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--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,
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--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
)abbrev package LGROBP LinGroebnerPackage
++ Given a Groebner basis B with respect to the total degree ordering for
++ a zero-dimensional ideal I, compute
++ a Groebner basis with respect to the lexicographical ordering by using
++ linear algebra.
LinGroebnerPackage(lv,F) : C == T
where
Z ==> Integer
lv : List Symbol
F : GcdDomain
DP ==> DirectProduct(#lv,NonNegativeInteger)
DPoly ==> DistributedMultivariatePolynomial(lv,F)
HDP ==> HomogeneousDirectProduct(#lv,NonNegativeInteger)
HDPoly ==> HomogeneousDistributedMultivariatePolynomial(lv,F)
OV ==> OrderedVariableList(lv)
NNI ==> NonNegativeInteger
LVals ==> Record(gblist : List DPoly,gvlist : List Z)
VF ==> Vector F
VV ==> Vector NNI
MF ==> Matrix F
cLVars ==> Record(glbase:List DPoly,glval:List Z)
C == with
linGenPos : List HDPoly -> LVals
++ linGenPos \undocumented
groebgen : List DPoly -> cLVars
++ groebgen \undocumented
totolex : List HDPoly -> List DPoly
++ totolex \undocumented
minPol : (List HDPoly,List HDPoly,OV) -> HDPoly
++ minPol \undocumented
minPol : (List HDPoly,OV) -> HDPoly
++ minPol \undocumented
computeBasis : List HDPoly -> List HDPoly
++ computeBasis \undocumented
coord : (HDPoly,List HDPoly) -> VF
++ coord \undocumented
anticoord : (List F,DPoly,List DPoly) -> DPoly
++ anticoord \undocumented
intcompBasis : (OV,List HDPoly,List HDPoly) -> List HDPoly
++ intcompBasis \undocumented
choosemon : (DPoly,List DPoly) -> DPoly
++ choosemon \undocumented
transform : DPoly -> HDPoly
++ transform \undocumented
T == add
import GroebnerPackage(F,DP,OV,DPoly)
import GroebnerPackage(F,HDP,OV,HDPoly)
import GroebnerInternalPackage(F,HDP,OV,HDPoly)
import GroebnerInternalPackage(F,DP,OV,DPoly)
lvar :=[variable(yx)::OV for yx in lv]
reduceRow(M:MF, v : VF, lastRow: Integer, pivots: Vector(Integer)) : VF ==
a1:F := 1
b:F := 0
dim := #v
for j in 1..lastRow repeat -- scan over rows
mj := row(M,j)
k:=pivots(j)
b:=mj.k
vk := v.k
for kk in 1..(k-1) repeat
v(kk) := ((-b*v(kk)) exquo a1) :: F
for kk in k..dim repeat
v(kk) := ((vk*mj(kk)-b*v(kk)) exquo a1)::F
a1 := b
v
rRedPol(f:HDPoly, B:List HDPoly):Record(poly:HDPoly, mult:F) ==
gm := redPo(f,B)
gm.poly = 0 => gm
gg := reductum(gm.poly)
ggm := rRedPol(gg,B)
[ggm.mult*(gm.poly - gg) + ggm.poly, ggm.mult*gm.mult]
----- transform the total basis B in lex basis -----
totolex(B : List HDPoly) : List DPoly ==
result:List DPoly :=[]
ltresult:List DPoly :=[]
vBasis:= computeBasis B
nBasis:List DPoly :=[1$DPoly]
ndim:=(#vBasis)::PositiveInteger
ndim1:NNI:=ndim+1
lm:VF
linmat:MF:=zero(ndim,2*ndim+1)
linmat(1,1):=1$F
linmat(1,ndim1):=1
pivots:Vector Integer := new(ndim,0)
pivots(1) := 1
firstmon:DPoly:=1$DPoly
ofirstmon:DPoly:=1$DPoly
orecfmon:Record(poly:HDPoly, mult:F) := [1,1]
