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speed up acb_elliptic_rf for high precision by using a variable length Taylor expansion

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This commit is contained in:
Fredrik Johansson 2017-08-17 23:51:03 +02:00
parent ac713b31aa
commit 2fb86a0383
2 changed files with 158 additions and 53 deletions
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206
acb_elliptic/rf.c
View file

@ -11,8 +11,111 @@
#include "acb_elliptic.h"
/* todo: fast handling of y == z (RC function) and other special cases */
/* todo: allow computing cauchy PV */
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)
@ -20,7 +123,7 @@ acb_elliptic_rf(acb_t res, const acb_t x, const acb_t y, const acb_t z,
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;
slong k, wp, accx, accy, accz, order;
if (!acb_is_finite(x) || !acb_is_finite(y) || !acb_is_finite(z))
{
@ -42,25 +145,26 @@ acb_elliptic_rf(acb_t res, const acb_t x, const acb_t y, const acb_t z,
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);
wp = prec + 20;
/* must do at least one iteration */
for (k = 0; k < prec; k++)
{
/* 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);
wp = FLINT_MAX(accx, accy);
wp = FLINT_MAX(wp, accz);
wp = FLINT_MAX(wp, 0);
wp = FLINT_MIN(wp, prec);
wp += 20;
/* 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);
@ -77,9 +181,29 @@ acb_elliptic_rf(acb_t res, const acb_t x, const acb_t y, const acb_t z,
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);
@ -87,8 +211,7 @@ acb_elliptic_rf(acb_t res, const acb_t x, const acb_t y, const acb_t z,
mag_max(err, err, err2);
acb_get_mag_lower(err2, xx);
mag_div(err, err, err2);
mag_pow_ui(err, err, 8);
mag_pow_ui(err, err, order);
if (mag_cmp_2exp_si(err, -prec) < 0 ||
(k > 2 && mag_cmp(err, prev_err) > 0))
break;
@ -97,35 +220,29 @@ acb_elliptic_rf(acb_t res, const acb_t x, const acb_t y, const acb_t z,
}
/* X = 1-x/t, Y = 1-y/t, Z = -X-Y, t = (x+y+z)/3 */
acb_add(t, xx, yy, prec);
acb_add(t, t, zz, prec);
acb_div_ui(t, t, 3, prec);
acb_add(t, xx, yy, wp);
acb_add(t, t, zz, wp);
acb_div_ui(t, t, 3, wp);
acb_div(X, xx, t, prec);
acb_sub_ui(X, X, 1, prec);
acb_div(X, xx, t, wp);
acb_sub_ui(X, X, 1, wp);
acb_neg(X, X);
acb_div(Y, yy, t, prec);
acb_sub_ui(Y, Y, 1, prec);
acb_div(Y, yy, t, wp);
acb_sub_ui(Y, Y, 1, wp);
acb_neg(Y, Y);
acb_add(Z, X, Y, prec);
acb_add(Z, X, Y, wp);
acb_neg(Z, Z);
/* E2 = XY-Z^2, E3 = XYZ */
acb_mul(E2, X, Y, prec);
acb_mul(E3, E2, Z, prec);
acb_submul(E2, Z, Z, prec);
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.
We truncate before n = 8 (note: for higher precision, we could gain by
computing more terms). The actual evaluation is done using
elementary symmetric polynomials, giving
1 + (-5775*E2**3 + 15015*E2**2*E3 + 10010*E2**2 - 16380*E2*E3 - 24024*E2
+ 6930*E3**2 + 17160*E3)/240240.
*/
/* Error bound. */
@ -134,32 +251,21 @@ acb_elliptic_rf(acb_t res, const acb_t x, const acb_t y, const acb_t z,
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, 8);
mag_geom_series(err, err, order);
mag_mul_2exp_si(err, err, 1);
/* 1 + (E2*(E2*(-5775*E2 + 15015*E3 + 10010) - 16380*E3 - 24024)
+ E3*(6930*E3 + 17160))/240240 */
acb_set_ui(sx, 10010);
acb_addmul_ui(sx, E3, 15015, prec);
acb_submul_ui(sx, E2, 5775, prec);
acb_mul(sx, sx, E2, prec);
acb_submul_ui(sx, E3, 16380, prec);
acb_sub_ui(sx, sx, 24024, prec);
acb_mul(sx, sx, E2, prec);
acb_mul_ui(sy, E3, 6930, prec);
acb_add_ui(sy, sy, 17160, prec);
acb_addmul(sx, sy, E3, prec);
acb_div_ui(sx, sx, 240240, prec);
acb_add_ui(sx, sx, 1, prec);
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, prec);
acb_rsqrt(t, t, wp);
acb_mul(res, sx, t, prec);
acb_clear(xx); acb_clear(yy); acb_clear(zz);

3
doc/source/acb_elliptic.rst
View file

@ -190,8 +190,7 @@ in [Car1995]_ and chapter 19 in [NIST2012]_.
still defined by analytic continuation.
In general, one or more duplication steps are applied until
`x,y,z` are close enough to use a multivariate Taylor polynomial
of total degree 7.
`x,y,z` are close enough to use a multivariate Taylor series.
The special case `R_C(x, y) = R_F(x, y, y) = \frac{1}{2} \int_0^{\infty} (t+x)^{-1/2} (t+y)^{-1} dt`
may be computed by