/usr/share/gap/pkg/Polycyclic/gap/matrix/hnf.gi is in gap-polycyclic 2.11-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 | #############################################################################
##
#F FindNiceRowOneNorm
#F FindNiceRowTwoNorm
#F FindNiceRowInfinityNorm
##
## Functions that select during an HNF computation a row from a matrix <M>
## such that the row is minimal with respect to a chosen norm and can
## function as pivot entry in position i,j.
##
#F FindNiceRowInfinityNormRowOps
##
## Does the same as FindNiceRowInfinityNorm() but records the row
## operations.
##
FindNiceRowOneNorm := function( M, i, j )
local m, n, k, a, r;
m := Length( M ); n := Length( M[1] );
for k in [i+1..m] do
a := AbsInt( M[k][j] );
if a <> 0 and
(a < AbsInt( M[i][j] )
or (a = AbsInt( M[i][j] )
and Sum( M[k], AbsInt ) < Sum( M[i], AbsInt ) ) ) then
r := M[i]; M[i] := M[k]; M[k] := r;
fi;
od;
return;
end;
FindNiceRowTwoNorm := function( M, i, j )
local m, n, k, a, r;
m := Length( M ); n := Length( M[1] );
for k in [i+1..m] do
a := AbsInt( M[k][j] );
if a <> 0 and
(a < AbsInt( M[i][j] )
or (a = AbsInt( M[i][j] )
and M[k]*M[k] < M[i]*M[i] ) ) then
r := M[i]; M[i] := M[k]; M[k] := r;
fi;
od;
return;
end;
FindNiceRowInfinityNorm := function( M, i, j )
local m, n, k, a, r;
m := Length( M ); n := Length( M[1] );
for k in [i+1..m] do
a := AbsInt( M[k][j] );
if a <> 0 and
(a < AbsInt( M[i][j] )
or (a = AbsInt( M[i][j] )
and Number( M[k], x->x<>0 ) < Number( M[i], x->x<>0 ) ) ) then
r := M[i]; M[i] := M[k]; M[k] := r;
fi;
od;
return;
end;
FindNiceRowInfinityNormRowOps := function( M, Q, i, j )
local m, n, k, a, r;
m := Length( M ); n := Length( M[1] );
for k in [i+1..m] do
a := AbsInt( M[k][j] );
if a <> 0 and
(a < AbsInt( M[i][j] )
or (a = AbsInt( M[i][j] )
and Number( M[k], x->x<>0 ) < Number( M[i], x->x<>0 ) ) ) then
r := M[i]; M[i] := M[k]; M[k] := r;
r := Q[i]; Q[i] := Q[k]; Q[k] := r;
fi;
od;
return;
end;
#############################################################################
##
#F HNFIntMat . . . . . . . . . . . . Hermite Normalform of an integer matrix
##
HNFIntMat := function( M )
local MM, m, n, i, j, k, r, Cleared, a;
if M = [] then return []; fi;
MM := M;
M := List( M, ShallowCopy );
m := Length( M ); n := Length( M[1] );
i := 1; j := 1;
while i <= m and j <= n do
# find first k with M[k][j] non-zero
k := i; while k <= m and M[k][j] = 0 do k := k+1; od;
if k <= m then
# swap rows
r := M[i]; M[i] := M[k]; M[k] := r;
# find nicest row with M[k][j] non-zero
FindNiceRowInfinityNorm( M, i, j );
if M[i][j] < 0 then M[i] := -1 * M[i]; fi;
# reduce all other entries in this columns with the pivot entry
Cleared := true;
for k in [i+1..m] do
a := QuoInt(M[k][j],M[i][j]);
if a <> 0 then
AddRowVector( M[k], M[i], -a, i, n );
fi;
if M[k][j] <> 0 then Cleared := false; fi;
od;
# if all entries below the pivot are zero, reduce above the
# pivot and then move on along the diagonal
if Cleared then
for k in [1..i-1] do
a := QuoInt(M[k][j],M[i][j]);
if M[k][j] < 0 and M[k][j] mod M[i][j] <> 0 then
a := a-1;
fi;
if a <> 0 then
AddRowVector( M[k], M[i], -a, 1, n );
fi;
od;
i := i+1; j := j+1;
fi;
else
# increase column counter if column has only zeroes
j := j+1;
fi;
od;
return M{[1..i-1]};
end;
#############################################################################
##
#F HNFIntMat . . . . . . . . . . . . Hermite Normal Form plus row operations
##
HNFIntMatRowOps := function( M )
local MM, m, n, Q, i, j, k, r, Cleared, a;
if M = [] then return []; fi;
MM := M;
M := List( M, ShallowCopy );
m := Length( M ); n := Length( M[1] );
Q := IdentityMat( Length(M) );
i := 1; j := 1;
while i <= m and j <= n do
# find first k with M[k][j] non-zero
k := i; while k <= m and M[k][j] = 0 do k := k+1; od;
if k <= m then
# swap rows
r := M[i]; M[i] := M[k]; M[k] := r;
r := Q[i]; Q[i] := Q[k]; Q[k] := r;
# find nicest row with M[k][j] non-zero
FindNiceRowInfinityNormRowOps( M, Q, i, j );
if M[i][j] < 0 then M[i] := -1 * M[i]; Q[i] := -1 * Q[i]; fi;
# reduce all other entries in this columns with the pivot entry
Cleared := true;
for k in [i+1..m] do
a := QuoInt(M[k][j],M[i][j]);
if a <> 0 then
AddRowVector( M[k], M[i], -a, i, n );
AddRowVector( Q[k], Q[i], -a, 1, m );