i:NNI:=2
while not one?(firstmon:=choosemon(firstmon,ltresult)) repeat
if (v:=firstmon exquo ofirstmon) case "failed" then
recfmon:=rRedPol(transform firstmon,B)
else
recfmon:=rRedPol(transform(v::DPoly) *orecfmon.poly,B)
recfmon.mult := recfmon.mult * orecfmon.mult
cc := gcd(content recfmon.poly, recfmon.mult)
recfmon.poly := (recfmon.poly exquo cc)::HDPoly
recfmon.mult := (recfmon.mult exquo cc)::F
veccoef:VF:=coord(recfmon.poly,vBasis)
ofirstmon:=firstmon
orecfmon := recfmon
lm:=zero(2*ndim+1)
for j in 1..ndim repeat lm(j):=veccoef(j)
lm(ndim+i):=recfmon.mult
lm := reduceRow(linmat, lm, i-1, pivots)
j : Integer
if i=ndim1 then j:=ndim1
else
j:=1
while lm(j) = 0 and j< ndim1 repeat j:=j+1
if j=ndim1 then
cordlist:List F:=[lm(k) for k in ndim1..ndim1+(#nBasis)]
antc:=+/[c*b for c in reverse cordlist
for b in concat(firstmon,nBasis)]
antc:=primitivePart antc
result:=concat(antc,result)
ltresult:=concat(antc-reductum antc,ltresult)
else
pivots(i) := j
setRow!(linmat,i,lm)
i:=i+1
nBasis:=cons(firstmon,nBasis)
result
---- Compute the univariate polynomial for x
----oldBasis is a total degree Groebner basis
minPol(oldBasis:List HDPoly,x:OV) :HDPoly ==
algBasis:= computeBasis oldBasis
minPol(oldBasis,algBasis,x)
---- Compute the univariate polynomial for x
---- oldBasis is total Groebner, algBasis is the basis as algebra
minPol(oldBasis:List HDPoly,algBasis:List HDPoly,x:OV) :HDPoly ==
nvp:HDPoly:=x::HDPoly
f:=1$HDPoly
omult:F :=1
ndim:=(#algBasis)::PositiveInteger
ndim1:NNI:=ndim+1
lm:VF
linmat:MF:=zero(ndim,2*ndim+1)
linmat(1,1):=1$F
linmat(1,ndim1):=1
pivots:Vector Integer := new(ndim,0)
pivots(1) := 1
for i in 2..ndim1 repeat
recf:=rRedPol(f*nvp,oldBasis)
omult := recf.mult * omult
f := recf.poly
cc := gcd(content f, omult)
f := (f exquo cc)::HDPoly
omult := (omult exquo cc)::F
veccoef:VF:=coord(f,algBasis)
lm:=zero(2*ndim+1)
for j in 1..ndim repeat lm(j) := veccoef(j)
lm(ndim+i):=omult
lm := reduceRow(linmat, lm, i-1, pivots)
j: Integer :=1
while lm(j)=0 and j<ndim1 repeat j:=j+1
if j=ndim1 then return
g:HDPoly:=0
for k in ndim1..2*ndim+1 repeat
g:=g+lm(k) * nvp**((k-ndim1):NNI)
primitivePart g
pivots(i) := j
setRow!(linmat,i,lm)
----- transform a DPoly in a HDPoly -----
transform(dpol:DPoly) : HDPoly ==
dpol=0 => 0$HDPoly
monomial(leadingCoefficient dpol,
directProduct(degree(dpol)::VV)$HDP)$HDPoly +
transform(reductum dpol)
----- compute the basis for the vector space determined by B -----
computeBasis(B:List HDPoly) : List HDPoly ==
mB:List HDPoly:=[monomial(1$F,degree f)$HDPoly for f in B]
result:List HDPoly := [1$HDPoly]
for var in lvar repeat
part:=intcompBasis(var,result,mB)
result:=concat(result,part)
result
----- internal function for computeBasis -----
intcompBasis(x:OV,lr:List HDPoly,mB : List HDPoly):List HDPoly ==
lr=[] => lr
part:List HDPoly :=[]
for f in lr repeat
g:=x::HDPoly * f
if not zero?(redPo(g,mB).poly) then part:=concat(g,part)
