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#############################################################################
##
#W linalg.gi GAP 4 package `cvec'
## Max Neunhoeffer
##
## Copyright (C) 2007 Max Neunhoeffer, Lehrstuhl D f. Math., RWTH Aachen
## This file is free software, see license information at the end.
##
## This file contains the higher levels for efficient implementation of
## some linear algebra routines for compact vectors over finite fields.
##
#############################################################################
# Creation of semi-echelonised basis:
#############################################################################
InstallMethod( SEBMaker, "generic method",
[ IsRowListMatrix, IsList],
function( vectors, pivots )
local b;
b := rec( vectors := vectors, pivots := pivots );
return Objectify(NewType(FamilyObj(b.vectors),SEBType and IsMutable),b);
end );
InstallMethod( SEBMaker, "for a cmat, and an integer list",
[ IsCMatRep, IsList ],
function( vectors, pivots )
local b,cl;
b := rec( vectors := vectors, pivots := pivots );
cl := CVEC_NewCVecClassSameField(b.vectors!.vecclass,1);
b.helper := CVEC_New(cl);
return Objectify(NewType(FamilyObj(b.vectors),
SEBType and IsMutable and IsCMatSEB),b);
end );
InstallOtherMethod( SEBMaker, "empty plist method",
[ IsList and IsEmpty, IsList and IsEmpty ],
function( vectors, pivots )
local b;
b := rec( vectors := vectors, pivots := pivots );
return Objectify(NewType(FamilyObj(b.vectors),SEBType and IsMutable),b);
end );
InstallMethod( SemiEchelonBasisMutable, "for a semi echelonised basis record",
[IsRecord],
function(r)
local i, b;
b := rec( vectors := r.vectors, pivots := [] );
for i in [1..NumberColumns(b.vectors)] do
if IsBound(r.heads[i]) and r.heads[i] > 0 then
b.pivots[r.heads[i]] := i;
fi;
od;
return SEBMaker(b.vectors,b.pivots);
end);
InstallMethod( EmptySemiEchelonBasis, "for a row list matrix",
[ IsRowListMatrix ],
function( m )
return SEBMaker( Matrix([],NumberColumns(m),m), [] );
end );
InstallMethod( EmptySemiEchelonBasis, "for a row vector",
[ IsRowVector ],
function( v )
return EmptySemiEchelonBasis(CMat([v]));
end );
InstallMethod( Vectors, "for a semi echelonsised basis", [ SEBType ],
function(b)
return b!.vectors;
end );
InstallMethod( BasisVectors, "for a semi echelonised basis", [ SEBType ],
function(b)
return b!.vectors;
end );
InstallMethod( Length, "for a semi echelonised basis", [ SEBType ],
function(b)
return Length(b!.vectors);
end );
InstallMethod( Pivots, "for a semi echelonised basis", [ SEBType ],
function(b)
return b!.pivots;
end );
InstallMethod( ViewObj, "for a semi echelonised basis",
[ SEBType ],
function(b)
Print("<");
if not(IsMutable(b)) or not(IsMutable(b!.vectors)) then
Print("immutable ");
fi;
if Length(b!.vectors) = 0 then
Print("empty semi echelonized basis>");
else
Print("semi echelonized basis over ",BaseDomain(b!.vectors)," of length ",
NumberRows(b!.vectors),">");
fi;
end);
InstallMethod( Display, "for a semi echelonised basis",
[ SEBType ],
function(b)
Print("<");
if not(IsMutable(b)) or not(IsMutable(b!.vectors)) then
Print("immutable ");
fi;
if Length(b!.vectors) = 0 then
Print("empty semi echelonized basis>");
else
Print("semi echelonized basis over ",BaseDomain(b!.vectors),
" of length ", NumberRows(b!.vectors),"\n");
Display(b!.vectors);
Print(">");
fi;
end );
#############################################################################
# High speed cleaning of vectors, semi-echelonised matrices:
#############################################################################
BindGlobal( "CVEC_CleanRow", function( basis, vec, extend, dec)
local c,firstnz,j,lc,len,lev,mo,newpiv,pivs;
