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#############################################################################
# Lazy grease:
#############################################################################
InstallMethod( LazyGreaser, "for a vector and a positive grease level",
[IsObject, IsPosInt],
function( vecs, lev )
local lg;
lg := rec( vecs := vecs, tab := vecs{[]}, ind := [], lev := lev,
fs := Size(BaseDomain(vecs)) );
if lg.fs > 65536 then
Error("Lazy grease only supported for fields with <= 65536 elements");
return;
fi;
return Objectify( NewType(LazyGreaserFamily, IsLazyGreaser), lg );
end );
InstallMethod( ViewObj, "for a lazy greaser",
[IsLazyGreaser],
function( lg )
Print("<lazy greaser level ",lg!.lev," with ",Length(lg!.tab)," vectors>");
end );
InstallMethod( GetLinearCombination,
"for a lazy greaser, a vector, an offset, and a list of integers",
[IsLazyGreaser, IsVector, IsPosInt, IsList],
function( lg, v, offset, pivs )
# offset must be congruent 1 mod lev
local group,i,pos,w;
pos := NumberFFVector(v{Reversed(pivs)},lg!.fs)+1;
if pos = 1 then return 0; fi;
group := (offset-1) / lg!.lev + 1;
#if not IsBound(lg!.ind[group]) then
# lg!.ind[group] := [];
#fi;
if not IsBound(lg!.ind[group][pos]) then
# Cut away all trailing zeroes:
i := Length(pivs);
while i > 0 and IsZero(v[pivs[i]]) do i := i - 1; od;
# i = 0 does not happen!
if i = 1 then
w := v[pivs[1]] * lg!.vecs[offset];
else
w := GetLinearCombination(lg,v,offset,pivs{[1..i-1]});
if w = 0 then
w := v[pivs[i]] * lg!.vecs[offset+i-1];
else
w := w + v[pivs[i]] * lg!.vecs[offset+i-1];
fi;
fi;
Add(lg!.tab,w);
lg!.ind[group][pos] := Length(lg!.tab);
else
w := lg!.tab[lg!.ind[group][pos]];
fi;
return w;
end );
InstallMethod( GetLinearCombination,
"for a lazy greaser, a cvec, an offset, and a list of integers",
[IsLazyGreaser, IsCVecRep, IsPosInt, IsList],
function( lg, v, offset, pivs )
# the vecs entry of the lazy greaser must be a cmat!
# offset must be congruent 1 mod lev
local group,i,pos,w;
pos := CVEC_GREASEPOS(v,pivs);
if pos = 1 then return 0; fi;
group := (offset-1) / lg!.lev + 1;
if not IsBound(lg!.ind[group]) then
lg!.ind[group] := [];
fi;
if not IsBound(lg!.ind[group][pos]) then
# Cut away all trailing zeroes:
i := Length(pivs);
while i > 0 and IsZero(v[pivs[i]]) do i := i - 1; od;