concat(part,intcompBasis(x,part,mB))
----- coordinate of f with respect to the basis B -----
----- f is a reduced polynomial -----
coord(f:HDPoly,B:List HDPoly) : VF ==
ndim := #B
vv:VF:=new(ndim,0$F)$VF
while not zero? f repeat
rf := reductum f
lf := f-rf
lcf := leadingCoefficient f
i:Z:=position(monomial(1$F,degree lf),B)
vv.i:=lcf
f := rf
vv
----- reconstruct the polynomial from its coordinate -----
anticoord(vv:List F,mf:DPoly,B:List DPoly) : DPoly ==
for f in B for c in vv repeat (mf:=mf-c*f)
mf
----- choose the next monom -----
choosemon(mf:DPoly,nB:List DPoly) : DPoly ==
nB = [] => ((lvar.last)::DPoly)*mf
for x in reverse lvar repeat
xx:=x ::DPoly
mf:=xx*mf
if redPo(mf,nB).poly ~= 0 then return mf
dx := degree(mf,x)
mf := (mf exquo (xx ** dx))::DPoly
mf
----- put B in general position, B is Groebner -----
linGenPos(B : List HDPoly) : LVals ==
result:List DPoly :=[]
ltresult:List DPoly :=[]
vBasis:= computeBasis B
nBasis:List DPoly :=[1$DPoly]
ndim:=#vBasis : PositiveInteger
ndim1:NNI:=ndim+1
lm:VF
linmat:MF:=zero(ndim,2*ndim+1)
linmat(1,1):=1$F
linmat(1,ndim1):=1
pivots:Vector Integer := new(ndim,0)
pivots(1) := 1
i:NNI:=2
rval:List Z :=[]
for ii in 1..(#lvar-1) repeat
c:Z:=0
while c=0 repeat c:=random()$Z rem 11
rval:=concat(c,rval)
nval:DPoly := (last.lvar)::DPoly -
(+/[r*(vv)::DPoly for r in rval for vv in lvar])
firstmon:DPoly:=1$DPoly
ofirstmon:DPoly:=1$DPoly
orecfmon:Record(poly:HDPoly, mult:F) := [1,1]
lx:= lvar.last
while not one?(firstmon:=choosemon(firstmon,ltresult)) repeat
if (v:=firstmon exquo ofirstmon) case "failed" then
recfmon:=rRedPol(transform(eval(firstmon,lx,nval)),B)
else
recfmon:=rRedPol(transform(eval(v,lx,nval))*orecfmon.poly,B)
recfmon.mult := recfmon.mult * orecfmon.mult
cc := gcd(content recfmon.poly, recfmon.mult)
recfmon.poly := (recfmon.poly exquo cc)::HDPoly
recfmon.mult := (recfmon.mult exquo cc)::F
veccoef:VF:=coord(recfmon.poly,vBasis)
ofirstmon:=firstmon
orecfmon := recfmon
lm:=zero(2*ndim+1)
for j in 1..ndim repeat lm(j):=veccoef(j)
lm(ndim+i):=recfmon.mult
lm := reduceRow(linmat, lm, i-1, pivots)
j: Integer :=1
while lm(j) = 0 and j<ndim1 repeat j:=j+1
if j=ndim1 then
cordlist:List F:=[lm(j) for j in ndim1..ndim1+(#nBasis)]
antc:=+/[c*b for c in reverse cordlist
for b in concat(firstmon,nBasis)]
result:=concat(primitivePart antc,result)
ltresult:=concat(antc-reductum antc,ltresult)
else
pivots(i) := j
setRow!(linmat,i,lm)
i:=i+1
nBasis:=concat(firstmon,nBasis)
[result,rval]$LVals
----- given a basis of a zero-dimensional ideal,
----- performs a random change of coordinates
----- computes a Groebner basis for the lex ordering
groebgen(L:List DPoly) : cLVars ==
xn:=lvar.last
val := xn::DPoly
nvar1:NNI:=(#lvar-1):NNI
ll: List Z :=[random()$Z rem 11 for i in 1..nvar1]
val:=val+ +/[ll.i*(lvar.i)::DPoly for i in 1..nvar1]
LL:=[elt(univariate(f,xn),val) for f in L]
LL:= groebner(LL)
[LL,ll]$cLVars
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