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/*
Copyright (C) 2010-2012 Fredrik Johansson
Copyright (C) 2014 Alex J. Best
This file is part of FLINT.
FLINT is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 3 of the License, or
(at your option) any later version. See <https://www.gnu.org/licenses/>.
*/
#include "ulong_extras.h"
#include "perm.h"
#include "nmod_mat.h"
#include "fmpz.h"
#include "fmpz_mat.h"
slong
fmpz_mat_rref_mul(fmpz_mat_t R, fmpz_t den, const fmpz_mat_t A)
{
nmod_mat_t Amod;
ulong p;
slong i, j, m, n, rank, * pivs, * P;
fmpz_mat_t B, C, D, E, E2, F, FD;
m = fmpz_mat_nrows(A);
n = fmpz_mat_ncols(A);
pivs = (slong *) flint_malloc(n * sizeof(slong));
P = _perm_init(m);
/* use 16 bit primes to ensure it is unlikely we hit a bad one and so that
the modular computations are not too long */
p = 1 << 16;
while (1)
{
p = n_nextprime(p, 1);
nmod_mat_init(Amod, m, n, p);
fmpz_mat_get_nmod_mat(Amod, A);
rank = _nmod_mat_rref(Amod, pivs, P);
nmod_mat_clear(Amod);
/* stop early if the rank is the number of columns */
if (rank == n)
{
fmpz_mat_one(R);
fmpz_one(den);
flint_free(pivs);
_perm_clear(P);
return rank;
}
fmpz_mat_init(B, rank, rank);
fmpz_mat_init(C, rank, n - rank);
/* set B to be the pivot columns and rows and C to be the non-pivot
columns in the pivot rows */
for (i = 0; i < rank; i++)
{
for (j = 0; j < rank; j++)
fmpz_set(fmpz_mat_entry(B, i, j),
fmpz_mat_entry(A, P[i], pivs[j]));
for (j = 0; j < n - rank; j++)
fmpz_set(fmpz_mat_entry(C, i, j),
fmpz_mat_entry(A, P[i], pivs[rank + j]));
}
/* solve B*E2 = den*C */
fmpz_mat_init(E2, rank, n - rank);
if (!fmpz_mat_solve(E2, den, B, C))
{
flint_throw(FLINT_ERROR, "(fmpz_mat_rref_mul): "
"Singular input matrix for solve.\n");
}
fmpz_mat_clear(B);
fmpz_mat_clear(C);
fmpz_mat_init(E, rank, n);
/* move columns of E2 and identity matrix into E so that it should be
in rref */
for (i = 0; i < rank; i++)
{
fmpz_set(fmpz_mat_entry(E, i, pivs[i]), den);
for (j = 0; j < n - rank; j++)
fmpz_set(fmpz_mat_entry(E, i, pivs[rank + j]),
fmpz_mat_entry(E2, i, j));
}
fmpz_mat_clear(E2);
if (!fmpz_mat_is_in_rref_with_rank(E, den, rank))
{
fmpz_mat_clear(E);
continue;
}
/* set D to be the nullspace basis matrix for E */
fmpz_mat_init(D, n, n - rank);
for (j = 0; j < n - rank; j++)
{
fmpz_set(fmpz_mat_entry(D, pivs[rank + j], j), den);
for (i = 0; i < rank; i++)
fmpz_neg(fmpz_mat_entry(D, pivs[i], j),
fmpz_mat_entry(E, i, pivs[rank + j]));
}
fmpz_mat_init(F, m - rank, n);
for (i = 0; i < m - rank; i++)
for (j = 0; j < n; j++)
fmpz_set(fmpz_mat_entry(F, i, j),
fmpz_mat_entry(A, P[rank + i], j));
fmpz_mat_init(FD, m - rank, n - rank);
fmpz_mat_mul(FD, F, D);
fmpz_mat_clear(F);
fmpz_mat_clear(D);
/* if FD = 0 we have computed the rref right so stop, otherwise try a
different p in the next iteration */
if (fmpz_mat_is_zero(FD))
break;
fmpz_mat_clear(E);
fmpz_mat_clear(FD);
}
/* write the entries of E into R and zeroes at the bottom */
for (i = 0; i < rank; i++)
for (j = 0; j < n; j++)
fmpz_set(fmpz_mat_entry(R, i, j), fmpz_mat_entry(E, i, j));
for (i = rank; i < m; i++)
for (j = 0; j < n; j++)
fmpz_zero(fmpz_mat_entry(R, i, j));
fmpz_mat_clear(E);
fmpz_mat_clear(FD);
flint_free(pivs);
_perm_clear(P);
return rank;
}