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F4.mw
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<?xml version="1.0" encoding="UTF-8"?>
<Worksheet>
<Version major="16" minor="1"/>
<Label-Scheme value="2" prefix=""/>
<View-Properties presentation="true"><Zoom percentage="75"/></View-Properties>
<MapleNet-Properties elisiondigitsbefore="100" labelling="true" indentamount="4" elisiontermsthreshold="10000" ansi="false" errorbreak="1" useclientjvm="true" echo="1" imaginaryunit="I" labelwidth="20" contextmenusize="automatic" plotdriver="opengl" elisiondigitsafter="100" plotoutput="terminal" helpbrowser="standard" rtablesize="10" elisiontermsbefore="100" elisiondigitsthreshold="10000" typesetting="standard" plotdevice="inline" verboseproc="1" showassumed="1" errorcursor="false" longdelim="true" plotoptions="" quiet="false" elisiontermsafter="100" screenwidth="79" preplot="" prettyprint="3" displayprecision="-1" screenpixelheight="1080" warnlevel="3" screenheight="25" latexwidth="6.0" postplot="" prompt="> " ShowLabels="true"/>
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<Metadata-table>
<Metadata-category name="<default>">
</Metadata-category>
<Metadata-tag id="0" category="<default>" name="Document Properties">
<Metadata-attribute name="Subject" value="<default>"/>
<Metadata-attribute name="Author" value="<default>"/>
<Metadata-attribute name="Title" value="<default>"/>
<Metadata-attribute name="Keywords" value="<default>"/>
</Metadata-tag>
</Metadata-table>
<Task-table>
<Task-category name="<default>">
</Task-category>
</Task-table>
<Task>
</Task><Presentation-Block>
<Group view="code" labelreference="L90" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal"># Implementation of the F4 Algorithm [1] in Maple
#
# By Pavel Trutman, pavel.trutman@fel.cvut.cz, 2015
#
# Center for Machine Perception
# Department of Cybernetics
# Faculty of Electrical Engineering
<Font encoding="UTF-8"># Czech Technical University
#
#
# [1] Jean-Charles Faugere. A new efficient algorithm for computing Groebner bases (F4). Journal of pure and applied algebra, 139(1\342\200\2233):61\342\200\22388, July 1999.
# [2] Thomas Becker and Volker Weispfenning. Groebner Bases, A Computational Approach to Commutative Algebra. Number 141 in Graduate Texts in Mathematics. Springer-Verlag, New York, NY, 1993.
</Font></Text-field>
</Input>
</Group>
<Group view="code" labelreference="L176" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">###################
# Init sequence #
###################
restart;
with(Groebner):
with(LinearAlgebra):
startTime := time[real]():
</Text-field>
</Input>
</Group>
<Group view="code" labelreference="L89" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">#######################################
# Main function of the F4 Algorithm #
#######################################
# [1] Improved Algorithm F4, page 9
F4 := proc (F::list, Sel, ordering)
local G, d, i, j, P, Pd, Ld, Fdtplus, FdTemp, FdtTemp, Fd, Fdt, GNew, h;
G := [];
d := 0;
P := {};
Fd := [{}];
Fdt := [{}];
#make pairs
for i in F do
G, P := Update(G, P, i, ordering);
end do;
while numelems(P) <> 0 do
d := d + 1;
<Font encoding="UTF-8">printf("\134n</Font>%12.2f: |P| = %d", time[real]() - startTime, numelems(P));
Pd := Sel(P);
P := `minus`(P, Pd);
Ld := {};
for i to numelems(Pd) do
Ld := `union`(Ld, {[Pd[i][2], Pd[i][3]], [Pd[i][4], Pd[i][5]]})
end do;
Fdtplus, FdTemp, FdtTemp := Reduction(Ld, G, Fd, Fdt, d, ordering);
Fd := [op(Fd), FdTemp];
Fdt := [op(Fdt), FdtTemp];
for i in Fdtplus do
#make new pairs
G, P := Update(G, P, i, ordering);
end do;
end do;
<Font encoding="UTF-8">printf("\134n%12.2f: Computing reduced Groebner basis\134n</Font>", time[real]() - startTime);
RedGroebner(G, ordering);
end proc:
</Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="code" labelreference="L102" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">########################################
