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\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}
<<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.
support: % -> List Pair(S,A)
++ \spad{support x} returns the list of terms in \spad{x}.
++ Each term is a pair of an index (the first component)
++ and the corresponding value (the second component).
@
\section{domain IDPO IndexedDirectProductObject}
<<domain IDPO IndexedDirectProductObject>>=
)abbrev domain IDPO IndexedDirectProductObject
++ Data Last Updated: June 28, 2010
++ 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:SetCategory,S:OrderedSet): IndexedDirectProductCategory(A,S)
== add
--representations
Term:= Record(k:S,c:A)
Rep:= List Term
--declarations
x,y: %
f: A -> A
s: S
--define
x = y ==
while not null x and not null y repeat
x.first.k ~= y.first.k => return false
x.first.c ~= y.first.c => return false
x:=x.rest
y:=y.rest
null x and null y
coerce(x:%):OutputForm ==
bracket [rarrow(t.k :: OutputForm, t.c :: OutputForm) for t in x]
-- sample():% == [[sample()$S,sample()$A]$Term]$Rep
monomial(r,s) == [[s,r]]
map(f,x) == [[tm.k,f(tm.c)] for tm in x]
reductum x ==
rest x
leadingCoefficient x ==
null x => error "Can't take leadingCoefficient of empty product element"
x.first.c
leadingSupport x ==
null x => error "Can't take leadingCoefficient of empty product element"
x.first.k
-- FIXME: We should define Rep directly as List Pair(S,R)
-- so that this gross pretend can go away.
support x == x pretend List Pair(S,A)
@
\section{domain IDPAM IndexedDirectProductAbelianMonoid}
<<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:= Record(k:S,c:A)
Rep:= List Term
x,y: %
r: A
n: NonNegativeInteger
f: A -> A
s: S
0 == []
zero? x == null x
-- 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 ==
-- null x => y
-- null y => x
-- y.first.k > x.first.k => cons(y.first,(x + y.rest))
-- x.first.k > y.first.k => cons(x.first,(x.rest + y))
-- r:= x.first.c + y.first.c
-- r = 0 => x.rest + y.rest
-- cons([x.first.k,r],(x.rest + y.rest))
qsetrest!: (Rep, Rep) -> Rep
qsetrest!(l: Rep, e: Rep): Rep == RPLACD(l, e)$Lisp
x + y ==
null x => y
null y => x
endcell: Rep := empty()
res: Rep := empty()
while not empty? x and not empty? y repeat
newcell := empty()
if x.first.k = y.first.k then
r:= x.first.c + y.first.c
if not zero? r then
newcell := cons([x.first.k, r], empty())
x := rest x
y := rest y
else if x.first.k > y.first.k then
newcell := cons(x.first, empty())
x := rest x
else
newcell := cons(y.first, empty())
y := rest y
if not empty? newcell then
if not empty? endcell then
qsetrest!(endcell, newcell)
endcell := newcell
else
res := newcell;
endcell := res
end :=
empty? x => y
x
if empty? res then res := end
else qsetrest!(endcell, end)
res
n * x ==
n = 0 => 0
n = 1 => x
[[u.k,a] for u in x | (a:=n*u.c) ~= 0$A]
monomial(r,s) == (r = 0 => 0; [[s,r]])
map(f,x) == [[tm.k,a] for tm in x | (a:=f(tm.c)) ~= 0$A]
reductum x == (null x => 0; rest x)
leadingCoefficient x == (null x => 0; x.first.c)
@
\section{domain IDPOAM IndexedDirectProductOrderedAbelianMonoid}
<<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<y ==
empty? y => 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}
<<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}
<<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))
== IndexedDirectProductAbelianMonoid(A,S) add
--representations
Term:= Record(k:S,c:A)
Rep:= List Term
x,y: %
r: A
n: Integer
f: A -> A
s: S
-x == [[u.k,-u.c] for u in x]
n * x ==
n = 0 => 0
n = 1 => x
[[u.k,a] for u in x | (a:=n*u.c) ~= 0$A]
qsetrest!: (Rep, Rep) -> Rep
qsetrest!(l: Rep, e: Rep): Rep == RPLACD(l, e)$Lisp
x - y ==
null x => -y
null y => x
endcell: Rep := empty()
res: Rep := empty()
while not empty? x and not empty? y repeat
newcell := empty()
if x.first.k = y.first.k then
r:= x.first.c - y.first.c
if not zero? r then
newcell := cons([x.first.k, r], empty())
x := rest x
y := rest y
else if x.first.k > y.first.k then
newcell := cons(x.first, empty())
x := rest x
else
newcell := cons([y.first.k,-y.first.c], 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
-- x - y ==
-- empty? x => - y
-- empty? y => x
-- y.first.k > x.first.k => cons([y.first.k,-y.first.c],(x - y.rest))
-- x.first.k > y.first.k => cons(x.first,(x.rest - y))
-- r:= x.first.c - y.first.c
-- r = 0 => x.rest - y.rest
-- cons([x.first.k,r],(x.rest - y.rest))
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--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.
@
<<*>>=
<<license>>
<<category IDPC IndexedDirectProductCategory>>
<<domain IDPO IndexedDirectProductObject>>
<<domain IDPAM IndexedDirectProductAbelianMonoid>>
<<domain IDPOAM IndexedDirectProductOrderedAbelianMonoid>>
<<domain IDPOAMS IndexedDirectProductOrderedAbelianMonoidSup>>
<<domain IDPAG IndexedDirectProductAbelianGroup>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|