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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra groebsol.spad}
\author{Patrizia Gianni}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package GROEBSOL GroebnerSolve}
<<package GROEBSOL GroebnerSolve>>=
)abbrev package GROEBSOL GroebnerSolve
++ Author : P.Gianni, Summer '88, revised November '89
++ Solve systems of polynomial equations using Groebner bases
++ Total order Groebner bases are computed and then converted to lex ones
++ This package is mostly intended for internal use.
GroebnerSolve(lv,F,R) : C == T
where
R : GcdDomain
F : GcdDomain
lv : List Symbol
NNI ==> NonNegativeInteger
I ==> Integer
S ==> Symbol
OV ==> OrderedVariableList(lv)
IES ==> IndexedExponents Symbol
DP ==> DirectProduct(#lv,NonNegativeInteger)
DPoly ==> DistributedMultivariatePolynomial(lv,F)
HDP ==> HomogeneousDirectProduct(#lv,NonNegativeInteger)
HDPoly ==> HomogeneousDistributedMultivariatePolynomial(lv,F)
SUP ==> SparseUnivariatePolynomial(DPoly)
L ==> List
P ==> Polynomial
C == with
groebSolve : (L DPoly,L OV) -> L L DPoly
++ groebSolve(lp,lv) reduces the polynomial system lp in variables lv
++ to triangular form. Algorithm based on groebner bases algorithm
++ with linear algebra for change of ordering.
++ Preprocessing for the general solver.
++ The polynomials in input are of type \spadtype{DMP}.
testDim : (L HDPoly,L OV) -> Union(L HDPoly,"failed")
++ testDim(lp,lv) tests if the polynomial system lp
++ in variables lv is zero dimensional.
genericPosition : (L DPoly, L OV) -> Record(dpolys:L DPoly, coords: L I)
++ genericPosition(lp,lv) puts a radical zero dimensional ideal
++ in general position, for system lp in variables lv.
T == add
import Boolean
import PolToPol(lv,F)
import GroebnerPackage(F,DP,OV,DPoly)
import GroebnerInternalPackage(F,DP,OV,DPoly)
import GroebnerPackage(F,HDP,OV,HDPoly)
import LinGroebnerPackage(lv,F)
nv:NNI:=#lv
---- test if f is power of a linear mod (rad lpol) ----
---- f is monic ----
testPower(uf:SUP,x:OV,lpol:L DPoly) : Union(DPoly,"failed") ==
df:=degree(uf)
trailp:DPoly := coefficient(uf,(df-1)::NNI)
(testquo := trailp exquo (df::F)) case "failed" => "failed"
trailp := testquo::DPoly
gg:=gcd(lc:=leadingCoefficient(uf),trailp)
trailp := (trailp exquo gg)::DPoly
lc := (lc exquo gg)::DPoly
linp:SUP:=monomial(lc,1$NNI)$SUP + monomial(trailp,0$NNI)$SUP
g:DPoly:=multivariate(uf-linp**df,x)
redPol(g,lpol) ~= 0 => "failed"
multivariate(linp,x)
-- is the 0-dimensional ideal I in general position ? --
---- internal function ----
testGenPos(lpol:L DPoly,lvar:L OV):Union(L DPoly,"failed") ==
rlpol:=reverse lpol
f:=rlpol.first
#lvar=1 => [f]
rlvar:=rest reverse lvar
newlpol:List(DPoly):=[f]
for f in rlpol.rest repeat
x:=first rlvar
fi:= univariate(f,x)
if (mainVariable leadingCoefficient fi case "failed") then
if ((g:= testPower(fi,x,newlpol)) case "failed")
then return "failed"
newlpol :=concat(redPol(g::DPoly,newlpol),newlpol)
rlvar:=rest rlvar
else if not zero? redPol(f,newlpol) then return"failed"
newlpol
-- change coordinates and out the ideal in general position ----
genPos(lp:L DPoly,lvar:L OV): Record(polys:L HDPoly, lpolys:L DPoly,
coord:L I, univp:HDPoly) ==
rlvar:=reverse lvar
lnp:=[dmpToHdmp(f) for f in lp]
