mirror of
https://github.com/vale981/arb
synced 2025-03-05 17:31:38 -05:00
91 lines
2.5 KiB
C
91 lines
2.5 KiB
C
/*
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Copyright (C) 2017 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.h"
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void
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acb_lambertw_bound_deriv(mag_t res, const acb_t z, const acb_t ez1, const fmpz_t k)
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{
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mag_t t, u;
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mag_init(t);
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mag_init(u);
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/* Main approximation: W'(z) ~ 1/|z| */
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if (fmpz_is_zero(k))
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{
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/* 1/(1+|z|) */
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acb_get_mag_lower(res, z);
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mag_one(t);
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mag_add_lower(res, res, t);
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mag_div(res, t, res);
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}
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else if (fmpz_is_pm1(k))
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{
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/* 2/|z| */
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mag_set_ui(t, 2);
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acb_get_mag_lower(res, z);
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mag_div(res, t, res);
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}
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else
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{
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/* |W'(z)| = |1/z| |W(z)/(1+W(z))|
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<= |1/z| (2pi) / (2pi-1) */
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mag_set_ui_2exp_si(t, 77, -6);
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acb_get_mag_lower(res, z);
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mag_div(res, t, res);
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}
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/* Compute correction near the branch point */
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if (fmpz_is_zero(k) ||
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(fmpz_equal_si(k,-1) && !arb_is_negative(acb_imagref(z))) ||
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(fmpz_is_one(k) && !arb_is_nonnegative(acb_imagref(z))))
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{
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acb_t b;
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acb_init(b);
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if (fmpz_is_zero(k))
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{
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/* [-4,1] + [-2,2]i */
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arf_set_si_2exp_si(arb_midref(acb_realref(b)), -3, -1);
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mag_set_ui_2exp_si(arb_radref(acb_realref(b)), 5, -1);
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arf_zero(arb_midref(acb_imagref(b)));
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mag_set_ui(arb_radref(acb_imagref(b)), 2);
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}
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else
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{
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/* k = 1 [-1/2,-1/4] + [-1/8,0) i */
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/* k = -1 [-1/2,-1/4] + [0,1/8] i */
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arf_set_si_2exp_si(arb_midref(acb_realref(b)), -3, -3);
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mag_set_ui_2exp_si(arb_radref(acb_realref(b)), 1, -3);
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if (fmpz_is_one(k))
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arf_set_si_2exp_si(arb_midref(acb_imagref(b)), -1, -4);
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else
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arf_set_ui_2exp_si(arb_midref(acb_imagref(b)), 1, -4);
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mag_set_ui_2exp_si(arb_radref(acb_imagref(b)), 1, -4);
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}
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if (acb_overlaps(z, b))
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{
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/* add 2/sqrt(|ez+1|) */
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acb_get_mag_lower(u, ez1);
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mag_rsqrt(u, u);
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mag_mul_2exp_si(u, u, 1);
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mag_add(res, res, u);
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}
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acb_clear(b);
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}
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mag_clear(t);
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mag_clear(u);
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}
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