# i = 0 does not happen!
if i = 1 then
w := v[pivs[1]] * lg!.vecs[offset];
else
w := GetLinearCombination(lg,v,offset,pivs{[1..i-1]});
if w = 0 then
w := v[pivs[i]] * lg!.vecs[offset+i-1];
else
w := w + v[pivs[i]] * lg!.vecs[offset+i-1];
fi;
fi;
Add(lg!.tab,w);
lg!.ind[group][pos] := Length(lg!.tab);
return w;
else
return lg!.tab[lg!.ind[group][pos]];
fi;
end );
CVEC_TESTLAZY := function(m,lev)
local erg,f,i,j,l,offset,newpos,poss,v,w;
v := ShallowCopy(m[1]);
l := LazyGreaser(m,lev);
f := BaseDomain(m);
offset := Random(1,Length(m)-lev+1);
offset := QuoInt(offset-1,lev)*lev + 1; # make it congruent 1 mod lev
for i in [1..10000] do
w := Zero(v);
poss := [];
for j in [1..lev] do
repeat
newpos := Random([1..Length(v)]);
until not newpos in poss;
Add(poss,newpos);
v[poss[j]] := Random(f);
w := w + v[poss[j]] * m[offset+j-1];
od;
erg := GetLinearCombination(l,v,offset,poss);
if not((erg = 0 and IsZero(w)) or w = erg) then
Error();
fi;
Print(i,"\r");
od;
Print("\n");
return l;
end;
if IsBound(basis.lazygreaser) and basis.lazygreaser <> fail then
# The grease case:
lev := basis.lazygreaser!.lev;
i := 1;
mo := -One(vec[1]);
while i+lev-1 <= len do
pivs := basis.pivots{[i..i+lev-1]};
j := lev;
while j > 0 and pivs[j] < firstnz do j := j - 1; od;
if j > 0 then # Otherwise, there is nothing to do at all
lc := GetLinearCombination(basis.lazygreaser,vec,i,pivs);
#lc := glc(basis.lazygreaser,vec,i,pivs);
if lc <> 0 then # 0 indicates the zero linear combination
if dec <> false and dec <> true then
for j in [i..i+lev-1] do
dec[j] := vec[basis.pivots[j]];
od;
fi;
AddRowVector( vec, lc, mo );
fi;
fi;
i := i + lev;
od;
# Now go to the standard case for the rest, starting at i
else
i := 1;
# The non-grease case starts from the beginning
fi;
# we have to work without the lazy greaser:
...
InstallMethod( EmptySemiEchelonBasis, "GAP method", [IsObject],
function( vec )
return rec( vectors := MatrixNC([],vec), pivots := [],
lazygreaser := fail, helper := vec{[1]} );
# The lazygreaser component must always be bound!
# The helper is needed for the kernel cleaner for CVecs
end );
InstallMethod( EmptySemiEchelonBasis, "GAP method", [IsObject, IsPosInt],
function( vec, greaselev )
local r;
r := rec( vectors := MatrixNC([],vec), pivots := [], helper := vec{[1]} );
# helper is needed for kernel cleaner
r.lazygreaser := LazyGreaser( r.vectors, greaselev );
return r;
end );
if IsBound(basis.lazygreaser) then
# In the grease case, we do full-echelon form within the grease block:
for j in [i..len] do
c := basis.vectors[j][newpiv];
if not IsZero(c) then
AddRowVector( basis.vectors[j], basis.vectors[len+1], -c );
fi;
od;
fi;
#############################################################################
# Lazy grease:
#############################################################################
BindGlobal( "LazyGreaserFamily", NewFamily("LazyGreaserFamily") );
DeclareCategory( "IsLazyGreaser", IsComponentObjectRep );
DeclareOperation( "LazyGreaser", [IsObject, IsPosInt] );
DeclareOperation( "GetLinearCombination",
[IsLazyGreaser, IsObject, IsPosInt, IsList] );
# From linalg.gi, early tries for minimal polynomial:
# The rest can probably go at some stage:
InstallGlobalFunction( CVEC_RelativeOrderPoly, function( m, v, subb, indetnr )
