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# don't invoke START_TEST, which resets the random number generators,
# so that we can re-read this .tst file multiple times to test more cases.
#gap> START_TEST("basechange.tst");
#
gap> MakeCanonicalOrthogonalBilinear := function(dim,F,rank,w)
> local A, e, i, offset;
> A:=NullMat(dim,dim,F);
> e:=One(F);
> if w = 0 then
> A[1,1] := e;
> if Size(F) mod 4 = 3 then
> A[2,2] := e;
> else
> A[2,2] := PrimitiveRoot(F);
> fi;
> elif w = 1 then
> A[1,1] := e;
> else
> Assert(0, w = 2);
> fi;
> offset := 2-w;
> for i in [1..QuoInt(rank - offset,2)] do
> A[offset+2*i,offset+2*i-1] := e/2;
> A[offset+2*i-1,offset+2*i] := e/2;
> od;
> return A;
> end;;
gap> TestBaseChangeOrthogonalBilinear := function(dim,q)
> local F, mat, bc, m, c;
> F := GF(q);
> # produce a symmetric matrix
> mat:=RandomMat(dim,dim,F);
> mat:=mat+TransposedMat(mat);
> if IsZero(mat) then return; fi;
> # compute base change
> bc:=BaseChangeOrthogonalBilinear(mat, F);
> Assert(0, dim = RankMat(bc[1]));
> Assert(0, bc[2]+1 = RankMat(mat)); # FIXME: why +1?
> # verify the base change really yields the canonical form
> m:=bc[1]*mat*TransposedMat(bc[1]);
> c := MakeCanonicalOrthogonalBilinear(dim,F,bc[2]+1,bc[3]);
> # HACK: 'normalize' both matrices
> Assert(0, m/First(m[1],x->not IsZero(x)) = c/First(c[1],x->not IsZero(x)));
> end;;
#
gap> for dim in [1,2,10,20] do
> for q in [3, 5, 7, 9, 25, 27, 17^2] do
> for i in [0..4] do
> TestBaseChangeOrthogonalBilinear(dim+i, q);
> od;
> od;
> od;
#
gap> MakeCanonicalOrthogonalQuadratic := function(dim,F,rank,w)
> local A, e, i, offset;
> A:=NullMat(dim,dim,F);
> e:=One(F);
> if w = 0 then
> A[1,1] := e;
> A[1,2] := e;
> A[2,2] := Forms_C1(F, DegreeOverPrimeField(F));
> elif w = 1 then
> A[1,1] := e;
> else
> Assert(0, w = 2);
> fi;
> offset := 2-w;
> for i in [1..QuoInt(rank - offset,2)] do
> A[offset+2*i-1,offset+2*i] := e;
> od;
> return A;
> end;;
gap> TestBaseChangeOrthogonalQuadratic := function(dim,q)
> local F, mat, i, j, bc, m, c;
> F := GF(q);
> # produce upper triangular matrix
> mat:=NullMat(dim,dim,F);
> for i in [1..dim] do
> for j in [i..dim] do
> mat[i,j] := Random(F);
> od;
> od;
> if IsZero(mat) then return; fi;
> # compute base change
> bc:=BaseChangeOrthogonalQuadratic(mat, F);
> Assert(0, dim = RankMat(bc[1]));
> #Assert(0, bc[2] = RankMat(mat)); # FIXME: what does bc[2] really mean?
> # verify the base change really yields the canonical form
> m:=bc[1]*mat*TransposedMat(bc[1]);
> c:=MakeCanonicalOrthogonalQuadratic(dim,F,bc[2]+1,bc[3]);
> Assert(0, Forms_RESET(m, dim, q) = c);
> end;;
#
gap> for dim in [1,2,10,20] do
> for q in [2, 4, 8, 16, 2^9] do
> for i in [0..4] do
> TestBaseChangeOrthogonalQuadratic(dim+1, q);
> od;
> od;
> od;
#
gap> MakeCanonicalHermitian := function(dim,F,rank)
> local A, e, i;
> A:=NullMat(dim,dim,F);
> e:=One(F);
> for i in [1..rank] do
> A[i,i] := e;
> od;
> return A;
> end;;
gap> TestBaseChangeHermitian := function(dim,q)
> local F, mat, bc, m, c;
> F := GF(q^2);
> # produce a symmetric matrix
> mat:=RandomInvertibleMat(dim,F);
> mat:=mat+Forms_HERM_CONJ(mat, q);
> if IsZero(mat) then return; fi;
> # compute base change
> bc:=BaseChangeHermitian(mat, F);
> Assert(0, dim = RankMat(bc[1]));
> # verify the base change really yields the canonical form
> m:=bc[1]*mat*Forms_HERM_CONJ(bc[1], q);
> c:=MakeCanonicalHermitian(dim,F,RankMat(mat));
> Assert(0, m = c);
> end;;
#
gap> for dim in [1,2,10,20] do
> for q in [2, 3, 4, 5, 7, 9, 16, 25, 27, 17^2] do
> for i in [0..4] do
> TestBaseChangeHermitian(dim+i, q);
> od;
> od;
> od;
#
gap> MakeCanonicalSymplectic := function(dim,F,rank)
> local A, e, i;
> A:=NullMat(dim,dim,F);
> e:=One(F);
> for i in [1..QuoInt(rank,2)] do
> A[2*i,2*i-1] := -e;
> A[2*i-1,2*i] := e;
> od;
> return A;
> end;;
gap> TestBaseChangeSymplectic := function(dim,q)
> local F, mat, bc, m, c;
> F := GF(q);
> # produce an alternating matrix
> mat:=RandomMat(dim,dim,F);
> mat:=mat-TransposedMat(mat);
> if IsZero(mat) then return; fi;
> # compute base change
> bc:=BaseChangeSymplectic(mat, F);
> Assert(0, dim = RankMat(bc[1]));
> # verify the base change really yields the canonical form
> m:=bc[1]*mat*TransposedMat(bc[1]);
> c:=MakeCanonicalSymplectic(dim,F,RankMat(mat));
> Assert(0, m = c);
> end;;
#
gap> for dim in [2,10,20] do
> for q in [2, 3, 4, 5, 7, 8, 9, 25, 27, 17^2] do
> for i in [0..4] do
> TestBaseChangeSymplectic(dim, q);
> od;
> od;
> od;
#
#gap> STOP_TEST("basechange.tst", 0);
[ Dauer der Verarbeitung: 0.31 Sekunden
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