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/*
Copyright (C) 2018 arbguest
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 "arb.h"
#include "arb_mat.h"
/*
* Helper function to compute a lower bound of 1 - inf_norm(I - A*B).
* Returns zero when this lower bound is zero.
*/
static int _mag_err_complement(mag_t m,
const arb_mat_t A, const arb_mat_t B, slong prec)
{
slong n;
arb_mat_t I, E;
mag_t err;
n = arb_mat_nrows(A);
arb_mat_init(I, n, n);
arb_mat_one(I);
arb_mat_init(E, n, n);
arb_mat_mul(E, A, B, prec);
arb_mat_sub(E, I, E, prec);
mag_init(err);
arb_mat_bound_inf_norm(err, E);
mag_one(m);
mag_sub_lower(m, m, err);
mag_clear(err);
arb_mat_clear(I);
arb_mat_clear(E);
return !mag_is_zero(m);
}
int arb_mat_solve_preapprox(arb_mat_t X, const arb_mat_t A,
const arb_mat_t B, const arb_mat_t R, const arb_mat_t T, slong prec)
{
int result;
slong m, n;
mag_t d;
result = 0;
n = arb_mat_nrows(A);
m = arb_mat_ncols(X);
if (n == 0 || m == 0)
return 1;
/* Use Theorem 10.2 of Rump in Acta Numerica 2010 */
mag_init(d);
if (_mag_err_complement(d, R, A, prec))
{
arb_mat_t C;
arb_mat_init(C, n, m);
arb_mat_mul(C, A, T, prec);
arb_mat_sub(C, C, B, prec);
arb_mat_mul(C, R, C, prec);
/* Each column gets its own error bound. */
arb_mat_set(X, T);
{
slong i, j;
mag_t e, err;
mag_init(e);
mag_init(err);
for (j = 0; j < m; j++)
{
mag_zero(err);
for (i = 0; i < n; i++)
{
arb_get_mag(e, arb_mat_entry(C, i, j));
mag_max(err, err, e);
}
mag_div(err, err, d);
for (i = 0; i < n; i++)
{
arb_add_error_mag(arb_mat_entry(X, i, j), err);
}
}
mag_clear(e);
mag_clear(err);
}
arb_mat_clear(C);
result = 1;
}
mag_clear(d);
return result;
}