mirror of
https://github.com/vale981/arb
synced 2025-03-04 17:01:40 -05:00
compute erf without cancellation problems for complex z
This commit is contained in:
Fredrik Johansson
parent
65d13e3cab
commit
d0a2c0cb0d
3 changed files with 137 additions and 19 deletions
|
|
@ -128,6 +128,7 @@ void acb_hypgeom_gamma_upper(acb_t res, const acb_t s, const acb_t z, int modifi
|
|||
|
||||
void acb_hypgeom_expint(acb_t res, const acb_t s, const acb_t z, slong prec);
|
||||
|
||||
void acb_hypgeom_erf_propagated_error(mag_t re, mag_t im, const acb_t z);
|
||||
void acb_hypgeom_erf_1f1a(acb_t res, const acb_t z, slong prec);
|
||||
void acb_hypgeom_erf_1f1b(acb_t res, const acb_t z, slong prec);
|
||||
void acb_hypgeom_erf_asymp(acb_t res, const acb_t z, slong prec, slong prec2);
|
||||
|
|
|
|||
|
|
@ -141,11 +141,108 @@ acb_hypgeom_erf_asymp(acb_t res, const acb_t z, slong prec, slong prec2)
|
|||
acb_clear(u);
|
||||
}
|
||||
|
||||
void
|
||||
acb_hypgeom_erf_propagated_error(mag_t re, mag_t im, const acb_t z)
|
||||
{
|
||||
mag_t x, y;
|
||||
|
||||
mag_init(x);
|
||||
mag_init(y);
|
||||
|
||||
/* |exp(-(x+y)^2)| = exp(y^2-x^2) */
|
||||
arb_get_mag(y, acb_imagref(z));
|
||||
mag_mul(y, y, y);
|
||||
|
||||
arb_get_mag_lower(x, acb_realref(z));
|
||||
mag_mul_lower(x, x, x);
|
||||
|
||||
if (mag_cmp(y, x) >= 0)
|
||||
{
|
||||
mag_sub(re, y, x);
|
||||
mag_exp(re, re);
|
||||
}
|
||||
else
|
||||
{
|
||||
mag_sub_lower(re, x, y);
|
||||
mag_expinv(re, re);
|
||||
}
|
||||
|
||||
/* Radius. */
|
||||
mag_hypot(x, arb_radref(acb_realref(z)), arb_radref(acb_imagref(z)));
|
||||
mag_mul(re, re, x);
|
||||
|
||||
/* 2/sqrt(pi) < 289/256 */
|
||||
mag_mul_ui(re, re, 289);
|
||||
mag_mul_2exp_si(re, re, -8);
|
||||
|
||||
if (arb_is_zero(acb_imagref(z)))
|
||||
{
|
||||
/* todo: could bound magnitude even for complex numbers */
|
||||
mag_set_ui(y, 2);
|
||||
mag_min(re, re, y);
|
||||
|
||||
mag_zero(im);
|
||||
}
|
||||
else if (arb_is_zero(acb_realref(z)))
|
||||
{
|
||||
mag_swap(im, re);
|
||||
mag_zero(re);
|
||||
}
|
||||
else
|
||||
{
|
||||
mag_set(im, re);
|
||||
}
|
||||
|
||||
mag_clear(x);
|
||||
mag_clear(y);
|
||||
}
|
||||
|
||||
void
|
||||
acb_hypgeom_erf_1f1(acb_t res, const acb_t z, slong prec,
|
||||
slong wp, int more_imaginary)
|
||||
{
|
||||
if (acb_rel_accuracy_bits(z) >= wp)
|
||||
{
|
||||
if (more_imaginary)
|
||||
acb_hypgeom_erf_1f1a(res, z, wp);
|
||||
else
|
||||
acb_hypgeom_erf_1f1b(res, z, wp);
|
||||
}
|
||||
else
|
||||
{
|
||||
acb_t zmid;
|
||||
mag_t re_err, im_err;
|
||||
|
||||
acb_init(zmid);
|
||||
mag_init(re_err);
|
||||
mag_init(im_err);
|
||||
|
||||
acb_hypgeom_erf_propagated_error(re_err, im_err, z);
|
||||
arf_set(arb_midref(acb_realref(zmid)), arb_midref(acb_realref(z)));
|
||||
arf_set(arb_midref(acb_imagref(zmid)), arb_midref(acb_imagref(z)));
|
||||
|
||||
if (more_imaginary)
|
||||
acb_hypgeom_erf_1f1a(res, zmid, wp);
|
||||
else
|
||||
acb_hypgeom_erf_1f1b(res, zmid, wp);
|
||||
|
||||
arb_add_error_mag(acb_realref(res), re_err);
|
||||
arb_add_error_mag(acb_imagref(res), im_err);
|
||||
|
||||
acb_clear(zmid);
|
||||
mag_clear(re_err);
|
||||
mag_clear(im_err);
|
||||
}
|
||||
|
||||
acb_set_round(res, res, prec);
|
||||
}
|
||||
|
||||
void
|
||||
acb_hypgeom_erf(acb_t res, const acb_t z, slong prec)
|
||||
{
|
||||
double x, y, absz2, logz;
|
||||
slong prec2;
|
||||
double x, y, abs_z2, log_z, log_erf_z_asymp;
|
||||
slong prec2, wp;
|
||||
int more_imaginary;
|
||||
|
||||
if (!acb_is_finite(z))
|
||||
{
|
||||
|
|
@ -159,10 +256,10 @@ acb_hypgeom_erf(acb_t res, const acb_t z, slong prec)
|
|||
return;
|
||||
}
|
||||
|
||||
if ((arf_cmpabs_2exp_si(arb_midref(acb_realref(z)), 0) < 0 &&
|
||||
arf_cmpabs_2exp_si(arb_midref(acb_imagref(z)), 0) < 0))
|
||||
if ((arf_cmpabs_2exp_si(arb_midref(acb_realref(z)), -64) < 0 &&
|
||||
arf_cmpabs_2exp_si(arb_midref(acb_imagref(z)), -64) < 0))
