arb/acb_mat/approx_solve_tril.c

/*
    Copyright (C) 2018 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 "acb_mat.h"

static void
acb_approx_set(acb_t z, const acb_t x)
{
    arf_set(arb_midref(acb_realref(z)), arb_midref(acb_realref(x)));
    arf_set(arb_midref(acb_imagref(z)), arb_midref(acb_imagref(x)));
}

static void
acb_approx_mul(acb_t res, const acb_t x, const acb_t y, slong prec)
{
    arf_complex_mul(arb_midref(acb_realref(res)), arb_midref(acb_imagref(res)),
        arb_midref(acb_realref(x)), arb_midref(acb_imagref(x)), 
        arb_midref(acb_realref(y)), arb_midref(acb_imagref(y)), prec, ARB_RND);
}

static void
acb_approx_sub(acb_t z, const acb_t x, const acb_t y, slong prec)
{
    arf_sub(arb_midref(acb_realref(z)), arb_midref(acb_realref(x)), arb_midref(acb_realref(y)), prec, ARF_RND_DOWN);
    arf_sub(arb_midref(acb_imagref(z)), arb_midref(acb_imagref(x)), arb_midref(acb_imagref(y)), prec, ARF_RND_DOWN);
}

static void
acb_approx_addmul(acb_t z, const acb_t x, const acb_t y, acb_t t, slong prec)
{
    acb_approx_mul(t, x, y, prec);

    arf_add(arb_midref(acb_realref(z)),
        arb_midref(acb_realref(z)), 
        arb_midref(acb_realref(t)), prec, ARB_RND);
    arf_add(arb_midref(acb_imagref(z)),
        arb_midref(acb_imagref(z)), 
        arb_midref(acb_imagref(t)), prec, ARB_RND);
}

/* note: the tmp variable t should have zero radius */
static void
acb_approx_div(acb_t z, const acb_t x, const acb_t y, acb_t t, slong prec)
{
    arf_set(arb_midref(acb_realref(t)), arb_midref(acb_realref(y)));
    arf_set(arb_midref(acb_imagref(t)), arb_midref(acb_imagref(y)));

    acb_inv(t, t, prec);

    mag_zero(arb_radref(acb_realref(t)));
    mag_zero(arb_radref(acb_imagref(t)));

    acb_approx_mul(z, x, t, prec);
}

void
acb_mat_approx_solve_tril_classical(acb_mat_t X,
        const acb_mat_t L, const acb_mat_t B, int unit, slong prec)
{
    slong i, j, k, n, m;
    acb_ptr tmp;
    acb_t s, t;

    n = L->r;
    m = B->c;

    acb_init(s);
    acb_init(t);
    tmp = _acb_vec_init(n);

    for (i = 0; i < m; i++)
    {
        for (j = 0; j < n; j++)
            acb_approx_set(tmp + j, acb_mat_entry(X, j, i));

        for (j = 0; j < n; j++)
        {
            acb_zero(s);
            for (k = 0; k < j; k++)
                acb_approx_addmul(s, L->rows[j] + k, tmp + k, t, prec);
            acb_approx_sub(s, acb_mat_entry(B, j, i), s, prec);
            if (!unit)
                acb_approx_div(s, s, acb_mat_entry(L, j, j), t, prec);
            acb_approx_set(tmp + j, s);
        }

        for (j = 0; j < n; j++)
            acb_approx_set(acb_mat_entry(X, j, i), tmp + j);
    }

    _acb_vec_clear(tmp, n);
    acb_clear(s);
    acb_clear(t);
}

void
acb_mat_approx_solve_tril_recursive(acb_mat_t X,
        const acb_mat_t L, const acb_mat_t B, int unit, slong prec)
{
    acb_mat_t LA, LC, LD, XX, XY, BX, BY, T;
    slong r, n, m;

    n = L->r;
    m = B->c;
    r = n / 2;

    if (n == 0 || m == 0)
        return;

    /*
    Denoting inv(M) by M^, we have:

    [A 0]^ [X]  ==  [A^          0 ] [X]  ==  [A^ X]
    [C D]  [Y]  ==  [-D^ C A^    D^] [Y]  ==  [D^ (Y - C A^ X)]
    */
    acb_mat_window_init(LA, L, 0, 0, r, r);
    acb_mat_window_init(LC, L, r, 0, n, r);
    acb_mat_window_init(LD, L, r, r, n, n);
    acb_mat_window_init(BX, B, 0, 0, r, m);
    acb_mat_window_init(BY, B, r, 0, n, m);
    acb_mat_window_init(XX, X, 0, 0, r, m);
    acb_mat_window_init(XY, X, r, 0, n, m);

    acb_mat_approx_solve_tril(XX, LA, BX, unit, prec);

    /* acb_mat_submul(XY, BY, LC, XX); */
    acb_mat_init(T, LC->r, BX->c);
    acb_mat_mul(T, LC, XX, prec);
    acb_mat_get_mid(T, T);
    acb_mat_sub(XY, BY, T, prec);
    acb_mat_get_mid(XY, XY);
    acb_mat_clear(T);

    acb_mat_approx_solve_tril(XY, LD, XY, unit, prec);

    acb_mat_window_clear(LA);
    acb_mat_window_clear(LC);
    acb_mat_window_clear(LD);
    acb_mat_window_clear(BX);
    acb_mat_window_clear(BY);
    acb_mat_window_clear(XX);
    acb_mat_window_clear(XY);
}

void
acb_mat_approx_solve_tril(acb_mat_t X, const acb_mat_t L,
                                    const acb_mat_t B, int unit, slong prec)
{
    if (B->r < 8 || B->c < 8)
        acb_mat_approx_solve_tril_classical(X, L, B, unit, prec);
    else
        acb_mat_approx_solve_tril_recursive(X, L, B, unit, prec);
}