mirror of
https://github.com/vale981/arb
synced 2025-03-04 17:01:40 -05:00
114 lines
2.8 KiB
C
114 lines
2.8 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_elliptic.h"
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/* Compute R_C(1,1+x) = atan(sqrt(x))/sqrt(x) = 2F1(1,1/2,3/2,-x) */
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static void
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_acb_elliptic_rc1(acb_t res, const acb_t x, slong prec)
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{
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acb_t t;
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acb_init(t);
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acb_sqrt(t, x, prec + 2);
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acb_atan(res, t, prec + 2);
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acb_div(res, res, t, prec);
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acb_clear(t);
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}
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void
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acb_elliptic_rc1(acb_t res, const acb_t x, slong prec)
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{
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mag_t xm;
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mag_init(xm);
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acb_get_mag(xm, x);
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if (mag_cmp_2exp_si(xm, 0) < 0)
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{
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slong k, n;
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for (n = 1; n <= 6; n++)
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{
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if (mag_cmp_2exp_si(xm, -prec / n) < 0)
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break;
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}
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/* Use Taylor series: 1 - x/3 + x^2/5 - x^3/7 + x^4/9 + ... */
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if (n <= 6)
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{
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const short coeffs[] = {3465, -1155, 693, -495, 385, -315};
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acb_t s;
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acb_init(s);
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for (k = n - 1; k >= 0; k--)
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{
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acb_mul(s, s, x, prec);
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acb_add_si(s, s, coeffs[k], prec);
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}
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acb_div_si(s, s, coeffs[0], prec);
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mag_geom_series(xm, xm, n);
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if (acb_is_real(x))
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arb_add_error_mag(acb_realref(s), xm);
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else
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acb_add_error_mag(s, xm);
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acb_set(res, s);
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acb_clear(s);
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}
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else if (acb_is_exact(x))
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{
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_acb_elliptic_rc1(res, x, prec);
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}
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else
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{
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acb_t w;
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mag_t err, rad;
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acb_init(w);
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mag_init(err);
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mag_init(rad);
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/* On the unit disc, |f'(x)| <= 0.5 / |1+x| */
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acb_add_ui(w, x, 1, MAG_BITS);
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acb_get_mag_lower(err, w);
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mag_one(rad);
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mag_mul_2exp_si(rad, rad, -1);
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mag_div(err, rad, err);
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mag_hypot(rad, arb_radref(acb_realref(x)), arb_radref(acb_imagref(x)));
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mag_mul(err, err, rad);
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acb_set(w, x);
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mag_zero(arb_radref(acb_realref(w)));
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mag_zero(arb_radref(acb_imagref(w)));
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_acb_elliptic_rc1(w, w, prec);
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if (acb_is_real(x))
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arb_add_error_mag(acb_realref(w), err);
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else
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acb_add_error_mag(w, err);
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acb_set(res, w);
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acb_clear(w);
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mag_clear(err);
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mag_clear(rad);
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}
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}
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else
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{
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_acb_elliptic_rc1(res, x, prec);
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}
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mag_clear(xm);
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}
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