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/*============================================================================= | ||
This file is part of ARB. | ||
ARB is free software; you can redistribute it and/or modify | ||
it under the terms of the GNU General Public License as published by | ||
the Free Software Foundation; either version 2 of the License, or | ||
(at your option) any later version. | ||
ARB is distributed in the hope that it will be useful, | ||
but WITHOUT ANY WARRANTY; without even the implied warranty of | ||
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | ||
GNU General Public License for more details. | ||
You should have received a copy of the GNU General Public License | ||
along with ARB; if not, write to the Free Software | ||
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA | ||
=============================================================================*/ | ||
/****************************************************************************** | ||
Copyright (C) 2015 Fredrik Johansson | ||
******************************************************************************/ | ||
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#include "acb_hypgeom.h" | ||
#include "double_extras.h" | ||
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#define LOG2 0.69314718055994530942 | ||
#define EXP1 2.7182818284590452354 | ||
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static const double small_log_tab[] = { | ||
0.0, 0.0, 0.693147180559945309, | ||
1.09861228866810969, 1.38629436111989062, 1.60943791243410037, | ||
1.791759469228055, 1.94591014905531331, 2.07944154167983593, | ||
2.19722457733621938, 2.30258509299404568, 2.39789527279837054, | ||
2.48490664978800031, 2.56494935746153674, 2.63905732961525861, | ||
2.70805020110221007, 2.77258872223978124, 2.83321334405621608, | ||
2.89037175789616469, 2.94443897916644046, 2.99573227355399099, | ||
3.044522437723423, 3.09104245335831585, 3.13549421592914969, | ||
3.17805383034794562, 3.21887582486820075, 3.25809653802148205, | ||
3.29583686600432907, 3.33220451017520392, 3.36729582998647403, | ||
3.40119738166215538, 3.43398720448514625, 3.46573590279972655, | ||
3.49650756146648024, 3.52636052461616139, 3.55534806148941368, | ||
3.58351893845611, 3.61091791264422444, 3.63758615972638577, | ||
3.66356164612964643, 3.6888794541139363, 3.7135720667043078, | ||
3.73766961828336831, 3.76120011569356242, 3.78418963391826116, | ||
3.80666248977031976, 3.828641396489095, 3.85014760171005859, | ||
3.87120101090789093, 3.89182029811062661, 3.91202300542814606, | ||
3.93182563272432577, 3.95124371858142735, 3.97029191355212183, | ||
3.98898404656427438, 4.00733318523247092, 4.02535169073514923, | ||
4.04305126783455015, 4.06044301054641934, 4.07753744390571945, | ||
4.09434456222210068, 4.11087386417331125, 4.12713438504509156, | ||
4.14313472639153269, | ||
}; | ||
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static slong | ||
asymp_pick_terms(double prec, double logz) | ||
{ | ||
double logk, log2term, log2term_prev; | ||
slong k; | ||
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log2term_prev = 0.0; | ||
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for (k = 1; ; k++) | ||
{ | ||
logk = k < 64 ? small_log_tab[k] : log(k); | ||
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log2term = -1.3257480647361593990 - 0.72134752044448170368*logk + | ||
k * (-1.8577325401678072259 + 1.4426950408889634074*logk | ||
- 2.1640425613334451110*logz); | ||
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if (log2term > log2term_prev - 0.5) | ||
return -1; | ||
