arb/acb_mat/exp.c

/*=============================================================================

    This file is part of ARB.

    ARB is free software; you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
    the Free Software Foundation; either version 2 of the License, or
    (at your option) any later version.

    ARB is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
    GNU General Public License for more details.

    You should have received a copy of the GNU General Public License
    along with ARB; if not, write to the Free Software
    Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA

=============================================================================*/
/******************************************************************************

    Copyright (C) 2012 Fredrik Johansson

******************************************************************************/

#include "double_extras.h"
#include "acb_mat.h"

slong _arb_mat_exp_choose_N(const mag_t norm, slong prec);
void _fmpz_mat_transitive_closure(fmpz_mat_t A, const fmpz_mat_t B);

int
_acb_mat_is_diagonal(const acb_mat_t A)
{
    slong i, j;
    for (i = 0; i < acb_mat_nrows(A); i++)
        for (j = 0; j < acb_mat_ncols(A); j++)
            if (i != j && !acb_is_zero(acb_mat_entry(A, i, j)))
                return 0;
    return 1;
}

int
_acb_mat_any_is_zero(const acb_mat_t A)
{
    slong i, j;
    for (i = 0; i < acb_mat_nrows(A); i++)
        for (j = 0; j < acb_mat_ncols(A); j++)
            if (acb_is_zero(acb_mat_entry(A, i, j)))
                return 1;
    return 0;
}

void
_acb_mat_exp_get_structure(fmpz_mat_t C, const acb_mat_t A)
{
    slong i, j, dim;

    dim = acb_mat_nrows(A);
    fmpz_mat_zero(C);
    for (i = 0; i < dim; i++)
    {
        for (j = 0; j < dim; j++)
        {
            if (!acb_is_zero(acb_mat_entry(A, i, j)))
            {
                fmpz_one(fmpz_mat_entry(C, i, j));
            }
        }
    }
    _fmpz_mat_transitive_closure(C, C);
}

void
_acb_mat_exp_set_structure(acb_mat_t B, const fmpz_mat_t C)
{
    slong i, j, dim;

    dim = acb_mat_nrows(B);
    for (i = 0; i < dim; i++)
    {
        for (j = 0; j < dim; j++)
        {
            if (fmpz_is_zero(fmpz_mat_entry(C, i, j)))
            {
                if (i == j)
                {
                    acb_one(acb_mat_entry(B, i, j));
                }
                else
                {
                    acb_zero(acb_mat_entry(B, i, j));
                }
            }
        }
    }
}

/* evaluates the truncated Taylor series (assumes no aliasing) */
void
_acb_mat_exp_taylor(acb_mat_t S, const acb_mat_t A, slong N, slong prec)
{
    if (N == 1)
    {
        acb_mat_one(S);
    }
    else if (N == 2)
    {
        acb_mat_one(S);
        acb_mat_add(S, S, A, prec);
    }
    else if (N == 3)
    {
        acb_mat_t T;
        acb_mat_init(T, acb_mat_nrows(A), acb_mat_nrows(A));
        acb_mat_sqr(T, A, prec);
        acb_mat_scalar_mul_2exp_si(T, T, -1);
        acb_mat_add(S, A, T, prec);
        acb_mat_one(T);
        acb_mat_add(S, S, T, prec);
        acb_mat_clear(T);
    }
    else
    {
        slong i, lo, hi, m, w, dim;
        acb_mat_struct * pows;
        acb_mat_t T, U;
        fmpz_t c, f;

        dim = acb_mat_nrows(A);
        m = n_sqrt(N);
        w = (N + m - 1) / m;

        fmpz_init(c);
        fmpz_init(f);
        pows = flint_malloc(sizeof(acb_mat_t) * (m + 1));
        acb_mat_init(T, dim, dim);
        acb_mat_init(U, dim, dim);

        for (i = 0; i <= m; i++)
        {
            acb_mat_init(pows + i, dim, dim);
            if (i == 0)
                acb_mat_one(pows + i);
            else if (i == 1)
                acb_mat_set(pows + i, A);
            else
                acb_mat_mul(pows + i, pows + i - 1, A, prec);
        }

        acb_mat_zero(S);
        fmpz_one(f);

        for (i = w - 1; i >= 0; i--)
        {
            lo = i * m;
            hi = FLINT_MIN(N - 1, lo + m - 1);

            acb_mat_zero(T);
            fmpz_one(c);

            while (hi >= lo)
            {
                acb_mat_scalar_addmul_fmpz(T, pows + hi - lo, c, prec);
                if (hi != 0)
                    fmpz_mul_ui(c, c, hi);
                hi--;
            }

            acb_mat_mul(U, pows + m, S, prec);
            acb_mat_scalar_mul_fmpz(S, T, f, prec);
            acb_mat_add(S, S, U, prec);
            fmpz_mul(f, f, c);
        }

        acb_mat_scalar_div_fmpz(S, S, f, prec);

        fmpz_clear(c);
        fmpz_clear(f);
        for (i = 0; i <= m; i++)
            acb_mat_clear(pows + i);
        flint_free(pows);
        acb_mat_clear(T);
        acb_mat_clear(U);
    }
}

void
acb_mat_exp(acb_mat_t B, const acb_mat_t A, slong prec)
{
    slong i, j, dim, wp, N, q, r;
    mag_t norm, err;
    acb_mat_t T;
    int is_real;

    dim = acb_mat_nrows(A);

    if (dim != acb_mat_ncols(A))
    {
        flint_printf("acb_mat_exp: a square matrix is required!\n");
        abort();
    }

    if (dim == 0)
    {
        return;
    }
    else if (dim == 1)
    {
        acb_exp(acb_mat_entry(B, 0, 0), acb_mat_entry(A, 0, 0), prec);
        return;
    }

    /* todo: generalize to (possibly permuted) block diagonal structure */
    if (_acb_mat_is_diagonal(A))
    {
        if (B != A)
        {
            acb_mat_zero(B);
        }
        for (i = 0; i < dim; i++)
        {
            acb_exp(acb_mat_entry(B, i, i), acb_mat_entry(A, i, i), prec);
        }
        return;
    }

    is_real = acb_mat_is_real(A);

    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);

    if (mag_is_zero(norm))
    {
        acb_mat_one(B);
    }
    else
    {
        fmpz_mat_t S;
        int using_structure;

        using_structure = _acb_mat_any_is_zero(A);
        if (using_structure)
        {
            fmpz_mat_init(S, dim, dim);
            _acb_mat_exp_get_structure(S, 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);
        mag_exp_tail(err, norm, N);

        _acb_mat_exp_taylor(B, T, N, wp);

        if (is_real)
        {
            for (i = 0; i < dim; i++)
                for (j = 0; j < dim; j++)
                    arb_add_error_mag(acb_realref(acb_mat_entry(B, i, j)), err);
        }
        else
        {
            for (i = 0; i < dim; i++)
                for (j = 0; j < dim; j++)
                    acb_add_error_mag(acb_mat_entry(B, i, j), err);
        }

        if (using_structure)
        {
            _acb_mat_exp_set_structure(B, S);
            fmpz_mat_clear(S);
        }

        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);
}