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/*
Copyright (C) 2017 Fredrik Johansson
This file is part of Arb.
Arb 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 2.1 of the License, or
(at your option) any later version. See <http://www.gnu.org/licenses/>.
*/
#include "acb_elliptic.h"
#include "acb_modular.h"
/* Evaluation on -pi/2 <= re(z) <= pi/2, no aliasing. */
/* s*RF(c^2, 1-m*s^2, 1) */
void
acb_elliptic_f_reduced(acb_t r, const acb_t z, const acb_t m, int times_pi, slong prec)
{
acb_t s, c, a;
acb_init(s);
acb_init(c);
acb_init(a);
if (times_pi)
acb_sin_cos_pi(s, c, z, prec);
else
acb_sin_cos(s, c, z, prec);
acb_mul(c, c, c, prec);
acb_mul(r, s, s, prec);
acb_mul(r, r, m, prec);
acb_sub_ui(r, r, 1, prec);
acb_neg(r, r);
acb_one(a);
acb_elliptic_rf(r, c, r, a, 0, prec);
acb_mul(r, r, s, prec);
acb_clear(s);
acb_clear(c);
acb_clear(a);
}
void
acb_elliptic_f(acb_t res, const acb_t phi, const acb_t m, int times_pi, slong prec)
{
arb_t x, d, pi;
acb_t z, w, r;
if (!acb_is_finite(phi) || !acb_is_finite(m))
{
acb_indeterminate(res);
return;
}
if (acb_is_zero(m))
{
if (times_pi)
{
arb_init(pi);
arb_const_pi(pi, prec);
acb_mul_arb(res, phi, pi, prec);
arb_clear(pi);
}
else
{
acb_set_round(res, phi, prec);
}
return;
}
if (acb_is_zero(phi))
{
acb_zero(res);
return;
}
if (times_pi && acb_is_int_2exp_si(phi, -1))
{
acb_t t;
acb_init(t);
acb_mul_2exp_si(t, phi, 1);
acb_elliptic_k(res, m, prec);
acb_mul(res, res, t, prec);
acb_clear(t);
return;
}
arb_init(x);
arb_init(d);
arb_init(pi);
acb_init(z);
acb_init(w);
acb_init(r);
arb_set(x, acb_realref(phi));
arb_const_pi(pi, prec);
if (times_pi)
arb_set(d, x);
else
arb_div(d, x, pi, prec);
arb_mul_2exp_si(d, d, 1);
arb_add_ui(d, d, 1, prec);
arb_mul_2exp_si(d, d, -1);
if (mag_cmp_2exp_si(arb_radref(d), -1) >= 0)
{
/* may span multiple periods... don't bother */
acb_indeterminate(res);
}
else if (arb_contains_int(d) && !arb_is_exact(d)) /* two adjacent d */
{
acb_t r2, w2;
int is_real;
acb_init(r2);
acb_init(w2);
arb_sub_ui(x, acb_realref(m), 1, prec);
is_real = acb_is_real(phi) && acb_is_real(m) && arb_is_negative(x);
/* left d */
acb_zero(z);
arf_set_mag(arb_midref(acb_realref(z)), arb_radref(d));
mag_zero(arb_radref(d));
arb_sub(d, d, acb_realref(z), 2 * prec + 100); /* meant to be exact */
arb_floor(d, d, prec);
/* w = 2 K(m) */
acb_elliptic_k(w, m, prec);
acb_mul_2exp_si(w, w, 1);
/* z = phi - d * pi */
if (times_pi)
{
acb_sub_arb(z, phi, d, prec);
}
else
{
arb_mul(acb_realref(z), pi, d, prec);
arb_sub(acb_realref(z), acb_realref(phi), acb_realref(z), prec);
arb_set(acb_imagref(z), acb_imagref(phi));
}
acb_elliptic_f_reduced(r, z, m, times_pi, prec);
acb_addmul_arb(r, w, d, prec);
/* z = phi - (d + 1) * pi */
if (times_pi)
acb_sub_ui(z, z, 1, prec);
else
acb_sub_arb(z, z, pi, prec);
acb_elliptic_f_reduced(r2, z, m, times_pi, prec);
arb_add_ui(d, d, 1, prec);
acb_addmul_arb(r2, w, d, prec);
arb_union(acb_realref(res), acb_realref(r), acb_realref(r2), prec);
arb_union(acb_imagref(res), acb_imagref(r), acb_imagref(r2), prec);
if (is_real)
arb_zero(acb_imagref(res));
acb_clear(r2);
acb_clear(w2);
}
else
{
/* this could still be inexact if d is large (which is fine) */
arb_floor(d, d, prec);
if (arb_is_zero(d))
{
acb_set(z, phi);
acb_zero(w);
}
else
{
/* z = phi - d*pi */
if (times_pi)
{
acb_sub_arb(z, phi, d, prec);
}
else
{
arb_mul(acb_realref(z), pi, d, prec);
arb_sub(acb_realref(z), acb_realref(phi), acb_realref(z), prec);
arb_set(acb_imagref(z), acb_imagref(phi));
}
/* w = 2 d K(m) */
acb_elliptic_k(w, m, prec);
acb_mul_arb(w, w, d, prec);
acb_mul_2exp_si(w, w, 1);
}
acb_elliptic_f_reduced(r, z, m, times_pi, prec);
acb_add(r, r, w, prec);
acb_set(res, r);
}
arb_clear(x);
arb_clear(d);
arb_clear(pi);
acb_clear(z);
acb_clear(w);
acb_clear(r);
}
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