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arb/acb_elliptic/rf.c

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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"
void
acb_elliptic_rf_taylor_sum(acb_t res, const acb_t E2, const acb_t E3, slong nterms, slong prec)
{
fmpz_t den, c, d, e;
acb_ptr E2pow;
arb_ptr E2powr;
acb_t s;
slong x, y, XMAX, YMAX, NMAX, N;
int real;
NMAX = nterms - 1;
YMAX = NMAX / 3;
XMAX = NMAX / 2;
real = acb_is_real(E2) && acb_is_real(E3);
fmpz_init(den);
fmpz_init(c);
fmpz_init(d);
fmpz_init(e);
acb_init(s);
if (real)
{
E2powr = _arb_vec_init(XMAX + 1);
E2pow = NULL;
_arb_vec_set_powers(E2powr, acb_realref(E2), XMAX + 1, prec);
}
else
{
E2pow = _acb_vec_init(XMAX + 1);
E2powr = NULL;
_acb_vec_set_powers(E2pow, E2, XMAX + 1, prec);
}
/* Compute universal denominator (could tighten this?). */
fmpz_one(den);
for (x = 3; x <= 2 * NMAX + 1; x += 2)
fmpz_mul_ui(den, den, x);
fmpz_mul_2exp(den, den, NMAX);
/* Compute initial coefficient rf(1/2,y) / y! */
fmpz_set(c, den);
for (y = 0; y < YMAX; y++)
{
fmpz_mul_ui(c, c, 2 * y + 1);
fmpz_divexact_ui(c, c, 2 * y + 2);
}
acb_zero(res);
for (y = YMAX; y >= 0; y--)
{
acb_zero(s);
if (y != YMAX)
{
fmpz_mul_ui(c, c, 2 * y + 2);
fmpz_divexact_ui(c, c, 2 * y + 1);
}
fmpz_set(d, c);
/* Use powers with respect to E2 */
for (x = 0; x <= XMAX; x++)
{
N = 2 * x + 3 * y;
if (N <= NMAX)
{
fmpz_divexact_ui(e, d, 2 * N + 1);
if (x % 2 == 1)
fmpz_neg(e, e);
if (x != 0 || y != 0)
{
if (real)
arb_addmul_fmpz(acb_realref(s), E2powr + x, e, prec);
else
acb_addmul_fmpz(s, E2pow + x, e, prec);
}
fmpz_mul_ui(d, d, 2 * x + 2 * y + 1);
fmpz_divexact_ui(d, d, 2 * x + 2);
}
}
/* Horner with respect to E3 */
acb_mul(res, res, E3, prec);
acb_add(res, res, s, prec);
}
acb_div_fmpz(res, res, den, prec);
acb_add_ui(res, res, 1, prec);
fmpz_clear(den);
fmpz_clear(c);
fmpz_clear(d);
fmpz_clear(e);
acb_clear(s);
if (real)
_arb_vec_clear(E2powr, XMAX + 1);
else
_acb_vec_clear(E2pow, XMAX + 1);
}
void
acb_elliptic_rf(acb_t res, const acb_t x, const acb_t y, const acb_t z,
int flags, slong prec)
{
acb_t xx, yy, zz, sx, sy, sz, t;
acb_t X, Y, Z, E2, E3;
mag_t err, err2, prev_err;
slong k, wp, accx, accy, accz, order;
if (!acb_is_finite(x) || !acb_is_finite(y) || !acb_is_finite(z))
{
acb_indeterminate(res);
return;
}
if (acb_contains_zero(x) + acb_contains_zero(y) + acb_contains_zero(z) > 1)
{
acb_indeterminate(res);
return;
}
acb_init(xx); acb_init(yy); acb_init(zz);
acb_init(sx); acb_init(sy); acb_init(sz);
acb_init(X); acb_init(Y); acb_init(Z); acb_init(E2); acb_init(E3);
acb_init(t);
mag_init(err);
mag_init(err2);
mag_init(prev_err);
order = 5; /* will be set later */
acb_set(xx, x);
acb_set(yy, y);
acb_set(zz, z);
/* First guess precision based on the inputs. */
/* This does not account for mixing. */
