flint-sys 0.9.0

Bindings to the FLINT C library
Documentation 
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/*
    Copyright (C) 2014 Fredrik Johansson

    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 "acb_poly.h"
#include "bernoulli.h"

void
_acb_poly_zeta_em_tail_naive(acb_ptr sum, const acb_t s, const acb_t Na, acb_srcptr Nasx, slong M, slong d, slong prec)
{
    acb_ptr u, term;
    acb_t Na2, splus, rec;
    arb_t x;
    fmpz_t c;
    int aint;
    slong r;

    BERNOULLI_ENSURE_CACHED(2 * M);

    u = _acb_vec_init(d);
    term = _acb_vec_init(d);
    acb_init(splus);
    acb_init(rec);
    acb_init(Na2);
    arb_init(x);
    fmpz_init(c);

    _acb_vec_zero(sum, d);

    /* u = 1/2 * Nasx */
    _acb_vec_scalar_mul_2exp_si(u, Nasx, d, -WORD(1));

    /* term = u * (s+x) / (N+a) */
    _acb_poly_mullow_cpx(u, u, d, s, d, prec);
    _acb_vec_scalar_div(term, u, d, Na, prec);

    /* (N+a)^2 or 1/(N+a)^2 */
    acb_mul(Na2, Na, Na, prec);
    aint = acb_is_int(Na2);

    if (!aint)
        acb_inv(Na2, Na2, prec);

    for (r = 1; r <= M; r++)
    {
        /* flint_printf("sum 2: %wd %wd\n", r, M); */

        /* sum += bernoulli number * term */
        arb_set_round_fmpz(x, fmpq_numref(bernoulli_cache + 2 * r), prec);
        arb_div_fmpz(x, x, fmpq_denref(bernoulli_cache + 2 * r), prec);

        _acb_vec_scalar_mul_arb(u, term, d, x, prec);
        _acb_vec_add(sum, sum, u, d, prec);

        /* multiply term by ((s+x)+2r-1)((s+x)+2r) / ((N+a)^2 * (2*r+1)*(2*r+2)) */
        acb_set(splus, s);
        arb_add_ui(acb_realref(splus), acb_realref(splus), 2*r-1, prec);
        _acb_poly_mullow_cpx(term, term, d, splus, d, prec);
        arb_add_ui(acb_realref(splus), acb_realref(splus), 1, prec);
        _acb_poly_mullow_cpx(term, term, d, splus, d, prec);

        /* TODO: combine with previous multiplication? */
        if (aint)
        {
            arb_mul_ui(x, acb_realref(Na2), 2*r+1, prec);
            arb_mul_ui(x, x, 2*r+2, prec);
            _acb_vec_scalar_div_arb(term, term, d, x, prec);
        }
        else
        {
            fmpz_set_ui(c, 2*r+1);
            fmpz_mul_ui(c, c, 2*r+2);
            acb_div_fmpz(rec, Na2, c, prec);
            _acb_vec_scalar_mul(term, term, d, rec, prec);
        }
    }

    _acb_vec_clear(u, d);
    _acb_vec_clear(term, d);
    acb_clear(splus);
    acb_clear(rec);
    acb_clear(Na2);
    arb_clear(x);
    fmpz_clear(c);
}