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Merge branch 'master' of github.com:gap-packages/forms
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@jdebeule
jdebeule committed Dec 5, 2023
2 parents cdcbd2a + 723b5da commit 326e4ec
Showing 1 changed file with 67 additions and 129 deletions.
196 changes: 67 additions & 129 deletions lib/forms.gi
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Original file line number Diff line number Diff line change
Expand Up @@ -1589,6 +1589,7 @@ else

fi;

# express v as sum of two squares of elements in gf
InstallGlobalFunction(Forms_SUM_OF_SQUARES,
function(v,gf)
local dummy,i,v1,v2, primroot;
Expand All @@ -1607,56 +1608,53 @@ InstallGlobalFunction(Forms_SUM_OF_SQUARES,
return [v1,v2];
end );

# Apply a 2x2 transformation matrix to rows 'p1' and 'p2' of the matrix 'D'.
# The matrix 'D' may be changed in-place by this. For efficiency the matrix
# entries are passed in as arguments 'a11', 'a12', 'a21', 'a22'.
BindGlobal("Forms_TRANSFORM_2_BY_2",
function(D,p1,p2,a11,a12,a21,a22)
local r1,r2;
r1 := D[p1];
r2 := D[p2];
D[p1] := a11 * r1 + a12 * r2;
D[p2] := a21 * r1 + a22 * r2;
end );

InstallGlobalFunction(Forms_REDUCE2,
function(start,stop,n,gf)
local t,i,P,half,primroot;
function(D,start,stop,n,gf)
local t,i,half,primroot;
primroot := PrimitiveRoot(gf);
half := One(gf) / 2;
t := primroot^(LogFFE(-One(gf),primroot)/2) / 2;
P := IdentityMat(n,gf);
i := start;
while i < stop do
P[i,i] := half;
P[i,i+1] := t;
P[i+1,i] := half;
P[i+1,i+1] := -t;
Forms_TRANSFORM_2_BY_2(D,i,i+1,half,t,half,-t);
i := i + 2;
od;
return P;
end );

InstallGlobalFunction(Forms_REDUCE4,
function(start,stop,n,gf)
local c,d,i,P,dummy;
P := IdentityMat(n,gf);
function(D,start,stop,n,gf)
local c,d,i,dummy;
i := start;
dummy := Forms_SUM_OF_SQUARES(-One(gf),gf);
c := dummy[1];
d := dummy[2];
while i < stop do
P[i+1,i+1] := c;
P[i+1,i+3] := d;
P[i+3,i+1] := d;
P[i+3,i+3] := -c;
Forms_TRANSFORM_2_BY_2(D,i+1,i+3,c,d,d,-c);
i := i + 4;
od;
return P;
end );

InstallGlobalFunction(Forms_DIFF_2_S,
function(start,stop,n,gf)
local i,P,half;
function(D,start,stop,n,gf)
local i,half;
i := start;
P := IdentityMat(n,gf);
half := One(gf) / 2;
while i < stop do
P[i,i] := half;
P[i,i+1] := half;
P[i+1,i] := half;
P[i+1,i+1] := -half;
Forms_TRANSFORM_2_BY_2(D,i,i+1,half,half,half,-half);
i := i + 2;
od;
return P;
end );

InstallGlobalFunction(Forms_HERM_CONJ,
Expand Down Expand Up @@ -1700,16 +1698,10 @@ InstallGlobalFunction(Forms_SQRT2,
end );

InstallGlobalFunction(Forms_PERM_VAR,
function(n,r)
local i,P;
P := IdentityMat(n);
for i in [1..r-1] do
P[i,i] := 0;
P[i+1,i] := 1;
od;
P[r,r] := 0;
P[1,r] := 1;
return P;
function(D,r)
local i;
i := Remove(D, r);
Add(D, i, 1);
end );

InstallGlobalFunction(Forms_C1,
Expand Down Expand Up @@ -1936,52 +1928,39 @@ InstallMethod( BaseChangeOrthogonalBilinear,
s := s - 1;
r := r - 1;

# We write first v=v1^2 + v2^2

dummy := Forms_SUM_OF_SQUARES(v,gf);
v1 := dummy[1];
v2 := dummy[2];