# Create a pair from two polynomials #
########################################
# [1] Definition 2.5, page 5
Pair := proc (f, g, ordering)
local ret, LTf, LTg, lcmfg, tf, tg;
LTf := LeadingMonomial(f, ordering);
LTg := LeadingMonomial(g, ordering);
lcmfg := lcm(LTf, LTg);
tf := lcmfg/LTf;
tg := lcmfg/LTg;
ret := [lcmfg, tf, f, tg, g]
end proc:
</Text-field>
</Input>
</Group>
<Group view="code" labelreference="L103" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">#######################################
# Selection function - select first #
#######################################
# [1] Remark 2.5, page 8
Sel := proc (P::set)
{P[1]}
end proc:
</Text-field>
</Input>
</Group>
<Group view="code" labelreference="L155" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">##############################################
# Selection function - the normal strategy #
##############################################
# [1] function Sel, page 13
SelMinDeg := proc (P::set)
local p, d, Pd;
Pd := {};
d := degree(P[1][1]);
for p in P do
if degree(p[1]) < d then
d := degree(p[1]);
Pd := {p};
elif degree(p[1]) = d then
Pd := Pd union {p};
end if;
end do;
<Font encoding="UTF-8">printf(", Selected %d of degree %d\134n", numelems(Pd), d);</Font>
Pd;
end proc:
</Text-field>
</Input>
</Group>
<Group view="code" labelreference="L106" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">###############
# Reduction #
###############
# [1] Reduction, page 9
Reduction := proc (L::set, G::list, Fd::list, Fdt::list, d, ordering)
local F, TF, HTF, Ft, FtPol, Ftp, i, j, k, l, tmp;
printf("<Font encoding="UTF-8">%12.2f: Symbolic Preprocessing: |L| = %d, |Fd| = %d\134n", </Font>time[real]() - startTime, numelems(L), add(numelems(Fd[i]), i=1..numelems(Fd)));
F, TF := SymbolicPreprocessing(L, G, Fd, Fdt, ordering);
#make matrix to reduce
HTF := LeadingMonomial(F, ordering);
Ft := [];
for i in F do
#iterate over polynomials
tmp := [];
for j in TF do
#iterate over monomials
tmp := [op(tmp), getCoeff(i, j, ordering)];
end do;
Ft := [op(Ft), tmp]
end do;
#reduce and get polynomials back
Ft := Matrix(Ft);
<Font encoding="UTF-8">printf("%12.2f: Reducing matrix of size %dx%d\134n", time[real]() - startTime, RowDimension(Ft), ColumnDimension(Ft));</Font>
Ft := ReducedRowEchelonForm(Ft);
printf("%12.2f: <Font encoding="UTF-8">Reduction finished\134n", </Font>time[real]() - startTime);
Ft := Multiply(Ft, Vector(TF));
Ftp := {};
FtPol := [];
for i in Ft do
#skip 0 polynomials
if i <> 0 then
FtPol := [op(FtPol), i];
if not `in`(LeadingMonomial(i, ordering), HTF) then
#add leading monomials if not in HTF
Ftp := `union`(Ftp, {i})
end if
end if
end do;
Ftp, F, FtPol
end proc:
</Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="code" labelreference="L105" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">############################
# Symbolic Preprocessing #
############################
# [1] Symbolic Preprocessing, page 10
SymbolicPreprocessing := proc (L::set, G::list, Fd::list, Fdt::list, ordering)
local F, TF, HTG, i, Done, m, mp, tmpF, tmp, tmpT, tmpl, Li, Ll;
#make F from pairs
F := {};
for Li in L do
Ll := [Simplify(Li[1], Li[2], Fd, Fdt, ordering)];
F := F union {expand(Ll[1]*Ll[2])};
end do;
HTG := LeadingMonomial(G, ordering);
#fill Done
Done := {op(LeadingMonomial(F, ordering))};
#take all monomial from F
TF := {};
tmpF := F;
while numelems(tmpF) <> 0 do
tmp := expand(tmpF[1]);
tmpF := `minus`(tmpF, {tmpF[1]});
if type(tmp, `+`) then
tmp := {op(tmp)};
while numelems(tmp) <> 0 do
tmpT := [LeadingTerm(tmp[1], ordering)];
TF := `union`(TF, {tmpT[2]});
tmp := `minus`(tmp, {tmp[1]})
end do
else
TF := `union`(TF, {tmp})
end if
end do;
while not verify(TF, Done, 'set') do
#pick the first monomial
m := sort([op(`minus`(TF, Done))], proc (a, b) options operator, arrow; not TestOrder(a, b, ordering) end proc)[1];
Done := `union`(Done, {m});
i := 1;
#find m' such that m = m'*HTG ...