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if (log2term < -prec) | ||
return k; | ||
} | ||
} | ||
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/* | ||
Accurate estimate of log2(|Ai(z)|) or log2(|Bi(z)|), given z = x + yi. | ||
Should subtract 0.25*log2(|z| + log2(2*sqrt(pi)) from the output. | ||
*/ | ||
static double | ||
estimate_airy(double x, double y, int ai) | ||
{ | ||
double r, t, sgn; | ||
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r = x; | ||
t = x * ((x * x) - 3.0 * (y * y)); | ||
y = y * (3.0 * (x * x) - (y * y)); | ||
x = t * (1.0 / 9.0); | ||
y = y * (1.0 / 9.0); | ||
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sgn = 1.0; | ||
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if (r > 0.0 && x > 0.0) | ||
{ | ||
t = sqrt(x * x + y * y) + x; | ||
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if (ai) sgn = -1.0; | ||
} | ||
else | ||
{ | ||
x = -x; | ||
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if (x > 0.0 && x > 1e6 * fabs(y)) | ||
t = y * y / (2.0 * x); | ||
else | ||
t = sqrt(x * x + y * y) - x; | ||
} | ||
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return sgn * sqrt(0.5 * t) * 2.8853900817779268147; | ||
} | ||
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/* error propagation based on derivatives */ | ||
void | ||
acb_hypgeom_airy_direct_prop(acb_t ai, acb_t aip, acb_t bi, acb_t bip, | ||
const acb_t z, slong n, long prec) | ||
{ | ||
mag_t aib, aipb, bib, bipb, zb, rad; | ||
acb_t zz; | ||
int real; | ||
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mag_init(aib); | ||
mag_init(aipb); | ||
mag_init(bib); | ||
mag_init(bipb); | ||
mag_init(zb); | ||
mag_init(rad); | ||
acb_init(zz); | ||
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real = acb_is_real(z); | ||
arf_set(arb_midref(acb_realref(zz)), arb_midref(acb_realref(z))); | ||
arf_set(arb_midref(acb_imagref(zz)), arb_midref(acb_imagref(z))); | ||
mag_hypot(rad, arb_radref(acb_realref(z)), arb_radref(acb_imagref(z))); | ||
acb_get_mag(zb, z); | ||
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acb_hypgeom_airy_bound(aib, aipb, bib, bipb, z); | ||
acb_hypgeom_airy_direct(ai, aip, bi, bip, zz, n, prec); | ||
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if (ai != NULL) | ||
{ | ||
mag_mul(aipb, aipb, rad); | ||
if (real) | ||
arb_add_error_mag(acb_realref(ai), aipb); | ||
else | ||
acb_add_error_mag(ai, aipb); | ||
} | ||
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if (aip != NULL) | ||
{ | ||
mag_mul(aib, aib, rad); | ||
mag_mul(aib, aib, zb); /* |Ai''(z)| = |z Ai(z)| */ | ||
if (real) | ||
arb_add_error_mag(acb_realref(aip), aib); | ||
else | ||
acb_add_error_mag(aip, aib); | ||
} | ||
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if (bi != NULL) | ||
{ | ||
mag_mul(bipb, bipb, rad); | ||
if (real) | ||
arb_add_error_mag(acb_realref(bi), bipb); | ||
else | ||
acb_add_error_mag(bi, bipb); | ||
} | ||
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if (bip != NULL) | ||
{ | ||
mag_mul(bib, bib, rad); | ||
mag_mul(bib, bib, zb); /* |Bi''(z)| = |z Bi(z)| */ | ||
if (real) | ||
arb_add_error_mag(acb_realref(bip), bib); | ||
else | ||
acb_add_error_mag(bip, bib); | ||
} | ||
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mag_clear(aib); | ||
mag_clear(aipb); | ||
mag_clear(bib); | ||
mag_clear(bipb); | ||
mag_clear(zb); | ||
mag_clear(rad); | ||
acb_clear(zz); | ||
} | ||
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void | ||
acb_hypgeom_airy(acb_t ai, acb_t aip, acb_t bi, acb_t bip, const acb_t z, long prec) | ||
{ | ||
arf_srcptr re, im; | ||
double x, y, t, zmag, z15, term_est, airy_est, abstol; | ||
