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

    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 "fmpcb.h"

void
fmpcb_invroot_newton(fmpcb_t r, const fmpcb_t a, ulong m,
    const fmpcb_t r0, long startprec, long prec)
{
    long iters, precs[FLINT_BITS];
    long i, extra, wp, rad_prec;

    fmpr_t en, enew, zlo, zhi, v;
    fmpcb_t t, z, z_exact;

    /* stay clear of overflow */
    if (m + 2 < m) abort();

    fmpr_init(en);
    fmpr_init(enew);
    fmpr_init(zlo);
    fmpr_init(zhi);
    fmpr_init(v);

    fmpcb_init(t);
    fmpcb_init(z);
    fmpcb_init(z_exact);

    fmpcb_set(z, r0);

    rad_prec = FMPRB_RAD_PREC;

    extra = 10 + FLINT_BIT_COUNT(m);
    precs[0] = prec + extra;
    iters = 1;

    while ((iters < FLINT_BITS) && (precs[iters-1] + extra > 2*startprec))
    {
        precs[iters] = (precs[iters-1] / 2) + extra;
        iters++;
    }

    for (i = iters - 1; i >= 0; i--)
    {
        wp = precs[i];

        /* printf("lifting to precision %ld\n", wp); */

        /* bounds for old error */
        fmpcb_get_rad_ubound_fmpr(en, z, rad_prec);
        fmpcb_get_abs_lbound_fmpr(zlo, z, rad_prec);
        fmpcb_get_abs_ubound_fmpr(zhi, z, rad_prec);

        /* to improve, we require |z| > 0, i.e. en < |z| */
        if (fmpr_cmp(en, zlo) >= 0)
            break;

        /* z[n+1] = z[n] * (m + 1 - a * z[n]^m) / m */
        fmpcb_set(z_exact, z);
        fmpr_zero(fmprb_radref(fmpcb_realref(z_exact)));
        fmpr_zero(fmprb_radref(fmpcb_imagref(z_exact)));

        fmpcb_pow_ui(t, z_exact, m, wp);
        fmpcb_mul(t, t, a, wp);
        fmpcb_neg(t, t);
        fmpcb_add_ui(t, t, m + 1, wp);
        fmpcb_mul(t, t, z_exact, wp);
        fmpcb_div_ui(t, t, m, wp);

        /* new error bound = en^2 * (m+1) / (|zn| / (|zn| - en))^(m+2) / |zn| */
        fmpr_mul(enew, en, en, rad_prec, FMPR_RND_UP);
        fmpr_mul_ui(enew, enew, m + 1, rad_prec, FMPR_RND_UP);
        fmpr_mul(enew, enew, zhi, rad_prec, FMPR_RND_UP);
        fmpr_sub(v, zlo, en, rad_prec, FMPR_RND_DOWN);
        fmpr_div(v, zhi, v, rad_prec, FMPR_RND_UP);
        fmpr_pow_sloppy_ui(v, v, m + 2, rad_prec, FMPR_RND_UP);
        fmpr_div(v, v, zlo, rad_prec, FMPR_RND_UP);
        fmpr_mul(enew, enew, v, rad_prec, FMPR_RND_UP);

        /* quit if there was no improvement */
        if (fmpr_cmp(enew, en) >= 0)
            break;

        fmpcb_set(z, t);

        /* add upper bound for new radius */
        fmprb_add_error_fmpr(fmpcb_realref(z), enew);
        fmprb_add_error_fmpr(fmpcb_imagref(z), enew);
    }

    fmpcb_set(r, z);

    fmpcb_clear(t);
    fmpcb_clear(z);
    fmpcb_clear(z_exact);

    fmpr_clear(en);
    fmpr_clear(enew);
    fmpr_clear(zlo);
    fmpr_clear(zhi);
    fmpr_clear(v);
}