while i <= numelems(HTG) do
if TestOrder(HTG[i], m, ordering) and degree(HTG[i])>0 then
if lcm(m, HTG[i]) = m then
mp:=m/HTG[i];
tmpl := [Simplify(mp, G[i], Fd, Fdt, ordering)];
tmp := expand(eval(tmpl[1]*tmpl[2]));
#add to polynomial to F
F := `union`({tmp}, F);
#add monomials to TF
if type(tmp, `+`) then
tmp := {op(tmp)};
while numelems(tmp) <> 0 do
tmpT := [LeadingTerm(tmp[1], ordering)];
TF := `union`(TF, {tmpT[2]});
tmp := `minus`(tmp, {tmp[1]})
end do
else
TF := `union`(TF, {tmp})
end if;
break
end if;
end if;
i := i+1;
end do;
end do;
F, sort([op(TF)], proc (a, b) options operator, arrow; not TestOrder(a, b, ordering) end proc)
end proc:
</Text-field>
</Input>
</Group>
<Group view="code" labelreference="L131" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">#################################################
# Get coefficient of polynomial at a monomial #
#################################################
getCoeff := proc(polynomial, monomial, ordering)
local variable, coe, mon, poly, monom;
poly := polynomial;
monom := monomial;
while monom <> 1 do
if type(monom, `*`) then
variable := [op(monom)][1];
poly := coeff(poly, variable);
monom := monom/variable;
else
poly := coeff(poly, monom);
monom := 1;
end if;
end do;
coe, mon := TrailingTerm(poly, ordering):
if mon <> 1 then
coe := 0:
end if;
coe;
end proc:
</Text-field>
</Input>
</Group>
<Group view="code" labelreference="L146" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">############
# Update #
############
# [2] UPDATE, Table 5.7, page 230
Update := proc(GOld::list, POld::set, h, ordering)
local C, D, E, PNew, GNew, g1, g2, p, HTh, disjoint, condition1, condition2;
HTh := LeadingMonomial(h, ordering);
C := {op(GOld)};
D := {};
while numelems(C) <> 0 do
g1 := C[1];
C := C minus {g1};
condition1 := true;
condition2 := true;
disjoint := evalb(gcd(HTh, LeadingMonomial(g1, ordering)) = 1);
for g2 in C do
if divide(lcm(HTh, LeadingMonomial(g1, ordering)), lcm(HTh, LeadingMonomial(g2, ordering))) then
condition1 := false;
break;
end if;
end do;
for g2 in D do
if divide(lcm(HTh, LeadingMonomial(g1, ordering)), lcm(HTh, LeadingMonomial(g2, ordering))) then
condition2 := false;
break;
end if;
end do;
if disjoint or (condition1 and condition2) then
D := D union {g1};
end if;
end do;
E := {};
for g1 in D do
if evalb(gcd(HTh, LeadingMonomial(g1, ordering)) <> 1) then
E := E union {Pair(h, g1, ordering)};
end if;
end do;
PNew := {};
for p in POld do
if (not divide(p[1], HTh)) or evalb(lcm(LeadingMonomial(p[3], ordering), HTh) = p[1]) or evalb(lcm(HTh, LeadingMonomial(p[5], ordering)) = p[1]) then
PNew := PNew union {p};
end if;
end do;
PNew := PNew union E;
GNew := [];
for g1 in GOld do
if not divide(LeadingMonomial(g1, ordering), HTh) then
GNew := [op(GNew), g1];
end if;
end do;
GNew := [op(GNew), h];
GNew, PNew;
end proc:
</Text-field>
</Input>
</Group>
<Group view="code" labelreference="L152" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">##############
# Simplify #
##############
# [1] Simplify, page 10
Simplify := proc(t, f, Fd::list, Fdt::list, ordering)
local u, j, p;
for u in getDivisors(t) do
for j from 1 to numelems(Fd) do
if expand(u*f) in Fd[j] then
for p in Fdt[j] do
if evalb(LeadingMonomial(p, ordering) = LeadingMonomial(expand(u*f), ordering)) then
if u <> t then
return Simplify(t/u, p, Fd, Fdt, ordering);
else
return 1, p
end if;
end if;
