flint-sys 0.9.0

Bindings to the FLINT C library
Documentation 
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/*
    Copyright (C) 2015, 2024 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 <math.h>
#include "ulong_extras.h"
#include "arb_poly.h"

/* Bound based on binomial theorem */
slong
_arb_poly_swinnerton_dyer_ui_prec(ulong n)
{
    slong i;
    double u, N;

    N = UWORD(1) << n;

    /* u = (sum of square roots)^(2^n) */
    u = 0;
    for (i = 0; i < n; i++)
        u += sqrt(n_nth_prime(1 + i));
    u = N * log(u) * 1.44269504088897;

    /* Central binomial coefficient C(N,N/2) < 2^N / sqrt(3*N/2) */
    u += N - 0.5*(n-1) - 0.792481250360578; /* log(sqrt(3)) */

    /* experimental heuristic: the bound is 2x too large */
    return u * 0.5 + 15;
}

void
_arb_poly_swinnerton_dyer_ui(arb_ptr T, ulong n, slong trunc, slong prec)
{
    arb_ptr square_roots, tmp1, tmp2, tmp3;
    arb_t one;
    slong i, j, k, N;

    if (n == 0)
    {
        arb_zero(T);
        arb_one(T + 1);
        return;
    }

    if (prec == 0)
        prec = _arb_poly_swinnerton_dyer_ui_prec(n);

    N = WORD(1) << n;
    trunc = FLINT_MIN(trunc, N + 1);

    arb_init(one);
    arb_one(one);

    square_roots = _arb_vec_init(n);
    tmp1 = flint_malloc((N / 4 + 1) * sizeof(arb_struct));
    tmp2 = flint_malloc((N / 4 + 1) * sizeof(arb_struct));
    tmp3 = _arb_vec_init(N / 2);

    for (i = 0; i < n; i++)
        arb_sqrt_ui(square_roots + i, n_nth_prime(i + 1), prec);

    /* Build deflated quadratic factors */
    for (i = 0; i < N / 2; i++)
    {
        arb_zero(T + i);

        for (j = 0; j < n; j++)
        {
            if ((i >> j) & 1)
                arb_add(T + i, T + i, square_roots + j, prec);
            else
                arb_sub(T + i, T + i, square_roots + j, prec);
        }

        arb_sqr(T + i, T + i, prec);
        arb_neg(T + i, T + i);
    }

    /* For each level... */
    for (i = 0; i < n - 1; i++)
    {
        slong stride = UWORD(1) << i;

        for (j = 0; j < N / 2; j += 2 * stride)
        {
            for (k = 0; k < stride; k++)
            {
                tmp1[k] = T[j + k];
                tmp2[k] = T[j + stride + k];
            }

            tmp1[stride] = *one;
            tmp2[stride] = *one;

            _arb_poly_mullow(tmp3, tmp1, stride + 1, tmp2, stride + 1,
                FLINT_MIN(2 * stride, trunc), prec);
            _arb_vec_swap(T + j, tmp3, FLINT_MIN(2 * stride, trunc));
        }
    }

    /* Inflate */
    for (i = N - 2; i >= 0; i -= 2)
    {
        if (i < trunc)
            arb_swap(T + i, T + i / 2);
        if (i + 1 < trunc)
            arb_zero(T + i + 1);
    }

    arb_one(T + N);
    _arb_vec_clear(square_roots, n);
    flint_free(tmp1);
    flint_free(tmp2);
    _arb_vec_clear(tmp3, N / 2);
    arb_clear(one);
}

void
arb_poly_swinnerton_dyer_ui(arb_poly_t poly, ulong n, slong prec)
{
    slong N = WORD(1) << n;

    arb_poly_fit_length(poly, N + 1);
    _arb_poly_swinnerton_dyer_ui(poly->coeffs, n, N + 1, prec);
    _arb_poly_set_length(poly, N + 1);
    _arb_poly_normalise(poly);
}