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/*
Copyright (C) 2016 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 "arb.h"
void
arb_bernoulli_poly_ui(arb_t res, ulong n, const arb_t x, slong prec)
{
arb_t s, t, c, x2;
ulong m, k;
int negate;
if (n == 0)
{
arb_one(res);
return;
}
if (n == 1)
{
arb_mul_2exp_si(res, x, 1);
arb_sub_ui(res, res, 1, prec);
arb_mul_2exp_si(res, res, -1);
return;
}
/* small integer x */
if (arb_is_int(x) && arf_cmpabs_ui(arb_midref(x), n) < 0 && n < WORD_MAX)
{
if (arf_sgn(arb_midref(x)) >= 0)
{
m = arf_get_si(arb_midref(x), ARF_RND_DOWN);
negate = 0;
}
else
{
m = UWORD(1) - arf_get_si(arb_midref(x), ARF_RND_DOWN);
negate = n % 2;
}
arb_init(t);
arb_zero(res);
/* todo: use a dedicated power sum function */
for (k = 1; k < m; k++)
{
arb_ui_pow_ui(t, k, n - 1, prec);
arb_add(res, res, t, prec);
}
arb_mul_ui(res, res, n, prec);
arb_bernoulli_ui(t, n, prec);
arb_add(res, res, t, prec);
if (negate)
arb_neg(res, res);
arb_clear(t);
return;
}
/* assuming small n simplifies the code that follows */
if (n >> (FLINT_BITS / 2) || !arb_is_finite(x))
{
arb_indeterminate(res);
return;
}
arb_init(s);
arb_init(t);
arb_init(c);
arb_init(x2);
arb_mul(x2, x, x, prec);
/* s = x^2 - x n / 2 */
arb_mul_ui(s, x, n, prec);
arb_mul_2exp_si(s, s, -1);
arb_sub(s, x2, s, prec);
/* c = n (n-1) / 2; s = s + c / 6 */
arb_set_ui(c, n * (n - 1));
arb_mul_2exp_si(c, c, -1);
arb_div_ui(t, c, 6, prec);
arb_add(s, s, t, prec);
for (k = 4; k <= n; k += 2)
{
/* c = binomial(n,k) */
arb_mul_ui(c, c, (n + 1 - k) * (n + 2 - k), prec);
arb_div_ui(c, c, k * (k - 1), prec);
/* s = s x^2 + b_k c */
arb_mul(s, s, x2, prec);
arb_bernoulli_ui(t, k, prec);
arb_addmul(s, t, c, prec);
}
if (n >= 3 && n % 2)
arb_mul(s, s, x, prec);
arb_swap(res, s);
arb_clear(s);
arb_clear(t);
arb_clear(c);
arb_clear(x2);
}