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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra nsregset.spad}
\author{Marc Moreno Maza}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{category SNTSCAT SquareFreeNormalizedTriangularSetCategory}
<<category SNTSCAT SquareFreeNormalizedTriangularSetCategory>>=
)abbrev category SNTSCAT SquareFreeNormalizedTriangularSetCategory
++ Author: Marc Moreno Maza
++ Date Created: 10/07/1998
++ Date Last Updated: 12/16/1998
++ Basic Functions:
++ Related Constructors:
++ Also See: essai Graphisme
++ AMS Classifications:
++ Keywords: polynomial, multivariate, ordered variables set
++ Description:
++ The category of square-free and normalized triangular sets.
++ Thus, up to the primitivity axiom of [1], these sets are Lazard
++ triangular sets.\newline
++ References :
++ [1] D. LAZARD "A new method for solving algebraic systems of
++ positive dimension" Discr. App. Math. 33:147-160,1991
SquareFreeNormalizedTriangularSetCategory(R:GcdDomain,E:OrderedAbelianMonoidSup,_
V:OrderedSet,P:RecursivePolynomialCategory(R,E,V)):
Category ==
Join(SquareFreeRegularTriangularSetCategory(R,E,V,P), NormalizedTriangularSetCategory(R,E,V,P))
@
\section{package LAZM3PK LazardSetSolvingPackage}
<<package LAZM3PK LazardSetSolvingPackage>>=
)abbrev package LAZM3PK LazardSetSolvingPackage
++ Author: Marc Moreno Maza
++ Date Created: 10/02/1998
++ Date Last Updated: 12/16/1998
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Description:
++ A package for solving polynomial systems by means of Lazard triangular
++ sets [1].
++ This package provides two operations. One for solving in the sense
++ of the regular zeros, and the other for solving in the sense of
++ the Zariski closure. Both produce square-free regular sets.
++ Moreover, the decompositions do not contain any redundant component.
++ However, only zero-dimensional regular sets are normalized, since
++ normalization may be time consumming in positive dimension.
++ The decomposition process is that of [2].\newline
++ References :
++ [1] D. LAZARD "A new method for solving algebraic systems of
++ positive dimension" Discr. App. Math. 33:147-160,1991
++ [2] M. MORENO MAZA "A new algorithm for computing triangular
++ decomposition of algebraic varieties" NAG Tech. Rep. 4/98.
++ Version: 1.
LazardSetSolvingPackage(R,E,V,P,TS,ST): Exports == Implementation where
R : GcdDomain
E : OrderedAbelianMonoidSup
V : OrderedSet
P : RecursivePolynomialCategory(R,E,V)
TS: RegularTriangularSetCategory(R,E,V,P)
ST : SquareFreeRegularTriangularSetCategory(R,E,V,P)
N ==> NonNegativeInteger
Z ==> Integer
B ==> Boolean
S ==> String
K ==> Fraction R
LP ==> List P
PWT ==> Record(val : P, tower : TS)
BWT ==> Record(val : Boolean, tower : TS)
LpWT ==> Record(val : (List P), tower : TS)
Split ==> List TS
--KeyGcd ==> Record(arg1: P, arg2: P, arg3: TS, arg4: B)
--EntryGcd ==> List PWT
--HGcd ==> TabulatedComputationPackage(KeyGcd, EntryGcd)
--KeyInvSet ==> Record(arg1: P, arg3: TS)
--EntryInvSet ==> List TS
--HInvSet ==> TabulatedComputationPackage(KeyInvSet, EntryInvSet)
polsetpack ==> PolynomialSetUtilitiesPackage(R,E,V,P)
regsetgcdpack ==> SquareFreeRegularTriangularSetGcdPackage(R,E,V,P,ST)
quasicomppack ==> SquareFreeQuasiComponentPackage(R,E,V,P,ST)
normalizpack ==> NormalizationPackage(R,E,V,P,ST)
Exports == with
normalizeIfCan: ST -> ST
++ \axiom{normalizeIfCan(ts)} returns \axiom{ts} in an normalized shape
++ if \axiom{ts} is zero-dimensional.
zeroSetSplit: (LP, B) -> List ST
++ \axiom{zeroSetSplit(lp,clos?)} has the same specifications as
++ \axiomOpFrom{zeroSetSplit(lp,clos?)}{RegularTriangularSetCategory}.
Implementation == add
convert(st: ST): TS ==
ts: TS := empty()
lp: LP := members(st)$ST
lp := sort(infRittWu?,lp)
for p in lp repeat
ts := internalAugment(p,ts)$TS
ts
squareFree(ts: TS): List ST ==
empty? ts => [empty()$ST]
lp: LP := members(ts)$TS
lp := sort(infRittWu?,lp)
newts: ST := empty()$ST
toSee: List ST := [newts]
toSave: List ST
for p in lp repeat
toSave := []
while (not empty? toSee) repeat
us := first toSee; toSee := rest toSee
lpwt := stoseSquareFreePart(p,us)$regsetgcdpack
for pwt in lpwt repeat
newus := internalAugment(pwt.val,pwt.tower)$ST
toSave := cons(newus,toSave)
toSee := toSave
toSave
normalizeIfCan(ts: ST): ST ==
empty? ts => ts
lp: LP := members(ts)$ST
lp := sort(infRittWu?,lp)
p: P := first lp
not univariate?(p)$polsetpack => ts
lp := rest lp
newts: ST := empty()$ST
newts := internalAugment(p,newts)$ST
while (not empty? lp) repeat
p := first lp
lv := variables(p)
for v in lv repeat
v = mvar(p) => "leave"
not algebraic?(v,newts) => return internalAugment(lp,newts)$ST
lp := rest lp
p := normalizedAssociate(p,newts)$normalizpack
newts := internalAugment(p,newts)$ST
newts
zeroSetSplit(lp:List(P), clos?:B): List ST ==
-- if clos? then SOLVE in the closure sense
toSee: Split := zeroSetSplit(lp, clos?)$TS
toSave: List ST := []
for ts in toSee repeat
toSave := concat(squareFree(ts),toSave)
toSave := removeSuperfluousQuasiComponents(toSave)$quasicomppack
[normalizeIfCan(ts) for ts in toSave]
@
\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 SNTSCAT SquareFreeNormalizedTriangularSetCategory>>
<<package LAZM3PK LazardSetSolvingPackage>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