# INPUT
# basis: component object with fields
# pivots : integer list of pivot columns of basis matrix
# vectors : matrix of basis vectors in semi echelon form
# vec : vector of same length as basis vectors
# extend : boolean value indicating whether the basis will we extended
# note that for the greased case also the basis vectors before
# the new one may be changed
# OUTPUT
# returns decomposition of vec in the basis, if possible.
# otherwise returns 'fail' and adds cleaned vec to basis and updates
# pivots
# NOTES
# destructive in both arguments
# Clear dec vector if given:
if dec <> fail then
MultVector(dec,Zero(dec[1]));
fi;
# First a little shortcut:
firstnz := PositionNonZero(vec);
if firstnz > Length(vec) then
return true;
fi;
len := Length(basis!.vectors);
if len = 1 and IsPlistRep(basis!.vectors) then
ConvertToMatrixRep(basis!.vectors);
fi;
for j in [1..len] do
if basis!.pivots[j] >= firstnz then
c := vec[ basis!.pivots[ j ] ];
if not IsZero( c ) then
if dec <> fail then
dec[ j ] := c;
fi;
AddRowVector( vec, basis!.vectors[ j ], -c );
fi;
#firstnz := basis!.pivots[j]+1;
fi;
od;
newpiv := PositionNonZero( vec );
if newpiv = Length( vec ) + 1 then
return true;
else
if extend then
c := vec[newpiv];
if dec <> fail then
dec[len+1] := c;
fi;
MultVector( vec, c^-1 );
Add( basis!.vectors, vec );
Add( basis!.pivots, newpiv );
fi;
return false;
fi;
end );
InstallMethod( CleanRow,
"GAP method for a semi echelon basis, a vector, and a boolean or vector",
[SEBType and IsMutable, IsVectorObj, IsBool, IsObject], CVEC_CleanRow );
InstallMethod( CleanRow,
"kernel method for a record, a cvec, and a boolean or cvec",
[SEBType and IsMutable and IsCMatSEB, IsCVecRep, IsBool, IsObject],
CVEC_CLEANROWKERNEL );
InstallMethod( IO_Pickle, "for a file and a cmat semi echelonized basis",
[ IsFile, IsCMatSEB and IsList and IsSemiEchelonized ],
function( f, seb )
return IO_GenericObjectPickler(f,"CSEB",[seb!.vectors,seb!.pivots],
seb,[],[],[]);
end );
IO_Unpicklers.CSEB :=
function( f )
local vectors, pivots, seb;
vectors := IO_Unpickle(f);
if vectors = IO_Error then return IO_Error; fi;
pivots := IO_Unpickle(f);
if vectors = IO_Error then return IO_Error; fi;
seb := SEBMaker(vectors,pivots);
IO_GenericObjectUnpickler(f,seb,[],[]);
return seb;
end;
InstallMethod( Coefficients, "for a semi echelonized basis, and a vector",
[SEBType, IsVectorObj],
function(b,v)
local dec;
dec := ZeroVector(Length(b),v);
if CleanRow(b,ShallowCopy(v),false,dec) then
return dec;
else
return fail;
fi;
end );
InstallMethod( LinearCombination, "for a semi echelonized basis, and a vector",
[SEBType, IsVectorObj],
function(b,v)
return v * b!.vectors;
end );
#############################################################################
# Computing semi-echelonised bases from matrices:
#############################################################################
InstallMethod( SemiEchelonBasisMutableX, "for a row list matrix",
[IsRowListMatrix and IsMutable],
function( m )
local b,v;
b := EmptySemiEchelonBasis(m);
for v in m do CleanRow(b,v,true,fail); od;
return b;
end );
InstallMethod( SemiEchelonBasisMutable, "for a row list matrix",
[IsRowListMatrix],
function( m )
return SemiEchelonBasisMutableX( MutableCopyMat( m ) );
end );
InstallMethod( SemiEchelonBasisMutableX, "for a row list matrix",
[IsRowListMatrix],
function( m )
return SemiEchelonBasisMutableX( MutableCopyMat( m ) );
end );
InstallMethod( SemiEchelonBasisMutableTX, "for a row list matrix",
[IsRowListMatrix and IsMutable],
function( m )
local b,coeffs,dec,i,j,mo,newcoeffs,newrelation,relations,rl,s,v,zerov;
rl := NumberColumns(m);
b := EmptySemiEchelonBasis( m );
zerov := ZeroVector(rl,m);
dec := ZeroVector(Length(m),zerov); # Maximal length of the basis
if Length(m) = 0 then
b!.coeffs := Matrix([],0,m);
b!.relations := rec( vectors := Matrix([],0,m), pivots := [] );
return b;
fi;
# In the end, we want: coeffs * m = b!.vectors:
coeffs := Matrix([],Length(m),m);
# In the end, we want relations to be a basis for the nullspace of m:
relations := Matrix([],Length(m),m);
i := 0; # this is the length of coeffs
mo := -One(BaseDomain(m));
for j in [1..Length(m)] do
v := m[j];
if not CleanRow(b,v,true,dec) then
# a new vector in the basis, we have to produce a coeff line:
# now dec * b!.vectors = v (original one)
# need: coeffs * mat = b!.vectors[Length(b!.vectors)]
# ==> we use b!.vectors[Length(b!.vectors)]
# = (v-dec{[1..i]}*b!.vectors)/dec[i+1]
if i > 0 then
s := dec[i+1]^-1;
newcoeffs := ((-s) * dec{[1..i]}) * coeffs;
newcoeffs[j] := s;
Add(coeffs,newcoeffs);
else
newcoeffs := ZeroMutable(dec);
newcoeffs[j] := dec[1]^-1;
Add(coeffs,newcoeffs);
fi;
i := i + 1;
else
if i > 0 then
newrelation := dec{[1..i]} * coeffs;
newrelation[j] := mo;
else
newrelation := ZeroMutable(dec);
newrelation[j] := mo;
fi;
Add(relations,newrelation);
fi;
od;
b!.coeffs := coeffs;
b!.relations := relations;
return b;
end);
InstallMethod( SemiEchelonBasisMutableT, "for a row list matrix",
[IsRowListMatrix],
function( m )
return SemiEchelonBasisMutableTX( MutableCopyMat( m ) );
end );
InstallMethod( SemiEchelonBasisMutableTX, "for an immutable row list matrix",
[IsRowListMatrix],
function( m )
return SemiEchelonBasisMutableTX( MutableCopyMat( m ) );
end );
InstallMethod( SemiEchelonBasisMutablePX, "for a row list matrix",
[IsRowListMatrix and IsMutable],
function( m )
# In the end we want b!.p * b!.vectors = m
local b,p,dec,j,v,zerov, decl;
b := EmptySemiEchelonBasis( m );
zerov := ZeroVector(NumberColumns(m),m);
decl := Minimum( 100, Length(m) );
dec := ZeroVector(decl,zerov);
if Length(m) = 0 then
b!.p := Matrix( [], 0, m );