x := first rlvar;rlvar:=rest rlvar
testfail:=true
ranvals: L I
gb: L HDPoly
gbt: L DPoly
gb1: Union(L DPoly,"failed")
for count in 1.. while testfail repeat
ranvals := [1+(random()$I rem (count*(# lvar))) for vv in rlvar]
val:=+/[rv*(vv::HDPoly)
for vv in rlvar for rv in ranvals]
val:=val+x::HDPoly
gb := [elt(univariate(p,x),val) for p in lnp]
gb:=groebner gb
gbt:=totolex gb
(gb1:=testGenPos(gbt,lvar)) case "failed"=>"try again"
testfail:=false
[gb,gbt,ranvals,dmpToHdmp(last (gb1::L DPoly))]
genericPosition(lp:L DPoly,lvar:L OV) ==
nans:=genPos(lp,lvar)
[nans.lpolys, nans.coord]
---- select the univariate factors
select(lup:L L HDPoly) : L L HDPoly ==
lup=[] => list []
[:[cons(f,lsel) for lsel in select lup.rest] for f in lup.first]
---- in the non generic case, we compute the prime ideals ----
---- associated to leq, basis is the algebra basis ----
findCompon(leq:L HDPoly,lvar:L OV):L L DPoly ==
teq:=totolex(leq)
#teq = #lvar => [teq]
-- not ((teq1:=testGenPos(teq,lvar)) case "failed") => [teq1::L DPoly]
gp:=genPos(teq,lvar)
lgp:= gp.polys
g:HDPoly:=gp.univp
fg:=(factor g)$GeneralizedMultivariateFactorize(OV,HDP,R,F,HDPoly)
lfact:=[ff.factor for ff in factors(fg::Factored(HDPoly))]
result: L L HDPoly := []
#lfact=1 => [teq]
for tfact in lfact repeat
tlfact:=concat(tfact,lgp)
result:=concat(tlfact,result)
ranvals:L I:=gp.coord
rlvar:=reverse lvar
x:=first rlvar
rlvar:=rest rlvar
val:=+/[rv*(vv::HDPoly) for vv in rlvar for rv in ranvals]
val:=(x::HDPoly)-val
ans:=[totolex groebner [elt(univariate(p,x),val) for p in lp]
for lp in result]
[ll for ll in ans | ll~=[1]]
zeroDim?(lp: List HDPoly,lvar:L OV) : Boolean ==
empty? lp => false
n:NNI := #lvar
#lp < n => false
lvint1 := lvar
for f in lp while not empty?(lvint1) repeat
g:= f - reductum f
x:=mainVariable(g)::OV
if ground?(leadingCoefficient(univariate(g,x))) then
lvint1 := remove(x, lvint1)
empty? lvint1
-- general solve, gives an error if the system not 0-dimensional
groebSolve(leq: L DPoly,lvar:L OV) : L L DPoly ==
lnp:=[dmpToHdmp(f) for f in leq]
leq1:=groebner lnp
#(leq1) = 1 and first(leq1) = 1 => list empty()
not (zeroDim?(leq1,lvar)) =>
error "system does not have a finite number of solutions"
-- add computation of dimension, for a more useful error
basis:=computeBasis(leq1)
lup:L HDPoly:=[]
llfact:L Factored(HDPoly):=[]
for x in lvar repeat
g:=minPol(leq1,basis,x)
fg:=(factor g)$GeneralizedMultivariateFactorize(OV,HDP,R,F,HDPoly)
llfact:=concat(fg::Factored(HDPoly),llfact)
if degree(g,x) = #basis then leave "stop factoring"
result: L L DPoly := []
-- selecting a factor from the lists of the univariate factors
lfact:=select [[ff.factor for ff in factors llf]
for llf in llfact]
for tfact in lfact repeat
tfact:=groebner concat(tfact,leq1)
tfact=[1] => "next value"
result:=concat(result,findCompon(tfact,lvar))
result
-- test if the system is zero dimensional
testDim(leq : L HDPoly,lvar : L OV) : Union(L HDPoly,"failed") ==
leq1:=groebner leq
#(leq1) = 1 and first(leq1) = 1 => empty()
not (zeroDim?(leq1,lvar)) => "failed"
leq1
@
\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 GROEBSOL GroebnerSolve>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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