arb/arb_mat/ldl.c

/*
    Copyright (C) 2016 Arb authors

    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 "arb_mat.h"

static void
_arb_sqr(arb_t dest, const arb_t src, slong prec)
{
    arb_mul(dest, src, src, prec);
}

int
_arb_mat_ldl_inplace(arb_mat_t A, slong prec)
{
    slong n, i, j, k;
    arb_t tmp;
    int result;

    n = arb_mat_nrows(A);
    arb_init(tmp);

    result = 1;
    for (i = 0; i < n && result; i++)
    {
        for (j = 0; j < i; j++)
        {
            for (k = 0; k < j; k++)
            {
                arb_mul(tmp,
                        arb_mat_entry(A, i, k),
                        arb_mat_entry(A, j, k), prec);
                arb_submul(arb_mat_entry(A, i, j),
                           arb_mat_entry(A, k, k), tmp, prec);
            }
            arb_div(arb_mat_entry(A, i, j),
                    arb_mat_entry(A, i, j),
                    arb_mat_entry(A, j, j), prec);
        }
        for (k = 0; k < i; k++)
        {
            _arb_sqr(tmp, arb_mat_entry(A, i, k), prec);
            arb_submul(arb_mat_entry(A, i, i),
                       arb_mat_entry(A, k, k), tmp, prec);
        }
        if (!arb_is_positive(arb_mat_entry(A, i, i)))
            result = 0;
    }

    arb_clear(tmp);

    return result;
}

int
_arb_mat_ldl_golub_and_van_loan(arb_mat_t A, slong prec)
{
    slong n, i, j, k;
    arb_struct *v;
    int result;

    n = arb_mat_nrows(A);
    v = _arb_vec_init(n);

    result = 1;
    for (j = 0; j < n; j++)
    {
        for (i = 0; i < j; i++)
        {
            arb_mul(v + i,
                    arb_mat_entry(A, j, i),
                    arb_mat_entry(A, i, i), prec);
        }

        arb_set(v + j, arb_mat_entry(A, j, j));
        for (i = 0; i < j; i++)
        {
            arb_submul(v + j, arb_mat_entry(A, j, i), v + i, prec);
        }

        if (!arb_is_positive(v + j))
        {
            result = 0;
            break;
        }

        arb_set(arb_mat_entry(A, j, j), v + j);
        for (i = j + 1; i < n; i++)
        {
            for (k = 0; k < j; k++)
            {
                arb_submul(arb_mat_entry(A, i, j),
                           arb_mat_entry(A, i, k), v + k, prec);
            }
            arb_div(arb_mat_entry(A, i, j),
                    arb_mat_entry(A, i, j), v + j, prec);
        }
    }

    _arb_vec_clear(v, n);

    return result;
}

int
arb_mat_ldl(arb_mat_t L, const arb_mat_t A, slong prec)
{
    slong n;
    int result;

    if (!arb_mat_is_square(A))
    {
        flint_printf("arb_mat_ldl: a square matrix is required\n");
        flint_abort();
    }

    if (arb_mat_nrows(L) != arb_mat_nrows(A) ||
        arb_mat_ncols(L) != arb_mat_ncols(A))
    {
        flint_printf("arb_mat_ldl: incompatible dimensions\n");
        flint_abort();
    }

    if (arb_mat_is_empty(A))
        return 1;

    n = arb_mat_nrows(A);

    arb_mat_set(L, A);

    if (n == 1)
        return arb_is_positive(arb_mat_entry(L, 0, 0));

    result = _arb_mat_ldl_golub_and_van_loan(L, prec);

    /* set the strictly upper triangular region of L to zero */
    {
        slong i, j;
        for (i = 0; i < n; i++)
            for (j = i+1; j < n; j++)
                arb_zero(arb_mat_entry(L, i, j));
    }

    return result;
}