return b;
fi;
p := []; # we collect the vectors in a list because we do not know
# their final length
for j in [1..Length(m)] do
v := m[j];
if Length( b!.vectors ) >= decl then
decl := Maximum( decl + 100, Length(m) );
dec := ZeroVector( decl, dec );
fi;
CleanRow(b,v,true,dec);
Add( p, ShallowCopy( dec ) );
od;
decl := Length( b!.vectors );
b!.p := Matrix([],decl,m);
j := 1;
while j <= Length( p ) and Length( p[ j ] ) < decl do
dec := ZeroVector( decl, dec );
CopySubVector( p[ j ], dec, [1..Length(p[j])],[1..Length(p[j])] );
Add(b!.p,dec);
j := j + 1;
od;
while j <= Length( p ) do
Add(b!.p,p[ j ]{[1..decl]});
j := j + 1;
od;
return b;
end);
InstallMethod( SemiEchelonBasisMutableP, "for a row list matrix",
[IsRowListMatrix],
function( m )
return SemiEchelonBasisMutablePX( MutableCopyMat( m ) );
end );
InstallMethod( SemiEchelonBasisMutablePX, "for an immutable row list matrix",
[IsRowListMatrix],
function( m )
return SemiEchelonBasisMutablePX( MutableCopyMat( m ) );
end );
InstallMethod( NullspaceMatMutableX, "for an immutable row list matrix",
[IsRowListMatrix],
function( m )
local b;
b := SemiEchelonBasisMutableTX( MutableCopyMat( m ) );
return b!.relations;
end );
InstallMethod( NullspaceMatMutableX, "for a row list matrix",
[IsRowListMatrix and IsMutable],
function( m )
local b;
b := SemiEchelonBasisMutableTX(m);
return b!.relations;
end );
InstallMethod( NullspaceMatMutable, "for a row list matrix",
[IsRowListMatrix],
function( m )
return NullspaceMatMutableX( MutableCopyMat( m ) );
end );
InstallOtherMethod( NullspaceMatDestructive, "for a cmat",
[IsCMatRep and IsOrdinaryMatrix],
function( m )
local res;
res := NullspaceMatMutableX( m );
MakeImmutable(res);
return res;
end );
InstallOtherMethod( NullspaceMat, "for a cmat",
[IsCMatRep and IsOrdinaryMatrix],
function( m )
local res;
res := NullspaceMatMutableX( MutableCopyMat( m ) );
MakeImmutable(res);
return res;
end );
InstallMethod( SemiEchelonBasisNullspaceX, "for a row list matrix",
[IsRowListMatrix],
function( m )
local n;
n := NullspaceMatMutableX( m );
return SemiEchelonBasisMutableX( n );
end );
InstallMethod( SemiEchelonBasisNullspace, "for a row list matrix",
[IsRowListMatrix],
function( m )
local n;
n := NullspaceMatMutable( m );
return SemiEchelonBasisMutableX( n );
end );
#############################################################################
# Intersections and sums of spaces given by bases:
#############################################################################
InstallMethod( IntersectionAndSumRowSpaces, "for two semi echelonised bases",
[ SEBType and IsMutable, SEBType and IsMutable ],
function( b1, b2 )
# Here is a proof why the following works. It is obviously better
# than Zassenhaus' algorithm as it only works with vectors of the
# original size rather than vectors of double length. It also
# can profit from b1 and b2 already being semi-echelonised.
#
# Let V be the full row space and W be the row space generated by b1.
# The semi-echenolised basis b1 defines a direct sum decomposition
# V = W + W_0 where W_0 is the subspace of V of vectors having zeros
# in the pivot columns of b1. Cleaning with b1 is an algorithm that
# computes the projection p_0 onto W_0. The resulting dec contains
# coefficients such that the corresponding linear combination of b1
# produces the result of the projection p onto W. Thus we can effectively
# compute both projections.
#
# Let U be the span of b2.
#
# If d1 is the dimension of b1 and d2 that of b2 and s the dimension
# of the sum, then the intersection has dimension d1+d2-s.
#
# By cleaning every vector of b2 with b1 we essentially compute
# the image of U under the projection p_0. Computing a semi-echelonised
# basis of the image in the variable "extension" computes this
# image p_0(U) and at the same time produces generators for the
# kernel, since every relation found in p_0(U) gives rise to an
# element of the kernel, which is exactly the intersection U \cap W.
# On the other hand these generators for the kernel are automatically
# linearly independent, since the last non-zero entry is in different
# columns. Thus they span the kernel by a dimension argument.
#
local d,dec,extension,images,res,v,w;
images := Matrix([],NumberColumns(b1!.vectors),b1!.vectors);
dec := ZeroVector( Length(b2), b1!.vectors );
d := Length(b1!.vectors);
# Calculate image:
for v in b2!.vectors do
w := ShallowCopy(v);
CleanRow(b1,w,false,fail);
Add(images,w);
od;
extension := SemiEchelonBasisMutableTX(images);
res := rec( intersection := extension!.relations * b2!.vectors,
sum := SEBMaker(ShallowCopy(b1!.vectors),
ShallowCopy(b1!.pivots)) );
Append(res.sum!.vectors,extension!.vectors);
Append(res.sum!.pivots,extension!.pivots);
return res;
end );
InstallMethod( IntersectionAndSumRowSpaces, "for two row list matrices",
[ IsRowListMatrix, IsRowListMatrix ],
function( m1, m2 )
local b1, b2;
b1 := SemiEchelonBasisMutable(m1);
b2 := SemiEchelonBasisMutable(m2);
return IntersectionAndSumRowSpaces(b1,b2);
end );
# Just an experiment:
BindGlobal( "SumIntersectionMatCMat",
function(m1,m2)
local i,int,mat,n,sum,v,w,zero;
n := NumberColumns(m1);
mat := Matrix([],2*n,m1);
zero := ZeroVector(2*n,m1);
for v in m1 do
w := ShallowCopy(zero);
CopySubVector(v,w,[1..n],[1..n]);
CopySubVector(v,w,[1..n],[n+1..2*n]);
Add(mat,w);
od;
for v in m2 do
w := ShallowCopy(zero);
CopySubVector(v,w,[1..n],[1..n]);
Add(mat,w);
od;
mat := SemiEchelonBasisMutableX(mat);
sum := Matrix([],n,m1);
int := Matrix([],n,m1);
for i in [1..Length(mat!.vectors)] do
if mat!.pivots[i] > n then
Add(int,mat!.vectors[i]{[n+1..2*n]});
else
