mirror of
https://github.com/vale981/arb
synced 2025-03-06 09:51:39 -05:00
500 lines
11 KiB
C
500 lines
11 KiB
C
/*=============================================================================
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This file is part of ARB.
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ARB is free software; you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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ARB is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with ARB; if not, write to the Free Software
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Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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=============================================================================*/
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/******************************************************************************
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Copyright (C) 2014 Fredrik Johansson
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******************************************************************************/
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#include "acb_poly.h"
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/* TODO: move and document helper functions */
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void
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acb_get_mag(mag_t u, const acb_t z)
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{
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if (arb_is_zero(acb_imagref(z)))
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{
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arb_get_mag(u, acb_realref(z));
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}
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else if (arb_is_zero(acb_realref(z)))
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{
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arb_get_mag(u, acb_imagref(z));
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}
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else
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{
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mag_t v;
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mag_init(v);
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arb_get_mag(u, acb_realref(z));
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arb_get_mag(v, acb_imagref(z));
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mag_mul(u, u, u);
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mag_addmul(u, v, v);
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mag_sqrt(u, u);
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mag_clear(v);
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}
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}
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void
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mag_log_ui(mag_t t, ulong n)
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{
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if (n <= 1)
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{
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if (n == 1)
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mag_zero(t);
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else
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mag_inf(t);
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}
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else
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{
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mag_set_ui(t, n-1);
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mag_log1p(t, t);
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}
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}
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/* TODO: needs reimplementing */
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void mag_sub_lower(mag_t z, const mag_t x, const mag_t y);
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long
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arb_get_si_lower(const arb_t x)
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{
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arf_t t;
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long v;
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arf_init(t);
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arf_set_mag(t, arb_radref(x));
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arf_sub(t, arb_midref(x), t, 2 * FLINT_BITS, ARF_RND_FLOOR);
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v = arf_get_si(t, ARF_RND_FLOOR);
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arf_clear(t);
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return v;
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}
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/* bound (1 + 1/m)^n, m > 0, n >= 0 */
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void
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mag_binpow_uiui(mag_t b, ulong m, ulong n)
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{
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mag_t t;
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mag_init(t);
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/* bound by exp(n/m) <= 1 + (n/m) + (n/m)^2 */
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if (m > n)
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{
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mag_set_ui(t, n); /* x = n/m */
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mag_div_ui(t, t, m);
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mag_mul(b, t, t); /* x^2 */
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mag_add(b, b, t); /* x */
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mag_one(t);
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mag_add(b, b, t); /* 1 */
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}
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else
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{
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mag_one(b);
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mag_div_ui(b, b, m);
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mag_one(t);
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mag_add(t, t, b);
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mag_pow_ui(b, t, n);
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}
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mag_clear(t);
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}
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void
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polylog_remainder_bound(mag_t u, const mag_t z, long sigma, ulong d, ulong N)
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{
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mag_t TN, UN, t;
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if (N < 2)
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{
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mag_inf(u);
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return;
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}
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mag_init(TN);
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mag_init(UN);
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mag_init(t);
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if (mag_cmp_2exp_si(z, 0) >= 0)
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{
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mag_inf(u);
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}
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else
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{
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/* Bound T(N) */
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mag_pow_ui(TN, z, N);
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/* multiply by log(N)^d */
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if (d > 0)
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{
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mag_log_ui(t, N);
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mag_pow_ui(t, t, d);
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mag_mul(TN, TN, t);
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}
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/* multiply by 1/k^s */
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if (sigma > 0)
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{
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mag_set_ui_lower(t, N);
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mag_pow_ui_lower(t, t, sigma);
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mag_div(TN, TN, t);
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}
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else if (sigma < 0)
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{
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mag_set_ui(t, N);
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mag_pow_ui(t, t, -sigma);
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mag_mul(TN, TN, t);
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}
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/* Bound U(N) */
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mag_set(UN, z);
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/* multiply by (1 + 1/N)**S */
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if (sigma < 0)
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{
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mag_binpow_uiui(t, N, -sigma);
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mag_mul(UN, UN, t);
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}
