x' if empty? res then res := end else qsetrest!(endcell, end) res pretend % @ \section{License} <>= --Copyright (C) 1991-2002, The Numerical ALgorithms Group Ltd. --All rights reserved. --Copyright (C) 2007-2010, Gabriel Dos Reis. --All rights reserved. -- --Redistribution and use in source and binary forms, with or without --modification, are permitted provided that the following conditions are --met: -- -- - Redistributions of source code must retain the above copyright -- notice, this list of conditions and the following disclaimer. -- -- - Redistributions in binary form must reproduce the above copyright -- notice, this list of conditions and the following disclaimer in -- the documentation and/or other materials provided with the -- distribution. -- -- - Neither the name of The Numerical ALgorithms Group Ltd. nor the -- names of its contributors may be used to endorse or promote products -- derived from this software without specific prior written permission. -- --THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS --IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED --TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A --PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER --OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, --EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, --PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR --PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF --LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING --NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS --SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. @ <<*>>= <> <> <> <> <> <> <> @ \eject \begin{thebibliography}{99} \bibitem{1} nothing \end{thebibliography} \end{document}