# Case by case:

P := IdentityMat(nplus1, gf);
if not (s = -1 or r = s ) then
# We write first v=v1^2 + v2^2
dummy := Forms_SUM_OF_SQUARES(v,gf);
v1 := dummy[1];
v2 := dummy[2];

if (r - s) mod 2 = 0 then
for i in [s+2..r+1] do
P[i,i] := v1/v;
od;
v1 := v1/v;
v2 := v2/v;
i := s + 2;
repeat
P[i,i+1] := -v2/v;
P[i+1,i] := v2/v;
Forms_TRANSFORM_2_BY_2(D,i,i+1,v1,-v2,v2,v1);
i := i + 2;
until i = r + 2;
D := P*D;
s := r;
else
if r mod 2 = 0 then
for i in [1..s+1] do
P[i,i] := v1;
od;
i := 1;
repeat
P[i,i+1] := v2;
P[i+1,i] := -v2;
Forms_TRANSFORM_2_BY_2(D,i,i+1,v1,v2,-v2,v1);
i := i + 2;
until i = s + 2;
D := P*D;
s := -1;
elif not (s = r - 1) then
for i in [s+2..r] do
P[i,i] := v1/v;
od;
v1 := v1/v;
v2 := v2/v;
i := s + 2;
repeat
P[i,i+1] := -v2/v;
P[i+1,i] := v2/v;
Forms_TRANSFORM_2_BY_2(D,i,i+1,v1,-v2,v2,v1);
i := i + 2;
until i = r + 1;
D := P*D;
s := r - 1;
fi;
fi;
Expand All @@ -1993,67 +1972,47 @@ InstallMethod( BaseChangeOrthogonalBilinear,
if r mod 2 <> 0 then
if s = -1 or s = r then
if q mod 4 = 1 then
P := Forms_REDUCE2(1,r+1,nplus1,gf);
D := P*D;
Forms_REDUCE2(D,1,r+1,nplus1,gf);
w := 2;
else
if ((r-1)/2) mod 2 <> 0 then
P := Forms_REDUCE4(1,r+1,nplus1,gf);
D := P*D;
P := Forms_DIFF_2_S(1,r+1,nplus1,gf);
D := P*D;
Forms_REDUCE4(D,1,r+1,nplus1,gf);
Forms_DIFF_2_S(D,1,r+1,nplus1,gf);
w := 2;
else
P := Forms_REDUCE4(3,r+1,nplus1,gf);
D := P*D;
P := Forms_DIFF_2_S(3,r+1,nplus1,gf);
D := P*D;
Forms_REDUCE4(D,3,r+1,nplus1,gf);
Forms_DIFF_2_S(D,3,r+1,nplus1,gf);
w := 0;
fi;
fi;
else
if q mod 4 = 1 then
if 1 < r then
Forms_SwapRows(D,2,r+1);
P := Forms_REDUCE2(3,r+1,nplus1,gf);
D := P*D;
Forms_REDUCE2(D,3,r+1,nplus1,gf);
fi;
w := 0;
else
if ((r-1)/2) mod 2 <> 0 then
Forms_SwapRows(D,4,r+1);
if 3 < r then
P := Forms_REDUCE4(5,r+1,nplus1,gf);
else
P := IdentityMat(nplus1,gf);
Forms_REDUCE4(D,5,r+1,nplus1,gf);
fi;
b := primroot^(LogFFE(-v, primroot)/2);
P[3,3] := (1/2)*one;
P[4,3] := (1/2)*one;
P[3,4] := -1/(2*b);
P[4,4] := 1/(2*b);
D := P*D;
b := primroot^(LogFFE(-v,primroot)/2);
Forms_TRANSFORM_2_BY_2(D,3,4,one/2,-b/2,one/2,b/2);
if 3 < r then
P := Forms_DIFF_2_S(5,r+1,nplus1,gf);
D := P*D;
Forms_DIFF_2_S(D,5,r+1,nplus1,gf);
fi;
w := 0;
else
Forms_SwapRows(D,2,r+1);
if 1 < r then
P := Forms_REDUCE4(3,r+1,nplus1,gf);
else
P := IdentityMat(nplus1,gf);
Forms_REDUCE4(D,3,r+1,nplus1,gf);
fi;
b := primroot^(LogFFE(-v,primroot)/2);
P[1,1] := (1/2)*one;
P[2,1] := (1/2)*one;
P[1,2] := -1/(2*b);
P[2,2] := 1/(2*b);
D := P*D;
Forms_TRANSFORM_2_BY_2(D,1,2,one/2,-b/2,one/2,b/2);
if 1 < r then
P := Forms_DIFF_2_S(3,r+1,nplus1,gf);
D := P*D;
Forms_DIFF_2_S(D,3,r+1,nplus1,gf);
fi;
w := 2;
fi;
Expand All @@ -2062,30 +2021,20 @@ InstallMethod( BaseChangeOrthogonalBilinear,
elif r <> 0 then
w := 1;
if q mod 4 = 1 then
P := Forms_REDUCE2(2,r+1,nplus1,gf);
D := P*D;
Forms_REDUCE2(D,2,r+1,nplus1,gf);
else
if r mod 4 = 0 then
P := Forms_REDUCE4(2,r+1,nplus1,gf);
D := P*D;
P := Forms_DIFF_2_S(2,r+1,nplus1,gf);
D := P*D;
Forms_REDUCE4(D,2,r+1,nplus1,gf);
Forms_DIFF_2_S(D,2,r+1,nplus1,gf);
else
if 3 < r then
P := Forms_REDUCE4(4,r+1,nplus1,gf);
else
P := IdentityMat(nplus1, gf);
Forms_REDUCE4(D,4,r+1,nplus1,gf);
fi;
dummy := Forms_SUM_OF_SQUARES(-1,gf);
c := dummy[1];
d := dummy[2];
P[1,1] := c;
P[1,3] := d;
P[3,1] := d;
P[3,3] := -c;
D := P*D;
P := Forms_DIFF_2_S(2,r+1,nplus1,gf);
D := P*D;
Forms_TRANSFORM_2_BY_2(D,1,3,c,d,d,-c);
Forms_DIFF_2_S(D,2,r+1,nplus1,gf);
i := 3;
while i <= r + 1 do
Forms_MultRow(D,i,-one);
Expand Down Expand Up @@ -2166,17 +2115,14 @@ InstallMethod(BaseChangeOrthogonalQuadratic, [ IsMatrix and IsFFECollColl, IsFie