# v must not lie in subb, a semi echelonised basis of a subspace W, m a matrix
# v must already be cleaned with subb!
# Returns a record:
# Gives back a new semi echelonised basis of V/W (component b), together
# with the relative order polynomial of v+W in V/W (component ordpol),
# v'=v*ordpol(m) in subb, and an invertible square matrix A with
# A*[v,vm,vm^2,...,vm^{d-1}] = b (to write vectors in V/W in terms of
# [v+W,vm+W,...]. Also a list vl containing [v,v*m,v*m^2,...,v*m^{d-1}]
# is bound.
# The original v is untouched!
local A,B,b,closed,d,dec,f,fam,i,l,lambda,o,ordpol,vcopy,vl,vv;
f := BaseDomain(m);
o := One(f);
fam := FamilyObj(o);
b := EmptySemiEchelonBasis(v);
dec := ZeroMutable(v);
CleanRow(b,ShallowCopy(v),true,dec); # this puts v into b as first vector
lambda := [dec{[1]}];
vl := [v];
while true do
v := v * m;
Add(vl,v);
vcopy := ShallowCopy(v);
CleanRow(subb,vcopy,false,fail);
closed := CleanRow(b,vcopy,true,dec);
if closed then break; fi;
# now dec{[1..Length(b.vectors)]} * b.vectors = v
# however, we need to express vectors in terms of the v's, so we
# have to invert the matrix of the dec's, this we do later on:
Add(lambda,dec{[1..Length(b.pivots)]});
od;
d := Length(b.pivots);
Unbind(vl[d+1]); # this is linear dependent from the first d vectors!
# 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:
B := CVEC_ZeroMat(d,CVecClass(lambda[d]));
for i in [1..d] do
CopySubVector(lambda[i],B[i],[1..i],[1..i]);
od;
A := B^-1;
l := - dec{[1..d]} * A;
l := Unpack(l);
Add(l,o);
ConvertToVectorRep(l,Size(f));
ordpol := UnivariatePolynomialByCoefficients(fam,l,indetnr);
vv := l[1] * vl[1];
for i in [2..Length(vl)] do
AddRowVector(vv,vl[i],l[i]);
od;
return rec( b := b, A := A, ordpol := ordpol, v := vv, vl := vl );
end );
InstallGlobalFunction( CVEC_CalcOrderPoly,
function( opi, v, i, indetnr )
local coeffs,g,h,j,ordpol,w;
ordpol := []; # comes factorised
while i >= 1 do
coeffs := v{opi.ranges[i]};
if IsZero(coeffs) then
i := i - 1;
continue;
fi;
coeffs := Unpack(coeffs);
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;
for j in [2..Length(coeffs)] do
v := v * opi.mm;
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 );
BindGlobal( "CVEC_CalcOrderPolyEstimate",
function( est, opi, v, i, indetnr )
local c,coeffs,g,h,j,k,op,ordpol,w;
ordpol := []; # comes factorised
op := UnivariatePolynomialByCoefficients(opi.fam,[opi.o],indetnr);
k := i;
while true do # break is used near the end
coeffs := Unpack(v{opi.ranges[k]});
h := UnivariatePolynomialByCoefficients(opi.fam,coeffs,indetnr);
c := Gcd(opi.rordpols[k],h);
g := opi.rordpols[k]/c;
Add(ordpol,g);
op := op * g;
#Print("Relative order pol: ",g,"\n");
# This is the part coming from the ith cyclic space, now go down:
coeffs := CoefficientsOfUnivariatePolynomial(g);
w := coeffs[1]*v;
for j in [2..Length(coeffs)] do
v := v * opi.mm;
AddRowVector(w,v,coeffs[j]);
od;
# Now w is one subspace lower
v := w;
k := k - 1;
if IsZero(v) then break; fi;
if k = 0 then break; fi;
#Print("Check:",Degree(est[i-1])," ", Degree(est[k]*op),"\n");
if IsZero(est[i-1] mod (est[k] * op)) then
Print("|\c");
return est[i-1];
fi;
#Print("i=",i," new vector: ");
#Display(v);
od;
return op;
end );
InstallGlobalFunction( CVEC_NewMinimalPolynomialMC,
function( m, eps, verify, indetnr )
# a new try using Cheryl's ideas for cyclic matrices generalised to
# an arbitrary matrix m
# eps is a cyclotomic which is an upper bound for the error probability
# verify is a boolean indicating, whether the result should be verified,
# thereby proving correctness
# indetnr is the number of an indeterminate
local A,b,col,d,g,i,irreds,j,l,lowbounds,mm,mult,multmin,nrunclear,ns,opi,
ordpol,p,pivs,prob,proof,re,v,vl,vli,w,zero;
zero := ZeroMutable(m[1]);
pivs := [1..Length(m)];
b := EmptySemiEchelonBasis(zero);
v := Matrix([],m[1]); # start vectors for the spinning
vl := Matrix([],m[1]); # start vectors together with their images under m
A := []; # base changes within each subquotient
l := []; # a list of semi echelon bases for the subquotients
# order polynomial infrastructure (grows over time):
opi := rec( f := BaseDomain(m),
d := [],
ranges := [],
rordpols := [], # list of relative order polynomials
ordpols := [], # list of real order polynomials
mm := fail, # will be base-changed m later on
);
opi.o := One(opi.f);
opi.fam := FamilyObj(opi.o);
Print(Length(m),":\c");
while Length(pivs) > 0 do
p := pivs[1];
w := ShallowCopy(zero);
w[p] := opi.o;
Add(v,w); # FIXME: needed?
