mirror of
https://github.com/vale981/arb
synced 2025-03-05 17:31:38 -05:00
117 lines
3 KiB
C
117 lines
3 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, v;
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mag_init(t);
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mag_init(u);
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mag_init(v);
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if (fmpz_is_zero(k))
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{
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acb_get_mag(t, z);
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/* |z| <= 64 */
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if (mag_cmp_2exp_si(t, 6) < 0)
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{
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/* 2.25 / sqrt(|t|) / sqrt(1+|t|), t = |ez+1| */
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acb_get_mag_lower(t, ez1);
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mag_one(u);
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mag_add_lower(u, u, t);
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mag_mul_lower(t, t, u);
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mag_rsqrt(t, t);
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if (arb_is_positive(acb_realref(ez1)))
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{
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mag_mul_ui(t, t, 135); /* x0.9375 for small improvement */
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mag_mul_2exp_si(t, t, -6);
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}
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else
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{
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mag_mul_ui(t, t, 9);
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mag_mul_2exp_si(t, t, -2);
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}
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mag_set(res, t);
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}
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else
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{
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acb_get_mag_lower(t, z);
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if (mag_cmp_2exp_si(t, 2) >= 0)
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{
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mag_one(u);
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mag_div(res, u, t);
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}
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else /* unlikely */
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{
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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_ui(u, u, 1);
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mag_mul_ui(u, u, 3);
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mag_div(res, u, t);
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}
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}
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}
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else if (fmpz_is_pm1(k))
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{
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if (arb_is_nonnegative(acb_realref(z)) ||
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(fmpz_is_one(k) && arb_is_nonnegative(acb_imagref(z))) ||
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(fmpz_equal_si(k, -1) && arb_is_negative(acb_imagref(z))))
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{
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/* (1 + 1/(4+|z|^2))/|z| */
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acb_get_mag_lower(t, z);
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mag_mul_lower(u, t, t);
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mag_set_ui_lower(v, 4);
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mag_add_lower(u, u, v);
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mag_one(v);
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mag_div(u, v, u);
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mag_add(u, u, v);
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mag_div(res, u, t);
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}
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else
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{
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/* (1 + 0.71875/sqrt(|e*z+1|)) / |z| */
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acb_get_mag_lower(t, ez1);
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mag_rsqrt(t, t);
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mag_mul_ui(t, t, 23);
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mag_mul_2exp_si(t, t, -5);
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mag_one(u);
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mag_add(t, t, u);
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acb_get_mag_lower(u, z);
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mag_div(res, t, u);
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}
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mag_clear(t);
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mag_clear(u);
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mag_clear(v);
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return;
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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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mag_clear(t);
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mag_clear(u);
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mag_clear(v);
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}
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