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/*
Copyright (C) 2012 Fredrik Johansson
This file is part of Arb.
Arb 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 2.1 of the License, or
(at your option) any later version. See <http://www.gnu.org/licenses/>.
*/
#include "flint/double_extras.h"
#include "acb_mat.h"
#include "bool_mat.h"
slong _arb_mat_exp_choose_N(const mag_t norm, slong prec);
static slong
_acb_mat_count_is_zero(const acb_mat_t A)
{
slong nz, i, j;
for (nz = 0, i = 0; i < acb_mat_nrows(A); i++)
for (j = 0; j < acb_mat_ncols(A); j++)
nz += acb_is_zero(acb_mat_entry(A, i, j));
return nz;
}
static void
_acb_mat_exp_diagonal(acb_mat_t B, const acb_mat_t A, slong prec)
{
slong n, i;
n = acb_mat_nrows(A);
if (B != A)
{
acb_mat_zero(B);
}
for (i = 0; i < n; i++)
{
acb_exp(acb_mat_entry(B, i, i), acb_mat_entry(A, i, i), prec);
}
}
void
acb_mat_exp(acb_mat_t B, const acb_mat_t A, slong prec)
{
slong i, j, dim, nz;
bool_mat_t S;
slong nildegree;
if (!acb_mat_is_square(A))
{
flint_printf("acb_mat_exp: a square matrix is required!\n");
flint_abort();
}
if (acb_mat_is_empty(A))
return;
dim = acb_mat_nrows(A);
if (dim == 1)
{
acb_exp(acb_mat_entry(B, 0, 0), acb_mat_entry(A, 0, 0), prec);
return;
}
if (acb_mat_is_real(A))
{
arb_mat_t R;
arb_mat_init(R, dim, dim);
for (i = 0; i < dim; i++)
for (j = 0; j < dim; j++)
arb_set(arb_mat_entry(R, i, j),
acb_realref(acb_mat_entry(A, i, j)));
arb_mat_exp(R, R, prec);
acb_mat_set_arb_mat(B, R);
arb_mat_clear(R);
return;
}
nz = _acb_mat_count_is_zero(A);
if (nz == dim * dim)
{
acb_mat_one(B);
return;
}
bool_mat_init(S, dim, dim);
if (nz == 0)
{
nildegree = -1;
bool_mat_complement(S, S);
}
else
{
for (i = 0; i < dim; i++)
for (j = 0; j < dim; j++)
bool_mat_set_entry(S, i, j, !acb_is_zero(acb_mat_entry(A, i, j)));
if (bool_mat_is_diagonal(S))
{
_acb_mat_exp_diagonal(B, A, prec);
bool_mat_clear(S);
return;
}
else
{
nildegree = bool_mat_nilpotency_degree(S);
}
}
/* evaluate using scaling and squaring of truncated taylor series */
{
slong wp, N, q, r;
mag_t norm, err;
acb_mat_t T;
wp = prec + 3 * FLINT_BIT_COUNT(prec);
mag_init(norm);
mag_init(err);
acb_mat_init(T, dim, dim);
acb_mat_bound_inf_norm(norm, A);
q = pow(wp, 0.25); /* wanted magnitude */
if (mag_cmp_2exp_si(norm, 2 * wp) > 0) /* too big */
r = 2 * wp;
else if (mag_cmp_2exp_si(norm, -q) < 0) /* tiny, no need to reduce */
r = 0;
else
r = FLINT_MAX(0, q + MAG_EXP(norm)); /* reduce to magnitude 2^(-r) */
acb_mat_scalar_mul_2exp_si(T, A, -r);
mag_mul_2exp_si(norm, norm, -r);
N = _arb_mat_exp_choose_N(norm, wp);
/* if positive, nildegree is an upper bound on nilpotency degree */
if (nildegree > 0)
N = FLINT_MIN(N, nildegree);
mag_exp_tail(err, norm, N);
acb_mat_exp_taylor_sum(B, T, N, wp);
/* add truncation error to entries for which it is not ruled out */
if (nz == 0)
{
for (i = 0; i < dim; i++)
for (j = 0; j < dim; j++)
acb_add_error_mag(acb_mat_entry(B, i, j), err);
}
else if (nildegree < 0 || N < nildegree)
{
slong w;
fmpz_mat_t W;
fmpz_mat_init(W, dim, dim);
w = bool_mat_all_pairs_longest_walk(W, S);
if (w + 1 != nildegree) flint_abort(); /* assert */
for (i = 0; i < dim; i++)
{
for (j = 0; j < dim; j++)
{
slong d = fmpz_get_si(fmpz_mat_entry(W, i, j)) + 1;
if (d < 0 || N < d)
{
acb_add_error_mag(acb_mat_entry(B, i, j), err);
}
}
}
fmpz_mat_clear(W);
}
for (i = 0; i < r; i++)
{
acb_mat_sqr(T, B, wp);
acb_mat_swap(T, B);
}
for (i = 0; i < dim; i++)
for (j = 0; j < dim; j++)
acb_set_round(acb_mat_entry(B, i, j),
acb_mat_entry(B, i, j), prec);
mag_clear(norm);
mag_clear(err);
acb_mat_clear(T);
}
bool_mat_clear(S);
}
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