end do;
end if;
end do;
end do;
t, f
end proc:
</Text-field>
</Input>
</Group>
<Group view="code" labelreference="L157" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">#####################################
# Reduction of the Groebner basis #
#####################################
# [2] REDGROEBNER, Table 5.5, page 216
RedGroebner := proc(G::list, ordering)
local H, F, f0, f, h, condition1, condition2, coeff, coefflcm, HNew;
H := [];
F := {op(G)};
while numelems(F) <> 0 do
f0 := F[1];
F := F minus {f0};
condition1 := true;
condition2 := true;
for f in F do
if divide(LeadingMonomial(f0, ordering), LeadingMonomial(f, ordering)) then
condition1 := false;
break;
end if;
end do;
for h in H do
if divide(LeadingMonomial(f0, ordering), LeadingMonomial(h, ordering)) then
condition2 := false;
break;
end if;
end do;
if condition1 and condition2 then
H := [op(H), f0];
end if;
end do;
#call Reduction on H
H := GeneralReduction(H, ordering);
# uncomment this if no fractions should appear in the resulting Groebner basis
#HNew := [];
#for h in H do
# coefflcm := 1;
# for coeff in coeffs(h) do
# if type(coeff, fraction) then
# coefflcm := lcm(coefflcm, op(coeff)[2]);
# end if;
# end do;
# h := expand(coefflcm*h);
# HNew := [op(HNew), h];
#end do;
#HNew
H
end proc:
</Text-field>
</Input>
</Group>
<Group view="code" labelreference="L158" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">########################################
# Reduction of the set of polynomial #
########################################
# [2] REDUCTION, Table 5.2, page 203
GeneralReduction := proc(P::list, ordering)
local Q, QFor, p, term, q, h, tmp, Tp, QNew, ToBreak;
Q := {op(P)};
QFor := Q;
#for polynomials in Q
while numelems(QFor) <> 0 do
ToBreak := false;
p := QFor[1];
QFor := QFor minus {p};
#get terms of p
tmp := p;
Tp := {};
if type(tmp, `+`) then
tmp := {op(tmp)};
while numelems(tmp) <> 0 do
Tp := `union`(Tp, {[LeadingTerm(tmp[1], ordering)]});
tmp := `minus`(tmp, {tmp[1]});
end do;
else
Tp := {tmp};
end if;
#for terms in p
for term in Tp do
#print("term", p, term);
#for left polynomials in Q
for q in Q minus {p} do
#reduce
if divide(term[2], LeadingMonomial(q, ordering)) then
h := expand(simplify(p - (term[1]/LeadingCoefficient(q, ordering))*(term[2]/LeadingMonomial(q, ordering))*q));
Q := Q minus {p};
if h <> 0 then
Q := Q union {h};
QFor := QFor union {h};
ToBreak := true;
break;
end if;
end if;
end do;
if ToBreak then
break;
end if;
end do;
end do;
QNew := [];
for q in Q do
if not evalb(q in QNew) then
QNew := [op(QNew), expand(q/LeadingCoefficient(q, ordering))];
end if;
end do;
QNew
end proc:
</Text-field>
</Input>
</Group>
<Group view="code" labelreference="L153" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">####################################
# Get all divisors of a monomial #
####################################
getDivisors := proc(t)
local term, vars, expvar, var, exp, ret, newret, i, j;
term := t;
if type(term, `*`) then
vars := [op(term)];
elif term <> 1 then
vars := [term];
else
return [];
end if;
ret := [1];
for expvar in vars do
if type(expvar, `^`) then
var, exp := op(expvar);
newret := [];
for i from 0 to exp do
for j in ret do
newret := [op(newret), j*var^i];
end do;
end do;
ret := newret;
else
newret := [];
for j in ret do
newret := [op(newret), j*expvar];
end do;
ret := [op(ret), op(newret)];
end if;
end do;
newret := [];