re := CVEC_RelativeOrderPoly(m,w,b,indetnr);
d := Degree(re.ordpol);
Print(d," \c");
Add(opi.rordpols,re.ordpol);
Add(opi.d,d);
Add(opi.ranges,[Length(b.vectors)+1..Length(b.vectors)+d]);
Add(l,re.b); # FIXME: needed?
Add(A,re.A); # FIXME: needed?
Append(vl,re.vl);
pivs := Difference(pivs,re.b.pivots);
# Add re.b to b:
Append(b.vectors,re.b.vectors);
Append(b.pivots,re.b.pivots);
# Note that the latter does not cost too much memory as the vectors
# are not copied.
od;
Print("\n");
if Degree(opi.rordpols[1]) = Length(m) then
return rec( minpoly := opi.rordpols[1],
charpoly := opi.rordpols[1],
opi := opi );
# FIXME: Return a consistent result!
fi;
Info(InfoCVec,2,"Doing basechange...");
# Now we have spun up the whole space, that is, vl is a basis of the full
# row space
# Now we do the base change:
vli := vl^-1;
opi.mm := vl*m*vli;
Unbind(vl); # no longer needed
Unbind(vli); # dito
Info(InfoCVec,2,"Factoring relative order polynomials...");
# Factor all parts:
col := List(opi.rordpols,f->Collected(Factors(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;
nrunclear := 0; # number of irreducible factors the multiplicity of which
# are not yet known
for i in [1..Length(irreds)] do
if mult[i] > lowbounds[i] then
nrunclear := nrunclear + 1;
fi;
od;
Info(InfoCVec,2,"Number of irreducible factors in charpoly: ",Length(irreds),
" mult. in minpoly unknown: ",nrunclear);
ordpol := Lcm(opi.rordpols); # this is a lower bound for the minpoly
# in particular, it is a multiple of rordpols[1]
i := 2;
p := 1/Size(opi.f);
proof := false;
repeat # until verification said "OK"
# This is an upper estimate of the probability not to find a generator
# of a cyclic space:
prob := p; # we already have for one vector the order polynomial
Info(InfoCVec,2,"Calculating order polynomials...");
while i <= Length(opi.rordpols) do
v := ShallowCopy(zero);
v[opi.ranges[i][1]] := opi.o;
g := CVEC_CalcOrderPoly(opi,v,i,indetnr);
if not(IsZero(ordpol mod g)) then
ordpol := Lcm(ordpol,g);
fi;
if Degree(ordpol) = Length(m) then # a cyclic matrix!
break;
fi;
prob := prob * p; # probability to have missed one Jordan block
Print( "Probability to have them all (%%): ",
Int((1-prob)^nrunclear*1000),"\n");
if 1-(1-prob)^nrunclear < eps then
break; # this is the probability to have missed one
fi;
i := i + 1;
od;
Info(InfoCVec,2,"Checking multiplicities...");
nrunclear := 0;
for j in [1..Length(irreds)] do
multmin[j] := CVEC_FactorMultiplicity(ordpol,irreds[j]);
if multmin[j] < mult[j] then
nrunclear := nrunclear + 1;
fi;
od;
if nrunclear = 0 then proof := true; break; fi; # result is correct!