fi;
if M[k][j] <> 0 then Cleared := false; fi;
od;
# if all entries below the pivot are zero, reduce above the
# pivot and then move on along the diagonal
if Cleared then
for k in [1..i-1] do
a := QuoInt(M[k][j],M[i][j]);
if M[k][j] < 0 and M[k][j] mod M[i][j] <> 0 then
a := a-1;
fi;
if a <> 0 then
AddRowVector( M[k], M[i], -a, 1, n );
AddRowVector( Q[k], Q[i], -a, 1, m );
fi;
od;
i := i+1; j := j+1;
fi;
else
# increase column counter if column has only zeroes
j := j+1;
fi;
od;
return [ M, Q ];
end;
#############################################################################
##
#F DiagonalFormIntMat . . . . diagonal form of an integer matrix plus column
#F operations
##
DiagonalFormIntMat := function( M )
local Q, pair;
M := HNFIntMat( M );
Q := IdentityMat( Length(M[1]) );
while not IsDiagonalMat( M ) do
M := TransposedMat( M );
pair := HNFIntMatRowOps( M );
Q := Q * TransposedMat( pair[2] );
M := TransposedMat( pair[1] );
if not IsDiagonalMat( M ) then
M := HNFIntMat( M );
fi;
od;
return [ M, Q ];
end;
##
## This function takes a matrix M in HNF and eliminates for each row whose
## leading entry is 1 the remaining entries of the row. This corresponds
## to a sequence of column operations. Note that all entries above and
## below the 1 are 0 since the matrix is in HNF.
##
## The function returns the transformed matrix M' together with the
## transforming matrix Q such that
## M * Q = M'
##
ClearOutWithOnes := function( M )
local Q, i, k, j, l;
M := List( M, ShallowCopy );
Q := IdentityMat( Length(M[1]) );
for i in [1..Length(M)] do
k := First( [1..Length(M[i])], e -> M[i][e] <> 0 );
if M[i][k] = 1 then
for j in [k+1..Length(M[i])] do
if M[i][j] <> 0 then
Q[j] := Q[j] - M[i][j] * Q[k];
M[i][j] := 0;
fi;
od;
fi;
od;
return [M, TransposedMat(Q)];
end;
##
## After we have cleared out those rows of the HNF whose leading entry is 1,
## we need to compute a diagonal form of the rest of the matrix. This
## routines cuts out the relevant part, computes a diagonal form of it, puts
## that back into the matrix and returns the performed columns operations.
##
CutOutNonOnes := function( M )
local rows, cols, nf, Q, i;
# Find all rows whose leading entry is 1
rows := Filtered( [1..Length(M)], i->First( M[i], e->e <> 0 ) = 1 );
if rows = [1..Length(M)] then
return IdentityMat( Length(M[1]) );
fi;
# Find those colums where the leading entry is
cols := List( rows, i->Position( M[i], 1 ) );
# The complement are those rows whose leading entry is not one and those
# colums that do not have a 1 in a leading position.
rows := Difference( [1..Length(M)], rows );
cols := Difference( [1..Length(M[1])], cols );
# skip leading zeroes
i := 1; while M[rows[1]][cols[i]] = 0 do i := i+1; od;
cols := cols{[i..Length(cols)]};
nf := DiagonalFormIntMat( M{rows}{cols} );
Q := IdentityMat( Length(M[1]) );
for i in cols do Q[i][i] := 0; od;
Q{cols}{cols} := nf[2];
M{rows}{cols} := nf[1];
return Q;
end;
##
## The HNF of a matrix that comes out of the consistency test for a
## central extension tends to have a lot of rows whose leading entry is 1.
## In particular, if we do not have an efficient strategy for computing
## tails, we have many generators which can be expressed by others.
##
## This is a simple consequence of the fact that we add about n^2/2 new
## generators to the polycyclic presentation if the the group has n
## generators. But it is clear that the rank of R/[R,F] is bounded from
## above by n. Therefore, about n^2/2 generators will be expressed by
## others.
##
## We return a diagonal form of M and the matrix of column operations in
## the same format as NormalFormIntMat()
##
## An example where this performs much better than NormalFormIntMat is
## given by
## G:=HeisenbergPcpGroup(2);
## NonAbelianTensorSquarePlusEpimorphism(G);
## Timing the call to NormalFormConsistencyRelations and comparing it to
## an equivalent NormalFormIntMat call yielded 50 msec vs. 1000 msec,
## i.e. a speedup by factor 20.
##
NormalFormConsistencyRelations := function( M )
local nf, Q, rows, cols, small, nfim, QQ;
M := HNFIntMat( M );
nf := ClearOutWithOnes( M );
M := nf[1];
Q := nf[2];
Q := Q * CutOutNonOnes( M );
return rec( normal := M, coltrans := Q );
end;
|