#exclude divisor 1
for i in ret do
if i <> 1 then
newret := [op(newret), i];
end if;
end do;
newret;
end proc:
</Text-field>
</Input>
</Group>
<Group view="code" labelreference="L174" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">#######################################################
# An example of calling of the implemented function #
#######################################################
startTime := time[real]():
F4([a*b*c*d - 1, a*b*c + a*b*d + a*c*d + b*c*d, a*b + b*c + a*d + c*d, a + b + c + d], SelMinDeg, tdeg(a, b, c, d));</Text-field>
</Input>
<Output>
<Text-field style="Line Printed Output" layout="Line Printed Output">
0.00: |P| = 3, Selected 1 of degree 2
0.01: Symbolic Preprocessing: |L| = 2, |Fd| = 0
0.02: Reducing matrix of size 3x7
0.02: Reduction finished
0.02: |P| = 2, Selected 1 of degree 3
0.02: Symbolic Preprocessing: |L| = 2, |Fd| = 3
0.03: Reducing matrix of size 4x9
0.03: Reduction finished
0.03: |P| = 2, Selected 2 of degree 4
0.03: Symbolic Preprocessing: |L| = 4, |Fd| = 7
0.04: Reducing matrix of size 8x12
0.05: Reduction finished
0.05: |P| = 2, Selected 2 of degree 5
0.05: Symbolic Preprocessing: |L| = 4, |Fd| = 15
0.06: Reducing matrix of size 6x12
0.06: Reduction finished
0.07: |P| = 3, Selected 3 of degree 6
0.07: Symbolic Preprocessing: |L| = 6, |Fd| = 21
0.08: Reducing matrix of size 10x14
0.09: Reduction finished
0.09: |P| = 2, Selected 2 of degree 7
0.09: Symbolic Preprocessing: |L| = 4, |Fd| = 31
0.10: Reducing matrix of size 7x10
0.10: Reduction finished
0.10: Computing reduced Groebner basis</Text-field>
</Output>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">NyksKCokSSJiRzYiIiIjIiIiKiZGJUYoSSJkR0YmRihGJyokRipGJ0YoLCpJImFHRiZGKEYlRihJImNHRiZGKEYqRigsKiomRiVGKEYuRidGKComRi5GJ0YqRihGKComRiVGKEYqRichIiIqJEYqIiIkRjMsKiomRiVGKEYqIiIlRigqJEYqIiImRihGJUYzRipGMywqKiZGLkY1RipGJ0YoKiZGLkYnRipGNUYoRi5GM0YqRjMsLComRipGOEYuRidGKComRiVGKEYuRihGKEYpRjMqJkYuRihGKkYoRihGKyEiIywuRjNGKCooRiVGKEYuRihGKkYnRigqJkYuRidGKkYnRigqJkYlRihGKkY1RjMqJkYuRihGKkY1RigqJEYqRjhGMw==</Equation></Text-field>
</Output>
</Group>
<Group view="code" labelreference="L170" drawlabel="true">
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">#################################################################################
# An example of calling of the default function implemented by J. Ch. Faugere #
#################################################################################
infolevel[GroebnerBasis] := 5:
Groebner[Basis]([a*b*c*d - 1, a*b*c + a*b*d + a*c*d + b*c*d, a*b + b*c + a*d + c*d, a + b + c + d], tdeg(a, b, c, d), method=fgb);</Text-field>
</Input>
<Output>
<Text-field style="Line Printed Output" layout="Line Printed Output">-> FGb
domain: rat_int_cof
Set offset primes to 0/20000
Lin Bk ignored [NEW lib]/S:0 -> 8/
[2](3x7)100%/
[3](5x10)100%/
[4](8x12)100%/
[5](6x12)100%/
[6](11x16)100%/
[7](8x11)100%/
Mingbasis2
(7x24)100%/restore Z1 Copy 80.00 for 24/30 exposants
{done}
SWAP Z1/2 Memory usage (estimate): 0.000
7 polynomials, 30 terms
total time: 0.012 sec
------------------------------</Text-field>
</Output>
<Output>
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</Output>
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<Text-field prompt="> " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=">JSFH</Equation></Text-field>
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