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/* multiply by (1 + 1/(N log(N)))^d */
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if (d > 0)
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{
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ulong nl;
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/* rounds down */
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nl = mag_d_log_lower_bound(N) * N * (1 - 1e-13);
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mag_binpow_uiui(t, nl, d);
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mag_mul(UN, UN, t);
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}
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/* T(N) / (1 - U(N)) */
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if (mag_cmp_2exp_si(UN, 0) >= 0)
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{
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mag_inf(u);
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}
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else
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{
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mag_one(t);
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mag_sub_lower(t, t, UN);
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mag_div(u, TN, t);
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}
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}
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mag_clear(TN);
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mag_clear(UN);
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mag_clear(t);
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}
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long
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polylog_choose_terms(mag_t err, long sigma, const mag_t z, long d, long prec)
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{
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long N;
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for (N = 3; ; N = FLINT_MAX(N+3, N*1.1))
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{
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polylog_remainder_bound(err, z, sigma, d, N);
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/* TODO: do something else when |Li_s(z)| is very small/very large? */
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if (mag_cmp_2exp_si(err, -prec) < 0)
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break;
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if (N > 100 * prec)
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{
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N = 3;
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mag_inf(err);
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break;
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}
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}
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return N;
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}
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int
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polylog_is_real(const acb_t s, const acb_t z)
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{
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if (!arb_is_zero(acb_imagref(s)))
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{
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return 0;
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}
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else if (!arb_is_zero(acb_imagref(z)))
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{
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return 0;
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}
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else
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{
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fmpz_t one;
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int res;
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fmpz_init(one);
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fmpz_one(one);
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if (arb_contains_fmpz(acb_realref(z), one))
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res = 0;
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else
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res = (arf_cmp_2exp_si(arb_midref(acb_realref(z)), 0) < 0);
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fmpz_clear(one);
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return res;
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}
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}
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void
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_acb_poly_polylog_cpx_zeta(acb_ptr w, const acb_t s, const acb_t z, long len, long prec)
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{
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acb_ptr e1, e2, z1, z2, e1z1, e2z2;
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acb_t t, u, v;
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long k, len2;
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int deflate_zeta, deflate_gamma;
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if (!acb_is_finite(s) || !acb_is_finite(z))
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{
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_acb_vec_indeterminate(w, len);
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return;
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}
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acb_init(t);
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acb_init(u);
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acb_init(v);
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/* v = 1-s */
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acb_one(v);
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acb_sub(v, v, s, prec);
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/* pole of zeta */
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deflate_zeta = acb_is_one(v);
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/* poles of gamma at nonpositive integer v */
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deflate_gamma = (arb_is_zero(acb_imagref(v)) &&
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arb_is_int(acb_realref(v)) &&
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arf_sgn(arb_midref(acb_realref(v))) <= 0);
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len2 = len + deflate_gamma;
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e1 = _acb_vec_init(len + 1);
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e2 = _acb_vec_init(len + 1);
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z1 = _acb_vec_init(len + 1);
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z2 = _acb_vec_init(len + 1);
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e1z1 = _acb_vec_init(len + 1);
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e2z2 = _acb_vec_init(len + 1);
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/* u = log(-z)/(pi*i) */
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acb_neg(t, z);
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acb_log(t, t, prec);
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acb_const_pi(u, prec);
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acb_mul_onei(u, u);
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acb_div(u, t, u, prec);
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/* z1 = zeta(v+x, 1/2 + log(-z)/(2*pi*i)) */
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acb_one(t);
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acb_add(t, t, u, prec);
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acb_mul_2exp_si(t, t, -1);
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_acb_poly_zeta_cpx_series(z1, v, t, deflate_zeta, len2, prec);
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/* z2 = zeta(v+x, 1/2 - log(-z)/(2*pi*i)) */
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acb_one(t);
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acb_sub(t, t, u, prec);
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acb_mul_2exp_si(t, t, -1);
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_acb_poly_zeta_cpx_series(z2, v, t, deflate_zeta, len2, prec);
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/* e1 = (i/(2pi))^(v+x) */
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acb_onei(t);
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acb_const_pi(u, prec);
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acb_div(t, t, u, prec);
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acb_mul_2exp_si(t, t, -1);
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_acb_poly_acb_pow_cpx(e1, t, v, len + (deflate_zeta || deflate_gamma), prec);
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/* e2 = (1/(2 pi i))^(v+x) */
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acb_conj(t, t);
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_acb_poly_acb_pow_cpx(e2, t, v, len + (deflate_zeta || deflate_gamma), prec);
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_acb_poly_mullow(e1z1, e1, len2, z1, len2, len2, prec);
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_acb_poly_mullow(e2z2, e2, len2, z2, len2, len2, prec);
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_acb_vec_add(z1, e1z1, e2z2, len2, prec);
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if (deflate_gamma)
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{
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/* gamma(v+x) = pi/sin(pi(v+x)) * 1/gamma(1-v-x) */
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/* TODO: write a csc function? */
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acb_zero(e1);