Add(sum,mat!.vectors[i]{[1..n]});
fi;
od;
return [sum,int];
end );
# Make it public:
InstallMethod( SumIntersectionMat, "for two cmats",
[ IsCMatRep, IsCMatRep ],
SumIntersectionMatCMat );
#############################################################################
# Characteristic polynomials:
#############################################################################
InstallGlobalFunction( CVEC_CharacteristicPolynomialFactors,
function(m,indetnr)
# m must be square
local L,b,b2,closed,d,dec,f,facs,fam,i,l,lambda,newlambda,o,p,
newpiv,pivs,subdim,v,vcopy,zero;
zero := ZeroVector(NumberColumns(m),m);
d := Length(m);
b := EmptySemiEchelonBasis(m);
pivs := [1..d];
facs := [];
f := BaseDomain(m);
o := One(f);
fam := FamilyObj(o);
dec := ShallowCopy(zero);
while Length(pivs) > 0 do
subdim := Length(b!.pivots);
p := pivs[1];
v := ShallowCopy(zero);
v[p] := o;
b2 := EmptySemiEchelonBasis(m);
Add(b2!.vectors,v);
Add(b2!.pivots,p);
lambda := [dec{[1]}];
lambda[1][1] := o;
RemoveSet(pivs,p);
while true do # break is used to jump out
v := v * m;
vcopy := ShallowCopy(v);
CleanRow(b,vcopy,false,fail);
closed := CleanRow(b2,vcopy,true,dec);
if closed then break; fi;
Add(lambda,dec{[1..Length(b2!.pivots)]});
RemoveSet(pivs,b2!.pivots[Length(b2!.pivots)]);
od;
d := Length(b2!.pivots);
# we have the lambdas, expressing v*m^(i-1) in terms of the semiechelon
# basis, now we have to express v*m^d in terms of the v*m^(i-1), that
# means inverting a matrix:
L := ZeroMatrix(d,Length(lambda[d]),m);
for i in [1..d] do
CopySubVector(lambda[i],L[i],[1..i],[1..i]);
od;
l := - dec{[1..d]} * L^-1;
l := Unpack(l);
Add(l,o);
ConvertToVectorRep(l,Size(f));
Add(facs,UnivariatePolynomialByCoefficients(fam,l,indetnr));
# Add b2 to b:
Append(b!.vectors,b2!.vectors);
Append(b!.pivots,b2!.pivots);
od;
return facs;
end );
InstallMethod( CharacteristicPolynomialOfMatrix,
"for a row list matrix and an indet nr",
[IsRowListMatrix, IsPosInt],
function( m, indetnr )
local facs;
facs := CVEC_CharacteristicPolynomialFactors(m, indetnr);
return rec( poly := Product(facs), factors := facs );
end );
InstallMethod( CharacteristicPolynomialOfMatrix, "for a row list matrix",
[IsRowListMatrix],
function( m )
local facs;
facs := CVEC_CharacteristicPolynomialFactors(m, 1);
return rec( poly := Product(facs), factors := facs );
end );
InstallMethod( FactorsOfCharacteristicPolynomial, "for a row list matrix",
[IsRowListMatrix],
function( m )
return FactorsOfCharacteristicPolynomial(m,1);
end );
InstallMethod( FactorsOfCharacteristicPolynomial,
"for a row list matrix and an indet nr",
[IsRowListMatrix, IsPosInt],
function( m, indetnr )
local f,facs,irrfacs,pr;
facs := CVEC_CharacteristicPolynomialFactors(m,indetnr);
pr := PolynomialRing(BaseDomain(m),[indetnr]);
irrfacs := [];
for f in facs do
Append(irrfacs,Factors(pr,f));
od;
Sort(irrfacs);
return irrfacs;
end );
BindGlobal( "CVEC_ActionOnQuotient", function( gens, basis )
local dimsubspc, dimfullspc, dimquotspc, diff, zerov, imgens, x, i, k;
# INPUT
# gens : List of matrices
# basis : basis of submodule ie record with fields
# pivots : integer list of pivot columns
# vectors : matrix of basis in semi-echelon form
# OUTPUT
# List of matrices representing the action of the module given by 'gens'
# on the quotient space induced by 'basis'
# NOTES
#
dimsubspc := Length( basis!.vectors );
dimfullspc := NumberColumns( basis!.vectors );
dimquotspc := dimfullspc - dimsubspc;
diff := Difference( [1..dimfullspc], basis!.pivots );
zerov := ZeroVector( dimquotspc, basis!.vectors ); #prepare vector type
imgens := []; # stores result
for i in [ 1 .. Length( gens ) ] do
imgens[ i ] := ZeroMatrix( dimquotspc, dimquotspc, basis!.vectors );
# grab rows corresponding to added basis vectors:
x := MutableCopyMat(gens[ i ]{ diff });
for k in [1..Length(x)] do # clean rows with subspace basis
CleanRow( basis, x[ k ], false, fail );
od;
# now remove zero columns
CopySubMatrix( x, imgens[ i ], [1..dimquotspc], [1..dimquotspc],
diff , [1..dimquotspc] );
od;
return imgens;
end );
InstallGlobalFunction( CVEC_MinimalPolynomial, function(m)
# m must be square
# This is the old algorithm, implemented for RowListMatrices
local L,b,b2,closed,d,dec,f,fam,i,l,lambda,o,p,pivs,poly,subdim,v,vcopy,zero;
zero := ZeroVector(NumberColumns(m),m);
d := NumberRows(m);
b := EmptySemiEchelonBasis(m);
pivs := [1..d];
f := BaseDomain(m);
poly := One(PolynomialRing(f,[1]));
o := One(f);
fam := FamilyObj(o);
dec := ShallowCopy(zero);
while Length(pivs) > 0 do
subdim := Length(b!.pivots);
p := pivs[1];
v := ShallowCopy(zero);
v[p] := o;
b2 := EmptySemiEchelonBasis(m);
Add(b2!.vectors,v);
Add(b2!.pivots,p);
lambda := [dec{[1]}];
lambda[1][1] := o;
RemoveSet(pivs,p);
while true do # break is used to jump out
v := v * m;
vcopy := ShallowCopy(v);
closed := CleanRow(b2,vcopy,true,dec);
if closed then break; fi;
Add(lambda,dec{[1..Length(b2!.pivots)]});
RemoveSet(pivs,b2!.pivots[Length(b2!.pivots)]);
od;
d := Length(b2!.pivots);
# we have the lambdas, expressing v*m^(i-1) in terms of the semiechelon
# basis, now we have to express v*m^d in terms of the v*m^(i-1), that
# means inverting a matrix:
L := ZeroMatrix(d,d,m);
for i in [1..d] do
CopySubVector(lambda[i],L[i],[1..i],[1..i]);
od;
l := - dec{[1..d]} * L^-1;
l := Unpack(l);
Add(l,o);
ConvertToVectorRep(l,Size(f));
poly := Lcm(poly,UnivariatePolynomialByCoefficients(fam,l,1));
# Add b2 to b:
for v in b2!.vectors do
if not(CleanRow(b,v,true,fail)) then
RemoveSet(pivs,b!.pivots[Length(b!.pivots)]);
fi;
od;
od;
return poly;
end );
InstallGlobalFunction( CVEC_CharAndMinimalPolynomial, function( m, indetnr )
