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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra facutil.spad}
\author{Barry Trager}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package FACUTIL FactoringUtilities}
<<package FACUTIL FactoringUtilities>>=
)abbrev package FACUTIL FactoringUtilities
++ Author: Barry Trager
++ Date Created: March 12, 1992
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This package provides utilities used by the factorizers
++ which operate on polynomials represented as univariate polynomials
++ with multivariate coefficients.
FactoringUtilities(E,OV,R,P) : C == T where
E : OrderedAbelianMonoidSup
OV : OrderedSet
R : Ring
P : PolynomialCategory(R,E,OV)
SUP ==> SparseUnivariatePolynomial
NNI ==> NonNegativeInteger
Z ==> Integer
C == with
completeEval : (SUP P,List OV,List R) -> SUP R
++ completeEval(upoly, lvar, lval) evaluates the polynomial upoly
++ with each variable in lvar replaced by the corresponding value
++ in lval. Substitutions are done for all variables in upoly
++ producing a univariate polynomial over R.
degree : (SUP P,List OV) -> List NNI
++ degree(upoly, lvar) returns a list containing the maximum
++ degree for each variable in lvar.
variables : SUP P -> List OV
++ variables(upoly) returns the list of variables for the coefficients
++ of upoly.
lowerPolynomial: SUP P -> SUP R
++ lowerPolynomial(upoly) converts upoly to be a univariate polynomial
++ over R. An error if the coefficients contain variables.
raisePolynomial: SUP R -> SUP P
++ raisePolynomial(rpoly) converts rpoly from a univariate polynomial
++ over r to be a univariate polynomial with polynomial coefficients.
normalDeriv : (SUP P,Z) -> SUP P
++ normalDeriv(poly,i) computes the ith derivative of poly divided
++ by i!.
ran : Z -> R
++ ran(k) computes a random integer between -k and k as a member of R.
T == add
lowerPolynomial(f:SUP P) : SUP R ==
zero? f => 0$SUP(R)
monomial(ground leadingCoefficient f, degree f)$SUP(R) +
lowerPolynomial(reductum f)
raisePolynomial(u:SUP R) : SUP P ==
zero? u => 0$SUP(P)
monomial(leadingCoefficient(u)::P, degree u)$SUP(P) +
raisePolynomial(reductum u)
completeEval(f:SUP P,lvar:List OV,lval:List R) : SUP R ==
zero? f => 0$SUP(R)
monomial(ground eval(leadingCoefficient f,lvar,lval),degree f)$SUP(R) +
completeEval(reductum f,lvar,lval)
degree(f:SUP P,lvar:List OV) : List NNI ==
coefs := coefficients f
ldeg:= ["max"/[degree(fc,xx) for fc in coefs] for xx in lvar]
variables(f:SUP P) : List OV ==
"setUnion"/[variables cf for cf in coefficients f]
if R has FiniteFieldCategory then
ran(k:Z):R == random()$R
else
ran(k:Z):R == (random(2*k+1)$Z -k)::R
-- Compute the normalized m derivative
normalDeriv(f:SUP P,m:Z) : SUP P==
(n1:Z:=degree f) < m => 0$SUP(P)
n1=m => (leadingCoefficient f)::SUP(P)
k:=binomial(n1,m)
ris:SUP:=0$SUP(P)
n:Z:=n1
while n>= m repeat
while n1>n repeat
k:=(k*(n1-m)) quo n1
n1:=n1-1
ris:=ris+monomial(k*leadingCoefficient f,(n-m)::NNI)
f:=reductum f
n:=degree f
ris
@
\section{package PUSHVAR PushVariables}
<<package PUSHVAR PushVariables>>=
)abbrev package PUSHVAR PushVariables
++ This package \undocumented{}
PushVariables(R,E,OV,PPR):C == T where
E : OrderedAbelianMonoidSup
OV: OrderedSet with
convert: % -> Symbol
++ convert(x) converts x to a symbol
variable: Symbol -> Union(%, "failed")
++ variable(s) makes an element from symbol s or fails
R : Ring
PR ==> Polynomial R
PPR: PolynomialCategory(PR,E,OV)
SUP ==> SparseUnivariatePolynomial
C == with
pushdown : (PPR, OV) -> PPR
++ pushdown(p,v) \undocumented{}
pushdown : (PPR, List OV) -> PPR
++ pushdown(p,lv) \undocumented{}
pushup : (PPR, OV) -> PPR
++ pushup(p,v) \undocumented{}
pushup : (PPR, List OV) -> PPR
++ pushup(p,lv) \undocumented{}
map : ((PR -> PPR), PPR) -> PPR
++ map(f,p) \undocumented{}
T == add
pushdown(g:PPR,x:OV) : PPR ==
eval(g,x,monomial(1,convert x,1)$PR)
pushdown(g:PPR, lv:List OV) : PPR ==
vals:=[monomial(1,convert x,1)$PR for x in lv]
eval(g,lv,vals)
map(f:(PR -> PPR), p: PPR) : PPR ==
ground? p => f(retract p)
v:=mainVariable(p)::OV
multivariate(map(map(f,#1),univariate(p,v)),v)
---- push back the variable ----
pushupCoef(c:PR, lv:List OV): PPR ==
ground? c => c::PPR
v:=mainVariable(c)::Symbol
v2 := variable(v)$OV
uc := univariate(c,v)
ppr : PPR := 0
v2 case OV =>
while not zero? uc repeat
ppr := ppr + monomial(1,v2,degree(uc))$PPR *
pushupCoef(leadingCoefficient uc, lv)
uc := reductum uc
ppr
while not zero? uc repeat
ppr := ppr + monomial(1,v,degree(uc))$PR *
pushupCoef(leadingCoefficient uc, lv)
uc := reductum uc
ppr
pushup(f:PPR,x:OV) :PPR ==
map(pushupCoef(#1,[x]), f)
pushup(g:PPR, lv:List OV) : PPR ==
map(pushupCoef(#1, lv), g)
@
\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>>
<<package FACUTIL FactoringUtilities>>
<<package PUSHVAR PushVariables>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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