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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra wtpol.spad}
\author{James Davenport}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain WP WeightedPolynomials}
<<domain WP WeightedPolynomials>>=
)abbrev domain WP WeightedPolynomials
++ Author: James Davenport
++ Date Created: 17 April 1992
++ Date Last Updated: 12 July 1992
++ Basic Functions: Ring, changeWeightLevel
++ Related Constructors: PolynomialRing
++ Also See: OrdinaryWeightedPolynomials
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This domain represents truncated weighted polynomials over a general
++ (not necessarily commutative) polynomial type. The variables must be
++ specified, as must the weights.
++ The representation is sparse
++ in the sense that only non-zero terms are represented.
WeightedPolynomials(R:Ring,VarSet: OrderedSet, E:OrderedAbelianMonoidSup,
P:PolynomialCategory(R,E,VarSet),
vl:List VarSet, wl:List NonNegativeInteger,
wtlevel:NonNegativeInteger):
Ring with
if R has CommutativeRing then Algebra(R)
coerce: $ -> P
++ convert back into a "P", ignoring weights
if R has Field then "/": ($,$) -> Union($,"failed")
++ x/y division (only works if minimum weight
++ of divisor is zero, and if R is a Field)
coerce: P -> $
++ coerce(p) coerces p into Weighted form, applying weights and ignoring terms
changeWeightLevel: NonNegativeInteger -> Void
++ changeWeightLevel(n) changes the weight level to the new value given:
++ NB: previously calculated terms are not affected
==
add
--representations
Rep := PolynomialRing(P,NonNegativeInteger)
p:P
w,x1,x2:$
n:NonNegativeInteger
z:Integer
changeWeightLevel(n) ==
wtlevel:=n
lookupList:List Record(var:VarSet, weight:NonNegativeInteger)
if #vl ~= #wl then error "incompatible length lists in WeightedPolynomial"
lookupList:=[[v,n] for v in vl for n in wl]
-- local operation
innercoerce: (P,Integer) -> $
lookup:VarSet -> NonNegativeInteger
lookup v ==
l:=lookupList
while l ~= [] repeat
v = l.first.var => return l.first.weight
l:=l.rest
0
innercoerce(p,z) ==
z<0 => 0
zero? p => 0
mv:= mainVariable p
mv case "failed" => monomial(p,0)
n:=lookup(mv)
up:=univariate(p,mv)
ans:$
ans:=0
while not zero? up repeat
d:=degree up
f:=n*d
lcup:=leadingCoefficient up
up:=up-leadingMonomial up
mon:=monomial(1,mv,d)
f<=z =>
tmp:= innercoerce(lcup,z-f)
while not zero? tmp repeat
ans:=ans+ monomial(mon*leadingCoefficient(tmp),degree(tmp)+f)
tmp:=reductum tmp
ans
coerce(p):$ == innercoerce(p,wtlevel)
coerce(w):P == "+"/[c for c in coefficients w]
coerce(p:$):OutputForm ==
zero? p => (0$Integer)::OutputForm
degree p = 0 => leadingCoefficient(p):: OutputForm
reduce("+",(reverse [paren(c::OutputForm) for c in coefficients p])
::List OutputForm)
0 == 0$Rep
1 == 1$Rep
x1 = x2 ==
-- Note that we must strip out any terms greater than wtlevel
while degree x1 > wtlevel repeat
x1 := reductum x1
while degree x2 > wtlevel repeat
x2 := reductum x2
x1 =$Rep x2
x1 + x2 == x1 +$Rep x2
-x1 == -(x1::Rep)
x1 * x2 ==
-- Note that this is probably an extremely inefficient definition
w:=x1 *$Rep x2
while degree(w) > wtlevel repeat
w:=reductum w
w
@
\section{domain OWP OrdinaryWeightedPolynomials}
<<domain OWP OrdinaryWeightedPolynomials>>=
)abbrev domain OWP OrdinaryWeightedPolynomials
++ Author: James Davenport
++ Date Created: 17 April 1992
++ Date Last Updated: 12 July 1992
++ Basic Functions: Ring, changeWeightLevel
++ Related Constructors: WeightedPolynomials
++ Also See: PolynomialRing
++ AMS classifications:
++ Keywords:
++ References:
++ Description:
++ This domain represents truncated weighted polynomials over the
++ "Polynomial" type. The variables must be
++ specified, as must the weights.
++ The representation is sparse
++ in the sense that only non-zero terms are represented.
OrdinaryWeightedPolynomials(R:Ring,
vl:List Symbol, wl:List NonNegativeInteger,
wtlevel:NonNegativeInteger):
Ring with
if R has CommutativeRing then Algebra(R)
coerce: $ -> Polynomial(R)
++ coerce(p) converts back into a Polynomial(R), ignoring weights
coerce: Polynomial(R) -> $
++ coerce(p) coerces a Polynomial(R) into Weighted form,
++ applying weights and ignoring terms
if R has Field then "/": ($,$) -> Union($,"failed")
++ x/y division (only works if minimum weight
++ of divisor is zero, and if R is a Field)
changeWeightLevel: NonNegativeInteger -> Void
++ changeWeightLevel(n) This changes the weight level to the new value given:
++ NB: previously calculated terms are not affected
== WeightedPolynomials(R,Symbol,IndexedExponents(Symbol),
Polynomial(R),
vl,wl,wtlevel)
@
\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>>
<<domain WP WeightedPolynomials>>
<<domain OWP OrdinaryWeightedPolynomials>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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