|
||||
{
|
||||
acb_hypgeom_erf_1f1a(res, z, prec);
|
||||
acb_hypgeom_erf_1f1(res, z, prec, prec, 1);
|
||||
return;
|
||||
}
|
||||
|
||||
|
|
@ -176,26 +273,34 @@ acb_hypgeom_erf(acb_t res, const acb_t z, slong prec)
|
|||
x = arf_get_d(arb_midref(acb_realref(z)), ARF_RND_DOWN);
|
||||
y = arf_get_d(arb_midref(acb_imagref(z)), ARF_RND_DOWN);
|
||||
|
||||
absz2 = x * x + y * y;
|
||||
logz = 0.5 * log(absz2);
|
||||
abs_z2 = x * x + y * y;
|
||||
log_z = 0.5 * log(abs_z2);
|
||||
/* estimate of log(erf(z)), disregarding csgn term */
|
||||
log_erf_z_asymp = y*y - x*x - log_z;
|
||||
|
||||
if (logz - absz2 < -(prec + 8) * 0.69314718055994530942)
|
||||
if (log_z - abs_z2 < -(prec + 8) * 0.69314718055994530942)
|
||||
{
|
||||
/* If the asymptotic term is small, we can
|
||||
compute with reduced precision */
|
||||
prec2 = FLINT_MIN(prec + 4 + (y*y - x*x - logz) * 1.4426950408889634074, (double) prec);
|
||||
compute with reduced precision. */
|
||||
prec2 = FLINT_MIN(prec + 4 + log_erf_z_asymp * 1.4426950408889634074, (double) prec);
|
||||
prec2 = FLINT_MAX(8, prec2);
|
||||
prec2 = FLINT_MIN(prec2, prec);
|
||||
|
||||
acb_hypgeom_erf_asymp(res, z, prec, prec2);
|
||||
}
|
||||
else if (arf_cmpabs(arb_midref(acb_imagref(z)), arb_midref(acb_realref(z))) > 0)
|
||||
{
|
||||
acb_hypgeom_erf_1f1a(res, z, prec);
|
||||
}
|
||||
else
|
||||
{
|
||||
acb_hypgeom_erf_1f1b(res, z, prec);
|
||||
more_imaginary = arf_cmpabs(arb_midref(acb_imagref(z)),
|
||||
arb_midref(acb_realref(z))) > 0;
|
||||
|
||||
/* Worst case: exp(|x|^2), computed: exp(x^2).
|
||||
(x^2+y^2) - (x^2-y^2) = 2y^2, etc. */
|
||||
if (more_imaginary)
|
||||
wp = prec + FLINT_MAX(2 * x * x, 0.0) * 1.4426950408889634074 + 5;
|
||||
else
|
||||
wp = prec + FLINT_MAX(2 * y * y, 0.0) * 1.4426950408889634074 + 5;
|
||||
|
||||
acb_hypgeom_erf_1f1(res, z, prec, wp, more_imaginary);
|
||||
}
|
||||
}
|
||||
|
||||
|
|
|
|||
|
|
@ -225,14 +225,19 @@ Confluent hypergeometric functions
|
|||
The error function
|
||||
-------------------------------------------------------------------------------
|
||||
|
||||
.. function:: void acb_hypgeom_erf_propagated_error(mag_t re, mag_t im, const acb_t z)
|
||||
|
||||
Sets *re* and *im* to upper bounds for the error in the real and imaginary
|
||||
part resulting from approximating the error function of *z* by
|
||||
the error function evaluated at the midpoint of *z*. Uses
|
||||
the first derivative.
|
||||
|
||||
.. function:: void acb_hypgeom_erf_1f1a(acb_t res, const acb_t z, slong prec)
|
||||
|
||||
.. function:: void acb_hypgeom_erf_1f1b(acb_t res, const acb_t z, slong prec)
|
||||
|
||||
.. function:: void acb_hypgeom_erf_asymp(acb_t res, const acb_t z, slong prec, slong prec2)
|
||||
|
||||
.. function:: void acb_hypgeom_erf(acb_t res, const acb_t z, slong prec)
|
||||
|
||||
Computes the error function respectively using
|
||||
|
||||
.. math ::
|
||||
|
|
@ -247,8 +252,15 @@ The error function
|
|||
\left(1 - \frac{e^{-z^2}}{\sqrt{\pi}}
|
||||
U(\tfrac{1}{2}, \tfrac{1}{2}, z^2)\right).
|
||||
|
||||
and an automatic algorithm choice. The *asymp* version takes a second
|
||||
precision to use for the *U* term.
|
||||
The *asymp* version takes a second precision to use for the *U* term.
|
||||
|
||||
.. function:: void acb_hypgeom_erf(acb_t res, const acb_t z, slong prec)
|
||||
|
||||
Computes the error function using an automatic algorithm choice.
|
||||
If *z* is too small to use the asymptotic expansion, a working precision
|
||||
sufficient to circumvent cancellation in the hypergeometric series is
|
||||
determined automatically, and a bound for the propagated error is
|
||||
computed with :func:`acb_hypgeom_erf_propagated_error`.
|
||||
|
||||
.. function:: void _acb_hypgeom_erf_series(acb_ptr res, acb_srcptr z, slong zlen, slong len, slong prec)
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue