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https://github.com/vale981/arb
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88 lines
2.1 KiB
C
88 lines
2.1 KiB
C
/*
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Copyright (C) 2016 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 "acb_dirichlet.h"
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void
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acb_dirichlet_hurwitz_precomp_bound(mag_t res, const acb_t s,
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slong A, slong K, slong N)
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{
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acb_t s1;
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mag_t x, t, TK, C;
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slong sigmaK;
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arf_t u;
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if (A < 1 || K < 1 || N < 1)
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{
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mag_inf(res);
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return;
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}
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/* sigmaK = re(s) + K, floor bound */
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arf_init(u);
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arf_set_mag(u, arb_radref(acb_realref(s)));
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arf_sub(u, arb_midref(acb_realref(s)), u, MAG_BITS, ARF_RND_FLOOR);
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arf_add_ui(u, u, K, MAG_BITS, ARF_RND_FLOOR);
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if (arf_cmp_ui(u, 2) < 0 || arf_cmp_2exp_si(u, FLINT_BITS - 2) > 0)
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{
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mag_inf(res);
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arf_clear(u);
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return;
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}
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sigmaK = arf_get_si(u, ARF_RND_FLOOR);
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arf_clear(u);
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acb_init(s1);
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mag_init(x);
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mag_init(t);
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mag_init(TK);
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mag_init(C);
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/* With N grid points, we will have |x| <= 1 / (2N). */
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mag_one(x);
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mag_div_ui(x, x, 2 * N);
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/* T(K) = |x|^K |(s)_K| / K! * [1/A^(sigma+K) + ...] */
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mag_pow_ui(TK, x, K);
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acb_rising_ui_get_mag(t, s, K);
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mag_mul(TK, TK, t);
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mag_rfac_ui(t, K);
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mag_mul(TK, TK, t);
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/* Note: here we assume that mag_hurwitz_zeta_uiui uses an error bound
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that is at least as large as the one used in the proof. */
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mag_hurwitz_zeta_uiui(t, sigmaK, A);
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mag_mul(TK, TK, t);
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/* C = |x|/A (1 + 1/(K+sigma+A-1)) (1 + |s-1|/(K+1)) */
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mag_div_ui(C, x, A);
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mag_one(t);
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mag_div_ui(t, t, sigmaK + A - 1);
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mag_add_ui(t, t, 1);
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mag_mul(C, C, t);
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acb_sub_ui(s1, s, 1, MAG_BITS);
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acb_get_mag(t, s1);
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mag_div_ui(t, t, K + 1);
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mag_add_ui(t, t, 1);
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mag_mul(C, C, t);
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mag_geom_series(t, C, 0);
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mag_mul(res, TK, t);
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acb_clear(s1);
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mag_clear(x);
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mag_clear(t);
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mag_clear(TK);
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mag_clear(C);
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}
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