slong n, wp; | ||
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if (!acb_is_finite(z)) | ||
{ | ||
if (ai != NULL) acb_indeterminate(ai); | ||
if (aip != NULL) acb_indeterminate(aip); | ||
if (bi != NULL) acb_indeterminate(bi); | ||
if (bip != NULL) acb_indeterminate(bip); | ||
return; | ||
} | ||
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re = arb_midref(acb_realref(z)); | ||
im = arb_midref(acb_imagref(z)); | ||
wp = prec * 1.03 + 15; | ||
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/* tiny input -- use direct method and pick n without underflowing */ | ||
if (arf_cmpabs_2exp_si(re, -64) < 0 && arf_cmpabs_2exp_si(im, -64) < 0) | ||
{ | ||
if (arf_cmpabs_2exp_si(re, -wp) < 0 && arf_cmpabs_2exp_si(im, -wp) < 0) | ||
{ | ||
n = 1; /* very tiny input */ | ||
} | ||
else | ||
{ | ||
if (arf_cmpabs(re, im) > 0) | ||
zmag = fmpz_get_d(ARF_EXPREF(re)); | ||
else | ||
zmag = fmpz_get_d(ARF_EXPREF(im)); | ||
zmag = (zmag + 1) * (1.0 / LOG2); | ||
n = wp / (-zmag) + 1; | ||
} | ||
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acb_hypgeom_airy_direct(ai, aip, bi, bip, z, n, wp); | ||
} /* huge input -- use asymptotics and pick n without overflowing */ | ||
else if ((arf_cmpabs_2exp_si(re, 64) > 0 || arf_cmpabs_2exp_si(im, 64) > 0)) | ||
{ | ||
if (arf_cmpabs_2exp_si(re, prec) > 0 || arf_cmpabs_2exp_si(im, prec) > 0) | ||
{ | ||
n = 1; /* very huge input */ | ||
} | ||
else | ||
{ | ||
x = fmpz_get_d(ARF_EXPREF(re)); | ||
y = fmpz_get_d(ARF_EXPREF(im)); | ||
zmag = (FLINT_MAX(x, y) - 2) * (1.0 / LOG2); | ||
n = asymp_pick_terms(wp, zmag); | ||
n = FLINT_MAX(n, 1); | ||
} | ||
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acb_hypgeom_airy_asymp(ai, aip, bi, bip, z, n, wp); | ||
} | ||
else /* moderate input */ | ||
{ | ||
x = arf_get_d(re, ARF_RND_DOWN); | ||
y = arf_get_d(im, ARF_RND_DOWN); | ||
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zmag = sqrt(x * x + y * y); | ||
z15 = zmag * sqrt(zmag); | ||
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if (zmag >= 4.0 && (n = asymp_pick_terms(wp, log(zmag))) != -1) | ||
{ | ||
acb_hypgeom_airy_asymp(ai, aip, bi, bip, z, n, wp); | ||
} | ||
else if (zmag <= 1.5) | ||
{ | ||
t = 3 * (wp * LOG2) / (2 * z15 * EXP1); | ||
t = (wp * LOG2) / (2 * d_lambertw(t)); | ||
n = FLINT_MAX(t + 1, 2); | ||
acb_hypgeom_airy_direct(ai, aip, bi, bip, z, n, wp); | ||
} | ||
else | ||
{ | ||
/* estimate largest term: log2(exp(2(z^3/9)^(1/2))) */ | ||
term_est = 0.96179669392597560491 * z15; | ||
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/* estimate the smaller of Ai and Bi */ | ||
airy_est = estimate_airy(x, y, (ai != NULL || aip != NULL)); | ||
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/* estimate absolute tolerance and necessary working precision */ | ||
abstol = airy_est - wp; | ||
wp = wp + term_est - airy_est; | ||
wp = FLINT_MAX(wp, 10); | ||
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t = 3 * (-abstol * LOG2) / (2 * z15 * EXP1); | ||
t = (-abstol * LOG2) / (2 * d_lambertw(t)); | ||
n = FLINT_MAX(t + 1, 2); | ||
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if (acb_is_exact(z)) | ||
acb_hypgeom_airy_direct(ai, aip, bi, bip, z, n, wp); | ||
else | ||
acb_hypgeom_airy_direct_prop(ai, aip, bi, bip, z, n, wp); | ||
} | ||
} | ||
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if (ai != NULL) acb_set_round(ai, ai, prec); | ||
if (aip != NULL) acb_set_round(aip, aip, prec); | ||
if (bi != NULL) acb_set_round(bi, bi, prec); | ||
if (bip != NULL) acb_set_round(bip, bip, prec); | ||
} | ||
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