mirror of
https://github.com/vale981/arb
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77 lines
1.8 KiB
C
77 lines
1.8 KiB
C
/*
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Copyright (C) 2013 Fredrik Johansson
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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 it under
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the terms of the GNU Lesser General Public License (LGPL) as published
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by the Free Software Foundation; either version 2.1 of the License, or
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(at your option) any later version. See <http://www.gnu.org/licenses/>.
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*/
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#include "partitions.h"
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static void
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_arf_sinh(arf_t y, const arf_t x, slong prec)
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{
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arb_t t;
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arb_init(t);
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arb_set_arf(t, x);
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arb_sinh(t, t, prec);
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arb_get_abs_ubound_arf(y, t, prec);
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arb_clear(t);
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}
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/* Equation (1.8) in the paper */
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void
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partitions_rademacher_bound(arf_t b, const fmpz_t n, ulong N)
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{
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arf_t A, B, C, t, u;
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fmpz_t n1;
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arf_init(A);
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arf_init(B);
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arf_init(C);
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arf_init(t);
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arf_init(u);
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fmpz_init(n1);
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/* bound for 44*pi^2/(225*sqrt(3)) */
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arf_set_si_2exp_si(A, 18695160, -24);
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/* bound for pi*sqrt(2)/75 */
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arf_set_si_2exp_si(B, 993857, -24);
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/* bound for pi*sqrt(2/3) */
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arf_set_si_2exp_si(C, 43035232, -24);
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/* first term: A / sqrt(N) */
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arf_sqrt_ui(t, N, MAG_BITS, ARF_RND_DOWN);
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arf_div(b, A, t, MAG_BITS, ARF_RND_UP);
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/* B * sqrt(N/(n-1)) */
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arf_set_ui(t, N);
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fmpz_sub_ui(n1, n, 1);
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arf_div_fmpz(t, t, n1, MAG_BITS, ARF_RND_UP);
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arf_sqrt(t, t, MAG_BITS, ARF_RND_UP);
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arf_mul(t, B, t, MAG_BITS, ARF_RND_UP);
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/* sinh(C*sqrt(n)/N) */
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arf_sqrt_fmpz(u, n, MAG_BITS, ARF_RND_UP);
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arf_div_ui(u, u, N, MAG_BITS, ARF_RND_UP);
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arf_mul(u, C, u, MAG_BITS, ARF_RND_UP);
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_arf_sinh(u, u, MAG_BITS);
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/* second term: B * ... * sinh... */
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arf_mul(t, t, u, MAG_BITS, ARF_RND_UP);
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arf_add(b, b, t, MAG_BITS, ARF_RND_UP);
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arf_clear(A);
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arf_clear(B);
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arf_clear(C);
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arf_clear(t);
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arf_clear(u);
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fmpz_clear(n1);
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}
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