# This is an early try to implement a deterministic, faster minimal
# polynomial algorithm. It now uses the RowListMatrix interface.
# Currently, this does not work properly, the method works in principle,
# but there is something wrong with the factoring out of subspaces and
# the corresponding lower and upper bounds. DO NOT USE!
local col,deg,facs,havedim,i,irreds,l,lowbound,lowbounds,mp,mult,
multfactoredout,multmin,nrblocks,ns,p,pr,targetmult,upbound,x;
# First the characteristic polynomial:
facs := CVEC_CharacteristicPolynomialFactors(m,indetnr);
if Length(facs) = 1 then
return rec( charpoly := facs[1], irreds := facs, mult := [1],
minpoly := facs[1], multmin := [1] );
fi;
Info(InfoCVec,2,
"More than 1 factor, factorising characteristic polynomial...");
# Factor all parts:
pr := PolynomialRing(BaseDomain(m),[indetnr]);
col := List(facs,f->Collected(Factors(pr,f)));
# Collect all irreducible factors:
irreds := [];
mult := [];
lowbounds := [];
multmin := [];
for l in col do
for i in l do
p := Position(irreds,i[1]);
if p = fail then
Add(irreds,i[1]);
Add(mult,i[2]);
Add(lowbounds,i[2]);
Add(multmin,0);
p := Sortex(irreds);
mult := Permuted(mult,p);
lowbounds := Permuted(lowbounds,p);
else
mult[p] := mult[p] + i[2];
if i[2] > lowbounds[p] then
lowbounds[p] := i[2];
fi;
fi;
od;
od;
mp := irreds[1]^0;
Info(InfoCVec,2,"Degrees of irreducible factors of charpoly:",
List(irreds,DegreeOfLaurentPolynomial));
for i in [1..Length(irreds)] do
deg := DegreeOfLaurentPolynomial(irreds[i]);
Info(InfoCVec,2,"Working on irreducible factor of degree ",deg,"...");
if mult[i] = lowbounds[i] then
Info(InfoCVec,2,"Found factor of degree ",deg," with multiplicity ",
mult[i]);
mp := mp * irreds[i]^mult[i];
multmin[i] := mult[i];
else
x := Value(irreds[i],m);
targetmult := mult[i]; # the multiplicity to reach
lowbound := lowbounds[i]; # from the calc. of the charpoly
upbound := targetmult; # an upper bound
Info(InfoCVec,2,"First lower bound: ",lowbound,
" upper bound: ",upbound);
multfactoredout := 0; # no quotient yet
# Note that when we divide something out, we adjust targetmult
# and record this in multfactoredout.
# This stores the number of dimensions each Jordan block is
# made smaller by our current quotient.
# We also adjust lowbound and upbound when we go to a quotient!
while true do # break is used to leave
# This loop tries to determine the size of the largest Jordan
# block of the matrix x and either exits with
# lowbound=upbound=that size
# or reduces the problem to a smaller one in some quotient,
# thereby adjusting multfactoredout by the number of rows/cols
# that are divided away by the quotient and going to the
# next iteration.
# I.e. in the end the right multiplicity is always equal to
# lowbound+multfactoredout
# First calculate a nullspace and get some estimates:
Info(InfoCVec,2,"Target multiplicity: ",targetmult,
", already factored out: ",multfactoredout);
ns := SemiEchelonBasisNullspace(x);
havedim := Length(ns!.vectors);
Info(InfoCVec,2,"Found nullspace of dimension ",havedim);
# We have a lower bound for the multiplicity in the minpoly
# from earlier and one from the number of generalized Jordan
# blocks we see:
nrblocks := havedim/deg; # this is in quotient!
# note that lowbound is absolute i.e. in the original space:
lowbound := Maximum(lowbound,
QuoInt(targetmult+nrblocks-1,nrblocks));
upbound := Minimum(upbound,targetmult-nrblocks+1);
Info(InfoCVec,2,"Lower bound: ",lowbound," upper bound: ",
upbound);
if lowbound = upbound then break; fi;
# Now we divide out the nullspace and go to lowbound:
lowbound := lowbound-1; # Adjustment because of quotient!