if verify then
Info(InfoCVec,2,"Verifying result (",nrunclear,
" unclear multiplicities) ...");
proof := true; # will be set to false once we discover something
for j in [1..Length(irreds)] do
if multmin[j] < mult[j] then
Info(InfoCVec,2,"Working on factor: ",irreds[j],
" multiplicity: ",multmin[j]);
mm := Value(irreds[j],opi.mm)^multmin[j];
ns := NullspaceMatMutableX(mm);
if Length(ns) < mult[j] then
proof := false;
break;
fi;
fi;
od;
fi;
until not(verify) or proof;
return rec(minpoly := ordpol,
charpoly := Product( opi.rordpols ),
opi := opi,
irreds := irreds,
mult := mult,
multmin := multmin);
end );
InstallGlobalFunction( CVEC_NewMinimalPolynomial, function( m, indetnr )
# a new try using Cheryl's ideas for cyclic matrices generalised to
# an arbitrary matrix m
local A,b,d,est,g,i,l,opi,ordpol,p,pivs,re,v,vl,vli,w,zero;
zero := ZeroMutable(m[1]);
pivs := [1..Length(m)];
b := EmptySemiEchelonBasis(zero);
v := Matrix([],m[1]); # start vectors for the spinning
vl := Matrix([],m[1]); # start vectors together with their images under m
A := []; # base changes within each subquotient
l := []; # a list of semi echelon bases for the subquotients
# order polynomial infrastructure (grows over time):
opi := rec( f := BaseDomain(m),
d := [],
ranges := [],
rordpols := [], # list of relative order polynomials
ordpols := [], # list of real order polynomials
mm := fail, # will be base-changed m later on
);
opi.o := One(opi.f);
opi.fam := FamilyObj(opi.o);
Print(Length(m),":\c");
while Length(pivs) > 0 do
p := pivs[1];
w := ShallowCopy(zero);
w[p] := opi.o;
Add(v,w); # FIXME: needed?
re := CVEC_RelativeOrderPoly(m,w,b,indetnr);
d := Degree(re.ordpol);
Print(d," \c");
Add(opi.rordpols,re.ordpol);
Add(opi.d,d);
Add(opi.ranges,[Length(b.vectors)+1..Length(b.vectors)+d]);
Add(l,re.b); # FIXME: needed?
Add(A,re.A); # FIXME: needed?
Append(vl,re.vl);
pivs := Difference(pivs,re.b.pivots);
# Add re.b to b:
Append(b.vectors,re.b.vectors);
Append(b.pivots,re.b.pivots);
# Note that the latter does not cost too much memory as the vectors
# are not copied.
od;
Print("\nDoing basechange...\c");
# Now we have spun up the whole space, that is, vl is a basis of the full
# row space
# Now we do the base change:
vli := vl^-1;
opi.mm := vl*m*vli;
Unbind(vl); # no longer needed
Unbind(vli); # dito
ordpol := opi.rordpols[1];
Print("done\nCalculating order polynomials...\c");
est := [opi.rordpols[1]];
for i in [2..Length(opi.rordpols)] do
v := ShallowCopy(zero);
v[opi.ranges[i][1]] := opi.o;
g := CVEC_CalcOrderPolyEstimate(est,opi,v,i,indetnr);
if g <> ordpol then
ordpol := Lcm(ordpol,g);
fi;
est[i] := ordpol;
Print(i,"(",Length(opi.rordpols),"):",Degree(ordpol)," \c");
if Degree(ordpol) = Length(m) then # a cyclic matrix!
break;
fi;
od;
return rec(minpoly := ordpol,charpoly := Product( opi.rordpols ),opi := opi);
end );
# From linalg.gd, early tries for the minimal polynomial:
# The rest can probably be deleted later on:
DeclareGlobalFunction( "CVEC_RelativeOrderPoly" );
DeclareGlobalFunction( "CVEC_CalcOrderPoly" );
DeclareGlobalFunction( "CVEC_NewMinimalPolynomial" );
DeclareGlobalFunction( "CVEC_NewMinimalPolynomialMC" );
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