# If all is zero, STOP
if dummy then
P := IdentityMat(nplus1, gf);
t := Forms_SQRT2(A[row,row],q);
P[row,row] := 1/t;
Forms_MultRow(D,row,1/t);
for i in [row + 1..r] do
P[i,row] := Forms_SQRT2(A[i,i],q)/t;
Forms_AddRows(D,i,row,Forms_SQRT2(A[i,i],q));
od;
D := P*D;
# Permutation of the variables, it is a parabolic
r := row;
P := Forms_PERM_VAR(nplus1,r);
D := P*D;
Forms_PERM_VAR(D,r);
w := 1;
r := r - 1;
return [D,r,w];
Expand Down Expand Up @@ -2234,9 +2180,10 @@ InstallMethod(BaseChangeOrthogonalQuadratic, [ IsMatrix and IsFFECollColl, IsFie
# check for zero row
dummy := true;
i := row + 1;
while (dummy and i <= nplus1) do
while i <= nplus1 do
if not IsZero( A[row,i] ) then
dummy := false;
break;
else
i := i + 1;
fi;
Expand Down Expand Up @@ -2291,8 +2238,7 @@ InstallMethod(BaseChangeOrthogonalQuadratic, [ IsMatrix and IsFFECollColl, IsFie
else
t := Forms_SQRT2(A[r,r],q);
Forms_MultRow(D,r,1/t);
P := Forms_PERM_VAR(nplus1,r);
D := P*D;
Forms_PERM_VAR(D,r);
w := 1;
fi;
else
Expand All @@ -2307,8 +2253,7 @@ InstallMethod(BaseChangeOrthogonalQuadratic, [ IsMatrix and IsFFECollColl, IsFie
w := 2;
else
Forms_MultRow(D,r,1/Forms_SQRT2(c,q));
P := Forms_PERM_VAR(nplus1,r);
D := P*D;
Forms_PERM_VAR(D,r);
r := r - 1;
w := 1;
fi;
Expand All @@ -2325,14 +2270,12 @@ InstallMethod(BaseChangeOrthogonalQuadratic, [ IsMatrix and IsFFECollColl, IsFie
if b = t then
if c = t then
Forms_MultRow(D,r-1,1/Forms_SQRT2(a,q));
P := Forms_PERM_VAR(nplus1,r-1);
D := P*D;
Forms_PERM_VAR(D,r-1);
r := r - 1;
else
Forms_MultRow(D,r-1,1/Forms_SQRT2(a,q));
Forms_AddRows(D,r,r-1,Forms_SQRT2(c,q));
P := Forms_PERM_VAR(nplus1,r-1);
D := P*D;
Forms_PERM_VAR(D,r-1);
r := r - 1;
fi;
w := 1;
Expand All @@ -2345,13 +2288,8 @@ InstallMethod(BaseChangeOrthogonalQuadratic, [ IsMatrix and IsFFECollColl, IsFie
d := (a*c)/(b^2);
if Trace(gf,d) = t then
e := Forms_SQRT2(a,q);
P := IdentityMat(nplus1, gf);
s := Forms_QUAD_EQ(d,gf,h);
P[r-1,r-1] := (s+one)/e;
P[r-1,r] := e/b;
P[r,r-1] := s/e;
P[r,r] := e/b;
D := P*D;
Forms_TRANSFORM_2_BY_2(D, r-1, r, (s+one)/e, e/b, s/e, e/b);
w := 2;
else
c := Forms_SQRT2(c,q);
Expand Down

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