accx = acb_rel_accuracy_bits(xx);
accy = acb_rel_accuracy_bits(yy);
accz = acb_rel_accuracy_bits(zz);
accx = FLINT_MAX(accx, accy);
accx = FLINT_MAX(accx, accz);
if (accx < prec - 20)
prec = FLINT_MAX(2, accx + 20);
wp = prec + 10 + FLINT_BIT_COUNT(prec);
/* Must do at least one iteration. */
for (k = 0; k < prec; k++)
{
acb_sqrt(sx, xx, wp);
acb_sqrt(sy, yy, wp);
acb_sqrt(sz, zz, wp);
acb_add(t, sy, sz, wp);
acb_mul(t, t, sx, wp);
acb_addmul(t, sy, sz, wp);
acb_add(xx, xx, t, wp);
acb_add(yy, yy, t, wp);
acb_add(zz, zz, t, wp);
acb_mul_2exp_si(xx, xx, -2);
acb_mul_2exp_si(yy, yy, -2);
acb_mul_2exp_si(zz, zz, -2);
/* Improve precision estimate and set expansion order. */
/* Should this done for other k also? */
if (k == 0)
{
accx = acb_rel_accuracy_bits(xx);
accy = acb_rel_accuracy_bits(yy);
accz = acb_rel_accuracy_bits(zz);
accx = FLINT_MAX(accx, accy);
accx = FLINT_MAX(accx, accz);
if (accx < prec - 20)
prec = FLINT_MAX(2, accx + 20);
wp = prec + 10 + FLINT_BIT_COUNT(prec);
if (acb_is_real(xx) && acb_is_real(yy) && acb_is_real(zz))
order = 2.0 * pow(prec, 0.4);
else
order = 2.5 * pow(prec, 0.4);
order = FLINT_MIN(order, 500);
order = FLINT_MAX(order, 2);
}
/* Close enough? Quick estimate based on |x-y|/|x| and |x-z|/|x| */
/* We also terminate if there is no improvement. */
acb_sub(t, xx, yy, wp);
acb_get_mag(err, t);
acb_sub(t, xx, zz, wp);
acb_get_mag(err2, t);
mag_max(err, err, err2);
acb_get_mag_lower(err2, xx);
mag_div(err, err, err2);
mag_pow_ui(err, err, order);
if (mag_cmp_2exp_si(err, -prec) < 0 ||
(k > 2 && mag_cmp(err, prev_err) > 0))
break;
mag_set(prev_err, err);
}
/* X = 1-x/t, Y = 1-y/t, Z = -X-Y, t = (x+y+z)/3 */
acb_add(t, xx, yy, wp);
acb_add(t, t, zz, wp);
acb_div_ui(t, t, 3, wp);
acb_div(X, xx, t, wp);
acb_sub_ui(X, X, 1, wp);
acb_neg(X, X);
acb_div(Y, yy, t, wp);
acb_sub_ui(Y, Y, 1, wp);
acb_neg(Y, Y);
acb_add(Z, X, Y, wp);
acb_neg(Z, Z);
/* E2 = XY-Z^2, E3 = XYZ */
acb_mul(E2, X, Y, wp);
acb_mul(E3, E2, Z, wp);
acb_submul(E2, Z, Z, wp);
/*
Crude bound for the coefficient of
X^n1 Y^n2 Z^n3 with n1+n2+n3=n: 2*(9/8)^n.
*/
/* Error bound. */
acb_get_mag(err, X);
acb_get_mag(err2, Y);
mag_max(err, err, err2);
acb_get_mag(err2, Z);
mag_max(err, err, err2);
mag_mul_ui(err, err, 9);
mag_mul_2exp_si(err, err, -3);
mag_geom_series(err, err, order);
mag_mul_2exp_si(err, err, 1);
acb_elliptic_rf_taylor_sum(sx, E2, E3, order, wp);
if (acb_is_real(X) && acb_is_real(Y))
arb_add_error_mag(acb_realref(sx), err);
else
acb_add_error_mag(sx, err);
acb_rsqrt(t, t, wp);
acb_mul(res, sx, t, prec);
acb_clear(xx); acb_clear(yy); acb_clear(zz);
acb_clear(sx); acb_clear(sy); acb_clear(sz);
acb_clear(X); acb_clear(Y); acb_clear(Z); acb_clear(E2); acb_clear(E3);
acb_clear(t);
mag_clear(err);
mag_clear(err2);
mag_clear(prev_err);
}
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