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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra npcoef.spad}
\author{Patrizia Gianni}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package NPCOEF NPCoef}
<<package NPCOEF NPCoef>>=
)abbrev package NPCOEF NPCoef
++ Author : P.Gianni, revised May 1990
++ Description:
++ Package for the determination of the coefficients in the lifting
++ process. Used by \spadtype{MultivariateLifting}.
++ This package will work for every euclidean domain R which has property
++ F, i.e. there exists a factor operation in \spad{R[x]}.
NPCoef(BP,E,OV,R,P) : C == T where
OV : OrderedSet
E : OrderedAbelianMonoidSup
R : EuclideanDomain -- with property "F"
BP : UnivariatePolynomialCategory R
P : PolynomialCategory(R,E,OV)
Z ==> Integer
NNI ==> NonNegativeInteger
USP ==> SparseUnivariatePolynomial(P)
Term ==> Record(expt:NNI,pcoef:P)
Detc ==> Record(valexp:NNI,valcoef:P,posit:NNI)
VTerm ==> List(Term)
DetCoef ==> Record(deter:List(USP),dterm:List(VTerm),
nfacts:List(BP),nlead:List(P))
TermC ==> Record(coefu:P,detfacts:List(VTerm))
TCoef ==> List(TermC)
C == with
npcoef : (USP,List(BP),List(P)) -> DetCoef
++ npcoef \undocumented
listexp : BP -> List(NNI)
++ listexp \undocumented
T == add
---- Local Functions ----
check : (TermC,Vector P) -> Union(Detc,"failed")
buildvect : (List(VTerm),NNI) -> Vector(List(VTerm))
buildtable : (Vector(P),List(List NNI),List P) -> TCoef
modify : (TCoef,Detc) -> TCoef
constructp : VTerm -> USP
npcoef(u:USP,factlist:List(BP),leadlist:List(P)) :DetCoef ==
detcoef:List(VTerm):=empty();detufact:List(USP):=empty()
lexp:List(List(NNI)):=[listexp(v) for v in factlist]
ulist :Vector(P):=vector [coefficient(u,i) for i in 0..degree u]
tablecoef:=buildtable(ulist,lexp,leadlist)
detcoef:=[[[ep.first,lcu]$Term] for ep in lexp for lcu in leadlist]
ldtcf:=detcoef
lexp:=[ep.rest for ep in lexp]
ndet:NNI:=#factlist
changed:Boolean:=true
ltochange:List(NNI):=empty()
ltodel:List(NNI):=empty()
while changed and ndet~=1 repeat
changed :=false
dt:=#tablecoef
for i in 1..dt while not changed repeat
(cf:=check(tablecoef.i,ulist)) case "failed" => "next i"
ltochange:=cons(i,ltochange)
celtf:Detc:=cf::Detc
tablecoef:=modify(tablecoef,celtf)
vpos:=celtf.posit
vexp:=celtf.valexp
nterm:=[vexp,celtf.valcoef]$Term
detcoef.vpos:=cons(nterm,detcoef.vpos)
lexp.vpos:=delete(lexp.vpos,position(vexp,lexp.vpos))
if lexp.vpos=[] then
ltodel:=cons(vpos,ltodel)
ndet:=(ndet-1):NNI
detufact:=cons(constructp(detcoef.vpos),detufact)
changed:=true
for i in ltochange repeat tablecoef:=delete(tablecoef,i)
ltochange:=[]
if ndet=1 then
uu:=u exquo */[pol for pol in detufact]
if uu case "failed" then return
[empty(),ldtcf,factlist,leadlist]$DetCoef
else detufact:=cons(uu::USP,detufact)
else
ltodel:=sort(#1>#2,ltodel)
for i in ltodel repeat
detcoef:=delete(detcoef,i)
factlist:=delete(factlist,i)
leadlist:=delete(leadlist,i)
[detufact,detcoef,factlist,leadlist]$DetCoef
check(tterm:TermC,ulist:Vector(P)) : Union(Detc,"failed") ==
cfu:P:=1$P;doit:NNI:=0;poselt:NNI:=0;pp:Union(P,"failed")
termlist:List(VTerm):=tterm.detfacts
vterm:VTerm:=empty()
#termlist=1 =>
vterm:=termlist.first
for elterm in vterm while doit<2 repeat
(cu1:=elterm.pcoef)~=0 => cfu:=cu1*cfu
doit:=doit+1
poselt:=position(elterm,vterm):NNI
doit=2 or (pp:=tterm.coefu exquo cfu) case "failed" => "failed"
[vterm.poselt.expt,pp::P,poselt]$Detc
"failed"
buildvect(lvterm:List(VTerm),n:NNI) : Vector(List(VTerm)) ==
vtable:Vector(List(VTerm)):=new(n,empty())
(#lvterm)=1 =>
for term in lvterm.first repeat vtable.(term.expt+1):=[[term]]
vtable
vtable:=buildvect(lvterm.rest,n)
ntable:Vector(List(VTerm)):=new(n,empty())
for term in lvterm.first repeat
nexp:=term.expt
for i in 1..n while (nexp+i)<(n+1) repeat
ntable.(nexp+i):=append(
[cons(term,lvterm) for lvterm in vtable.i],
ntable.(nexp+i))
ntable
buildtable(vu:Vector(P),lvect:List(List(NNI)),leadlist:List(P)):TCoef==
nfact:NNI:=#leadlist
table:TCoef:=empty()
degu:=(#vu-1)::NNI
prelim:List(VTerm):=[[[e,0$P]$Term for e in lv] for lv in lvect]
for i in 1..nfact repeat prelim.i.first.pcoef:=leadlist.i
partialv:Vector(List(VTerm)):=new(nfact,empty())
partialv:=buildvect(prelim,degu)
for i in 1..degu repeat
empty? partialv.i => "next i"
table:=cons([vu.i,partialv.i]$TermC, table)
table
modify(tablecoef:TCoef,cfter:Detc) : TCoef ==
cfexp:=cfter.valexp;cfcoef:=cfter.valcoef;cfpos:=cfter.posit
lterase:List(NNI):=empty()
for cterm in tablecoef | not empty?(ctdet:=cterm.detfacts) repeat
(+/[term.expt for term in ctdet.first])<cfexp => "next term"
for celt in ctdet repeat
if celt.cfpos.expt=cfexp then
celt.cfpos.pcoef:=cfcoef
if (and/[cc.pcoef ~=0 for cc in celt]) then
k:=position(celt,ctdet):NNI
lterase:=cons(k,lterase)
cterm.coefu:=(cterm.coefu - */[cc.pcoef for cc in celt])
if not empty? lterase then
lterase:=sort(#1>#2,lterase)
for i in lterase repeat ctdet:=delete(ctdet,i)
cterm.detfacts:=ctdet
lterase:=empty()
tablecoef
listexp(up:BP) :List(NNI) ==
degree up=0 => [0]
[degree up,:listexp(reductum up)]
constructp(lterm:VTerm):USP ==
+/[monomial(term.pcoef,term.expt) for term in lterm]
@
\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 NPCOEF NPCoef>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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