Info(InfoCVec,2,"Factoring out nullspace of dimension ",havedim,
" and going to power ",lowbound);
x := CVEC_ActionOnQuotient([x],ns)[1];
multfactoredout := multfactoredout + 1;
targetmult := targetmult - nrblocks;
x := x^lowbound;
Info(InfoCVec,2,"Target multiplicity: ",targetmult);
ns := SemiEchelonBasisNullspace(x);
havedim := Length(ns!.vectors);
Info(InfoCVec,2,"Found nullspace of dimension ",havedim);
# Check, whether we have the complete generalized eigenspace:
if havedim/deg = targetmult then
# lowbound is correct!
upbound := lowbound;
break;
fi;
# Now we want to go to the quotient and redo everything:
Info(InfoCVec,2,"Factoring out nullspace of dimension ",havedim,
" and going to power ",lowbound);
x := CVEC_ActionOnQuotient([x],ns)[1];
multfactoredout := multfactoredout + 1 + lowbound;
targetmult := targetmult - Length(ns!.vectors)/deg;
lowbound := 0; # we do not know anything about this quotient
upbound := targetmult;
od;
Info(InfoCVec,2,"Done! Multiplicity is ",lowbound+multfactoredout);
mp := mp * irreds[i]^(lowbound+multfactoredout);
multmin[i] := (lowbound+multfactoredout);
fi;
od;
return rec(charpoly := Product(facs), irreds := irreds, mult := mult,
minpoly := mp, multmin := multmin);
end );
InstallMethod( CharAndMinimalPolynomialOfMatrix, "for a matrix and an indet nr",
[IsCMatRep, IsPosInt],
function( m, indetnr )
return MinimalPolynomialOfMatrixMC(m,0,indetnr);
end );
InstallMethod( CharAndMinimalPolynomialOfMatrix, "for a matrix", [IsCMatRep],
function( m )
return MinimalPolynomialOfMatrixMC(m,0);
end );
InstallMethod( MinimalPolynomialOfMatrix, "for a matrix and an indet nr",
[IsCMatRep, IsPosInt],
function( m, indetnr )
local res;
res := CVEC_CharAndMinimalPolynomial(m,indetnr);
return res.minpoly;
end );
InstallMethod( MinimalPolynomialOfMatrix, "for a matrix", [IsCMatRep],
function( m )
local res;
res := CVEC_CharAndMinimalPolynomial(m,1);
return res.minpoly;
end );
InstallGlobalFunction( CVEC_GlueMatrices, function(l)
# all elements of the list l must be CMats over the same field
# l must not be empty
local d,g,i,m,n,pr,pos,x;
n := Sum(l,NumberRows);
m := ZeroMatrix(n,n,l[1]);
pos := 1;
for i in [1..Length(l)] do
d := NumberRows(l[i]);
CopySubMatrix(l[i],m,[1..d],[pos..pos+d-1],
[1..d],[pos..pos+d-1]);
pos := pos + d;
od;
if InfoLevel(InfoCVec) >= 2 then
OverviewMat(m);
Print("\n");
fi;
return m;
end );
InstallGlobalFunction( CVEC_ScrambleMatrices,
function( l )
local n,p,d,g,pr,x,xi;
n := NumberRows(l[1]);
p := Characteristic(l[1]);
d := DegreeFFE(l[1]);
g := GL(n,p^d);
g := List(GeneratorsOfGroup(g),CMat);
pr := ProductReplacer(g,rec(scramble := Maximum(QuoInt(n,5),200)));
x := Next(pr);
xi := x^-1;
l := List(l,y->x * y * xi);
if InfoLevel(InfoCVec) >= 2 and Length(l) = 1 then
OverviewMat(l[1]);
Print("\n");
fi;
return l;
end );
InstallGlobalFunction( CVEC_MakeJordanBlock, function(f,pol,s)
local c,cl,d,deg,i,m,n,o,p,pos;
p := Characteristic(f);
d := DegreeOverPrimeField(f);
deg := DegreeOfLaurentPolynomial(pol);
n := s * deg;
m := CVEC_ZeroMat(n,n,p,d);
c := CompanionMat(pol);
c := CMat(List(c,CVec));
o := OneMutable(c);
pos := 1;
for i in [1..s] do
CopySubMatrix(c,m,[1..deg],[pos..pos+deg-1],[1..deg],[pos..pos+deg-1]);
pos := pos + deg;
od;
pos := 1;
for i in [1..s-1] do
CopySubMatrix(o,m,[1..deg],[pos..pos+deg-1],[1..deg],
[pos+deg..pos+2*deg-1]);
pos := pos + deg;
od;
return m;
end );
InstallGlobalFunction( CVEC_MakeExample, function(f,p,l)
# p a list of irredcible polynomials
# l a list of lists of the same length than p, each a list of sizes of
# generalized Jordan blocks
local i,ll,s,x;
ll := [];
for i in [1..Length(p)] do
for s in l[i] do
Add(ll,CVEC_MakeJordanBlock(f,p[i],s));
od;
od;
x := CVEC_ScrambleMatrices([CVEC_GlueMatrices(ll)]);
return x[1];
end );
# The following function is used in the Monte Carlo minimal polynomial
# algorithm:
InstallGlobalFunction( CVEC_CalcOrderPolyTuned,
function( opi, v, i, indetnr )
local coeffs,g,h,j,k,ordpol,vv,w,Top,top;
Top := range -> range[Length(range)];
ordpol := []; # comes factorised
while i >= 1 do
coeffs := v{opi.ranges[i]};
if IsZero(coeffs) then
i := i - 1;
continue;
fi;
coeffs := Unpack(coeffs);
ConvertToVectorRep(coeffs,Size(opi.f));
h := UnivariatePolynomialByCoefficients(opi.fam,coeffs,indetnr);
g := opi.rordpols[i]/Gcd(opi.rordpols[i],h);
Add(ordpol,g);
# This is the part coming from the ith cyclic space, now go down:
coeffs := CoefficientsOfUnivariatePolynomial(g);
w := coeffs[1]*v;
top := Top(opi.ranges[i]);
for j in [2..Length(coeffs)] do
# Now apply base changed matrix to v:
# v := v * opi.mm;
# but remember that we only store the interesting rows of mm:
vv := ZeroMutable(v);
CopySubVector(v,vv,[1..top-1],[2..top]);
for k in [2..i] do
vv[opi.ranges[k][1]] := opi.z;
od;
for k in [1..i] do
AddRowVector(vv,opi.mm[k],v[Top(opi.ranges[k])],
1,Top(opi.ranges[k]));
od;
v := vv;
