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/*
Copyright (C) 2017 Fredrik Johansson
This file is part of FLINT.
FLINT is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 3 of the License, or
(at your option) any later version. See <https://www.gnu.org/licenses/>.
*/
#include "acb.h"
void
acb_lambertw_bound_deriv(mag_t res, const acb_t z, const acb_t ez1, const fmpz_t k)
{
mag_t t, u, v;
mag_init(t);
mag_init(u);
mag_init(v);
if (fmpz_is_zero(k))
{
acb_get_mag(t, z);
/* |z| <= 64 */
if (mag_cmp_2exp_si(t, 6) < 0)
{
/* 2.25 / sqrt(|t|) / sqrt(1+|t|), t = |ez+1| */
acb_get_mag_lower(t, ez1);
mag_one(u);
mag_add_lower(u, u, t);
mag_mul_lower(t, t, u);
mag_rsqrt(t, t);
if (arb_is_positive(acb_realref(ez1)))
{
mag_mul_ui(t, t, 135); /* x0.9375 for small improvement */
mag_mul_2exp_si(t, t, -6);
}
else
{
mag_mul_ui(t, t, 9);
mag_mul_2exp_si(t, t, -2);
}
mag_set(res, t);
}
else
{
acb_get_mag_lower(t, z);
if (mag_cmp_2exp_si(t, 2) >= 0)
{
mag_one(u);
mag_div(res, u, t);
}
else /* unlikely */
{
acb_get_mag_lower(u, ez1);
mag_rsqrt(u, u);
mag_mul_2exp_si(u, u, -1);
mag_add_ui(u, u, 1);
mag_mul_ui(u, u, 3);
mag_div(res, u, t);
}
}
}
else if (fmpz_is_pm1(k))
{
if (arb_is_nonnegative(acb_realref(z)) ||
(fmpz_is_one(k) && arb_is_nonnegative(acb_imagref(z))) ||
(fmpz_equal_si(k, -1) && arb_is_negative(acb_imagref(z))))
{
/* (1 + 1/(4+|z|^2))/|z| */
acb_get_mag_lower(t, z);
mag_mul_lower(u, t, t);
mag_set_ui_lower(v, 4);
mag_add_lower(u, u, v);
mag_one(v);
mag_div(u, v, u);
mag_add(u, u, v);
mag_div(res, u, t);
}
else
{
/* (1 + 0.71875/sqrt(|e*z+1|)) / |z| */
acb_get_mag_lower(t, ez1);
mag_rsqrt(t, t);
mag_mul_ui(t, t, 23);
mag_mul_2exp_si(t, t, -5);
mag_one(u);
mag_add(t, t, u);
acb_get_mag_lower(u, z);
mag_div(res, t, u);
}
mag_clear(t);
mag_clear(u);
mag_clear(v);
return;
}
else
{
/* |W'(z)| = |1/z| |W(z)/(1+W(z))|
<= |1/z| (2pi) / (2pi-1) */
mag_set_ui_2exp_si(t, 77, -6);
acb_get_mag_lower(res, z);
mag_div(res, t, res);
}
mag_clear(t);
mag_clear(u);
mag_clear(v);
}