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acb_const_pi(e1 + 1, prec);
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acb_mul_2exp_si(e2, v, -1);
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if (!arb_is_int(acb_realref(e2)))
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acb_neg(e1 + 1, e1 + 1);
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_acb_poly_sin_series(e2, e1, 2, len2, prec);
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_acb_poly_inv_series(e1, e2 + 1, len, len, prec);
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acb_const_pi(e2, prec);
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_acb_vec_scalar_mul(e1, e1, len, e2, prec);
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acb_set(z2, s);
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acb_set_si(z2 + 1, -1);
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_acb_poly_rgamma_series(e2, z2, 2, len, prec);
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_acb_poly_mullow(z2, e1, len, e2, len, len, prec);
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_acb_poly_mullow(w, z1 + 1, len, z2, len, len, prec);
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}
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else
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{
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if (deflate_zeta)
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{
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for (k = 0; k < len; k++)
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{
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arb_mul_2exp_si(acb_realref(e1 + k + 1), acb_realref(e1 + k + 1), 1);
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arb_add(acb_realref(z1 + k), acb_realref(z1 + k), acb_realref(e1 + k + 1), prec);
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}
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}
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/* gamma(v+x) */
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acb_set(e1, v);
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if (len > 1)
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acb_one(e1 + 1);
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_acb_poly_gamma_series(z2, e1, FLINT_MIN(len, 2), len, prec);
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_acb_poly_mullow(w, z2, len, z1, len, len, prec);
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}
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/* correct signs (from s -> 1-s) */
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for (k = 1; k < len; k += 2)
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acb_neg(w + k, w + k);
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_acb_vec_clear(e1, len + 1);
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_acb_vec_clear(e2, len + 1);
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_acb_vec_clear(z1, len + 1);
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_acb_vec_clear(z2, len + 1);
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_acb_vec_clear(e1z1, len + 1);
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_acb_vec_clear(e2z2, len + 1);
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acb_clear(t);
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acb_clear(u);
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acb_clear(v);
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}
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void
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_acb_poly_polylog_cpx_small(acb_ptr w, const acb_t s, const acb_t z, long len, long prec)
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{
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long k, N, sigma;
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int is_real;
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mag_t zmag, err, errf;
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acb_t a;
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acb_init(a);
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mag_init(zmag);
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mag_init(err);
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mag_init(errf);
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is_real = polylog_is_real(s, z);
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acb_get_mag(zmag, z);
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sigma = arb_get_si_lower(acb_realref(s));
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N = polylog_choose_terms(err, sigma, zmag, len - 1, prec);
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/* TODO: allow threading */
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acb_one(a);
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_acb_poly_powsum_series_naive(w, s, a, z, N - 1, len, prec);
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_acb_vec_scalar_mul(w, w, len, z, prec);
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for (k = 0; k < len; k++)
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{
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polylog_remainder_bound(err, zmag, sigma, k, N);
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mag_rfac_ui(errf, k);
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mag_mul(err, err, errf);
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if (is_real)
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arb_add_error_mag(acb_realref(w + k), err);
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else
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acb_add_error_mag(w + k, err);
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}
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acb_clear(a);
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mag_clear(zmag);
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mag_clear(err);
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mag_clear(errf);
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}
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void
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_acb_poly_polylog_cpx(acb_ptr w, const acb_t s, const acb_t z, long len, long prec)
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{
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mag_t zmag;
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mag_init(zmag);
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acb_get_mag(zmag, z);
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if (mag_cmp_2exp_si(zmag, -1) < 0)
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_acb_poly_polylog_cpx_small(w, s, z, len, prec);
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else
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_acb_poly_polylog_cpx_zeta(w, s, z, len, prec);
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mag_clear(zmag);
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}
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void
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_acb_poly_polylog_series(acb_ptr res, acb_srcptr s, long slen, const acb_t z, long len, long prec)
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{
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acb_ptr t, u;
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slen = FLINT_MIN(slen, len);
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t = _acb_vec_init(len);
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u = _acb_vec_init(len);
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_acb_poly_polylog_cpx(t, s, z, len, prec);
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/* compose with nonconstant part */
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acb_zero(u);
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_acb_vec_set(u + 1, s + 1, slen - 1);
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_acb_poly_compose_series(res, t, len, u, slen, len, prec);
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_acb_vec_clear(t, len);
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_acb_vec_clear(u, len);
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}
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void
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acb_poly_polylog_series(acb_poly_t res, const acb_poly_t s, const acb_t z, long n, long prec)
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{
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if (n == 0)
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{
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acb_poly_zero(res);
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return;
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}
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acb_poly_fit_length(res, n);
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if (s->length == 0)
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{
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acb_t t;
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acb_init(t);
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_acb_poly_polylog_series(res->coeffs, t, 1, z, n, prec);
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acb_clear(t);
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}
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else
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{
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_acb_poly_polylog_series(res->coeffs, s->coeffs, s->length, z, n, prec);
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}
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_acb_poly_set_length(res, n);
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_acb_poly_normalise(res);
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}
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