# Done.
AddRowVector(w,v,coeffs[j]);
od;
# Now w is one subspace lower
v := w;
i := i - 1;
if IsZero(v) then break; fi;
#Print("i=",i," new vector: ");
#Display(v);
od;
return Product(ordpol);
end );
# The following function is used in the Monte Carlo minimal polynomial
# algorithm:
InstallGlobalFunction( CVEC_FactorMultiplicity,
function( p, f )
local m,r;
m := 0;
while true do # we use break
r := QuotientRemainder( p, f );
if not(IsZero(r[2])) then break; fi;
m := m + 1;
p := r[1];
od;
return m;
end );
InstallGlobalFunction( MinimalPolynomialOfMatrixMC,
function( arg )
# The new algorithm of Cheryl and Max. Can be used as Monte Carlo algorithm
# or as deterministic algorithm with verification.
# Neunhöffer, Max; Praeger, Cheryl E., Computing minimal polynomials
# of matrices. LMS J. Comput. Math. 11 (2008), 252–279.
# Arguments: m, eps [,indetnr]
# m a matrix
# eps is a cyclotomic which is an upper bound for the error probability
# if it is set to 0 then deterministic verification is done
# indetnr is the number of an indeterminate, if omitted 1 is taken
local A,B,S,coeffs,col,d,dec,eps,g,i,indetnr,irreds,j,l,lcm,lowbounds,m,bound,
mm,mult,multmin,newBrow,nrunclear,opi,ordpol,ordpolsinv,p,pivs, est,
pr,prob,proof,res,rl,se,ti,ti2,veri,verify,w,wcopy,ww,zero, wred, c;
if Length(arg) < 2 or Length(arg) > 3 then
Error("Usage: m, eps [,indetnr]");
fi;
m := arg[1];
eps := arg[2];
if Length(arg) = 3 then
indetnr := arg[3];
else
indetnr := 1;
fi;
verify := IsZero(eps);
if verify then
eps := 1/10000; # a good compromise to start deterministic verification?
fi;
ti := Runtime();
rl := NumberColumns(m);
zero := ZeroVector(rl,m);
pivs := [1..rl]; # those are the columns we still want pivots for
S := EmptySemiEchelonBasis(m);
# order polynomial infrastructure (grows over time):
opi := rec( f := BaseDomain(m),
d := [], # Degrees of the relative cyclic spaces
ranges := [], # numbers of basis vectors
rordpols := [], # list of relative order polynomials
mm := Matrix([],rl,m), # will be crucial rows of base-changed m
);
opi.o := One(opi.f);
opi.z := Zero(opi.f);
opi.fam := FamilyObj(opi.o);
pr := PolynomialRing(opi.f,[indetnr]);
# We keep the base change between the basis
# Y = [x_1,x_1*m,x_1*m^2,...,x_1*(m^(d_1-1)),x_2,...,x_2*m^(d_2-1),...]
# and the semi echelonised basis S by keeping Y=A*S and S=B*Y at the same
# time. We are only interested in B, but we get A as a byproduct.
A := Matrix([],rl,m);
B := Matrix([],rl,m);
ordpolsinv := []; # here we collect information to be used for the order pols
Info(InfoCVec,2,"Spinning up vectors...");
while Length(S!.vectors) < Length(m) do
# random vector cleaned with known subspace
wred := ShallowCopy(zero);
while IsZero(wred) do
Randomize(wred);
c := wred{S!.pivots};
wred{S!.pivots} := 0*c;
od;
# add random vector from subspace
if Length(c) > 0 then
w := wred + c*S!.vectors;
else
w := wred;
fi;
d := Length(S!.vectors); # dimension of subspace:
l := d; # is always equal to Length(S!.vectors)
while true do
dec := ShallowCopy(zero);
wcopy := ShallowCopy(w);
if CleanRow(S,wcopy,true,dec) then break; fi;
l := l + 1;
Add(A,dec);
# Now update B:
# We know: with l=Length(S) we have dec{[1..l]}*S = Y[l]
# Thus: dec[l]*S[l] = Y[l] - dec{[1..l-1]}*S{[1..l-1]}
# = Y[l] - dec{[1..l-1]}*B*Y{[1..l-1]}
# by a slight abuse of "*" because dec{[1..l-1]}*B has full length.
if l=1 then
newBrow := ShallowCopy(zero);
else
newBrow := dec{[1..l-1]}*B;
fi;
MultVector(newBrow,-opi.o);
newBrow[l] := opi.o;
MultVector(newBrow,dec[l]^-1);
Add(B,newBrow);
w := w * m;
od;
# Now we have extended the basis S together with A and B, such that
# we still have Y = A*S and S = B*Y. The latest dec expresses
# x*m^something in terms of S, first express it in terms of Y, then
# we can read off the relative order polynomial from components
# components d+1 .. Length(S!.vectors).
dec := dec{[1..l]} * B;
Add(opi.mm,dec);
coeffs := -dec{[d+1..l]};
coeffs := Unpack(coeffs);
Add(coeffs,opi.o);
ConvertToVectorRep(coeffs,Size(opi.f));
Add(opi.rordpols,
UnivariatePolynomialByCoefficients(opi.fam,coeffs,indetnr));
Add(opi.d,l-d); # the degree of the order polynomial
Add(opi.ranges,[d+1..l]);
SubtractSet(pivs,S!.pivots{[d+1..l]});
od;
ti2 := Runtime();
Info(InfoCVec,2,"Time until now: ",ti2-ti," lap: ",ti2-ti);
Info(InfoCVec,2,"Factoring relative order polynomials...");
# Factor all parts:
col := List(opi.rordpols,f->Collected(Factors(pr,f)));
Info(InfoCVec,2,"Time until now: ",Runtime()-ti," lap: ",Runtime()-ti2);
ti2 := Runtime();
Info(InfoCVec,2,"Sorting and collecting factors...");
# Collect all irreducible factors:
irreds := [];
mult := [];
lowbounds := [];
multmin := [];
for l in col do
for i in l do
p := PositionSorted(irreds,i[1]);
if p > Length(irreds) or irreds[p] <> i[1] then
Add(irreds,i[1]);
Add(mult,i[2]);
Add(lowbounds,i[2]);
Add(multmin,0);
p := Sortex(irreds);
mult := Permuted(mult,p);
lowbounds := Permuted(lowbounds,p);
else
mult[p] := mult[p] + i[2];
if i[2] > lowbounds[p] then
lowbounds[p] := i[2];
fi;
fi;
od;
od;
repeat # this is used via break to jump to the end where we return
if Length(opi.rordpols)^2 < Length(m) then # a quick check for cyclicity:
lcm := Lcm(opi.rordpols);
if Degree(lcm) = rl then
Info(InfoCVec,2,"Cyclic matrix, this proves minpoly=charpoly!");
proof := true;
multmin := ShallowCopy(mult);
ordpol := lcm;
break; # go to the end, report success
fi;
fi;
nrunclear := 0; # number of irreducible factors the multiplicity of which
# are not yet known
est := []; # for estimate that we have not yet found full
# multiplicities, see Prop. 6.1, 6.3 (only need to
# consider irreds not known to have same multiplicity
# as in characteristic polynomial)
for i in [1..Length(irreds)] do
if mult[i] > lowbounds[i] then
nrunclear := nrunclear + 1;
Add(est, Degree(irreds[i]));
fi;
od;
Info(InfoCVec,2,"Time until now: ",Runtime()-ti," lap: ",Runtime()-ti2);
ti2 := Runtime();
Info(InfoCVec,2,"Number of irreducible factors in charpoly: ",
Length(irreds)," mult. in minpoly unknown: ",nrunclear);
if nrunclear = 0 then
proof := true;
multmin := ShallowCopy(mult);
ordpol := Product([1..Length(irreds)],i->irreds[i]^multmin[i]);
break;
fi;
ordpol := Lcm(Set(opi.rordpols)); # this is a lower bound for the minpoly
# in particular, it is a multiple of rordpols[1]
i := 2; # here we count, which abs. order polynomial to do next
veri := 1; # here we count, which irred. factor to verify next
p := 1/Size(opi.f); # probability to miss a big Jordan block with the
# order polynomial of one vector (upper bound)
proof := false;
# This is an upper estimate of the probability not to find a generator
# of a cyclic space:
prob := p; # we already have the order polynomial for one vector
while not(proof) do
Info(InfoCVec,2,"Calculating order polynomials...");
while i <= Length(opi.rordpols) do
w := ShallowCopy(zero);
w[opi.ranges[i][1]] := opi.o;
g := CVEC_CalcOrderPolyTuned(opi,w,i,indetnr);
if not(IsZero(ordpol mod g)) then
ordpol := Lcm(ordpol,g);
fi;
if Degree(ordpol) = Length(m) then # a cyclic matrix!
proof := true;
multmin := ShallowCopy(mult);
Info(InfoCVec,2,"Cyclic matrix, found proof.");
break;
fi;
prob := prob * p; # = (1/q)^u in Prop. 6.3
bound := Sum(est, d-> prob^d);
i := i + 1;
Info( InfoCVec, 2, "Probability to have missed a factor: ",
1.0*bound);
if bound < eps then
break; # this is the probability to have missed one
fi;
od;
Info(InfoCVec,2,"Checking multiplicities...");
nrunclear := 0;
est := [];
for j in [1..Length(irreds)] do
multmin[j] := CVEC_FactorMultiplicity(ordpol,irreds[j]);
if j >= veri and multmin[j] < mult[j] then
nrunclear := nrunclear + 1;
Add(est, Degree(irreds[j]));
fi;
od;
if nrunclear = 0 then proof := true; break; fi; # result is correct!
if proof or not(verify) then break; fi;
if i > Length(opi.rordpols) then
proof := true;
Info(InfoCVec,2,"Have found proof by computing all absolute ",
"order polynomials!");
break;
fi;
Info(InfoCVec,2,"Time until now: ",Runtime()-ti,
" lap: ",Runtime()-ti2);
ti2 := Runtime();
Info(InfoCVec,2,"Verifying result (",nrunclear,
" unclear multiplicities) ...");
while veri <= Length(irreds) do
if multmin[veri] < mult[veri] then
Info(InfoCVec,2,"Working on factor: ",irreds[veri],
" multiplicity: ",multmin[veri]);
mm := Value(irreds[veri],m)^multmin[veri];
se := SemiEchelonBasisMutableX(mm);
if Length(mm)-Length(se!.vectors) < mult[veri] then
Info(InfoCVec,2,"Found too small multiplicity!");
break;
fi;
fi;
veri := veri + 1;
od;
if veri > Length(irreds) then
Info(InfoCVec,2,"Verified all irreducible factors, found proof.");
proof := true;
fi;
od; # until proof found or break hit
until true; # also used to jump out early
Info(InfoCVec,2,"Time until now: ",Runtime()-ti," lap: ",Runtime()-ti2);
res := rec(minpoly := ordpol, charpoly := Product( opi.rordpols ),
opi := opi, irreds := irreds, mult := mult, multmin := multmin,
proof := proof, prob := eps,
A := A, B := B, S := S);
res.iscyclic := Degree(res.minpoly) = Degree(res.charpoly);
return res;
end );
InstallMethod( MinimalPolynomial, "new MC method with verification",
[ IsField, IsRowListMatrix and IsOrdinaryMatrix, IsPosInt ],
function(f,m,indet)
if f <> BaseDomain(m) then TryNextMethod(); fi;
return MinimalPolynomialOfMatrixMC(m,0,indet).minpoly;
end );
InstallMethod( MinimalPolynomialMatrixNC, "new MC method with verification",
[ IsField, IsRowListMatrix and IsOrdinaryMatrix, IsPosInt ],
function(f,m,indet)
return MinimalPolynomialOfMatrixMC(m,0,indet).minpoly;
end );
##
## This program is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License,
## or (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with this program; if not, write to the Free Software
## Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
##
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2026-03-28
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