void
acb_hypgeom_airy_asymp2(acb_t ai, acb_t aip, acb_t bi, acb_t bip,
const acb_t z, slong n, slong prec)
{
/* avoid singularity in asymptotic expansion near 0 */
if (acb_rel_accuracy_bits(z) > 3)
acb_hypgeom_airy_asymp(ai, aip, bi, bip, z, n, prec);
else
acb_hypgeom_airy_prop(ai, aip, bi, bip, z, n, 1, prec);
}
void
acb_hypgeom_airy(acb_t ai, acb_t aip, acb_t bi, acb_t bip, const acb_t z, slong prec)
{
arf_srcptr re, im;
double x, y, t, zmag, z15, term_est, airy_est, abstol;
slong n, wp;
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;
}
re = arb_midref(acb_realref(z));
im = arb_midref(acb_imagref(z));
wp = prec * 1.03 + 15;
/* 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;
}
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);
}
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);
zmag = sqrt(x * x + y * y);
z15 = zmag * sqrt(zmag);
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;
/* estimate the smaller of Ai and Bi */
airy_est = estimate_airy(x, y, (ai != NULL || aip != NULL));
/* estimate absolute tolerance and necessary working precision */
abstol = airy_est - wp;
wp = wp + term_est - airy_est;
wp = FLINT_MAX(wp, 10);
t = 3 * (-abstol * LOG2) / (2 * z15 * EXP1);
t = (-abstol * LOG2) / (2 * d_lambertw(t));
n = FLINT_MAX(t + 1, 2);
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);
}
}
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);
}
/* error propagation based on derivatives */
void
acb_hypgeom_airy_prop(acb_t ai, acb_t aip, acb_t bi, acb_t bip,
const acb_t z, slong n, int algo, slong prec)
{
mag_t aib, aipb, bib, bipb, zb, rad;
acb_t zz;
int real;
mag_init(aib);
mag_init(aipb);
mag_init(bib);
mag_init(bipb);
mag_init(zb);
mag_init(rad);
acb_init(zz);
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);
acb_hypgeom_airy_bound(aib, aipb, bib, bipb, z);
if (algo == 0)
acb_hypgeom_airy_direct(ai, aip, bi, bip, zz, n, prec);
else
acb_hypgeom_airy_asymp(ai, aip, bi, bip, zz, n, prec);
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);
}
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);
}
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);
}
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);
}
mag_clear(aib);
mag_clear(aipb);
mag_clear(bib);
mag_clear(bipb);
mag_clear(zb);
mag_clear(rad);
acb_clear(zz);
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("airy....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 10000 * arb_test_multiplier(); iter++)
{
acb_t z, t, w;
acb_t ai1, aip1, bi1, bip1;
acb_t ai2, aip2, bi2, bip2;
slong n1, n2, prec1, prec2;
unsigned int mask;
acb_init(z); acb_init(t); acb_init(w);
acb_init(ai1); acb_init(aip1); acb_init(bi1); acb_init(bip1);
acb_init(ai2); acb_init(aip2); acb_init(bi2); acb_init(bip2);
prec1 = 2 + n_randint(state, 1000);
prec2 = 2 + n_randint(state, 1000);
n1 = n_randint(state, 300);
n2 = n_randint(state, 300);
acb_randtest_param(z, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100));
acb_randtest_param(t, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100));
acb_add(z, z, t, 1000);
acb_sub(z, z, t, 1000);
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_airy_direct(ai1, aip1, bi1, bip1, z, n1, prec1);
break;
case 1:
acb_hypgeom_airy_asymp(ai1, aip1, bi1, bip1, z, n1, prec1);
break;
default:
acb_hypgeom_airy(ai1, aip1, bi1, bip1, z, prec1);
break;
}
switch (n_randint(state, 3))
{
case 0:
acb_hypgeom_airy_direct(ai2, aip2, bi2, bip2, z, n2, prec2);
break;
case 1:
acb_hypgeom_airy_asymp(ai2, aip2, bi2, bip2, z, n2, prec2);
break;
default:
acb_hypgeom_airy(ai2, aip2, bi2, bip2, z, prec2);
break;
}
if (!acb_overlaps(ai1, ai2))
{
flint_printf("FAIL: consistency (Ai)\n\n");
flint_printf("z = "); acb_printd(z, 30); flint_printf("\n\n");
flint_printf("ai1 = "); acb_printd(ai1, 30); flint_printf("\n\n");
flint_printf("ai2 = "); acb_printd(ai2, 30); flint_printf("\n\n");
abort();
}
if (!acb_overlaps(aip1, aip2))
{
flint_printf("FAIL: consistency (Ai')\n\n");
flint_printf("z = "); acb_printd(z, 30); flint_printf("\n\n");
flint_printf("aip1 = "); acb_printd(aip1, 30); flint_printf("\n\n");
flint_printf("aip2 = "); acb_printd(aip2, 30); flint_printf("\n\n");
abort();
}
if (!acb_overlaps(bi1, bi2))
{
flint_printf("FAIL: consistency (Bi)\n\n");
flint_printf("z = "); acb_printd(z, 30); flint_printf("\n\n");
flint_printf("bi1 = "); acb_printd(bi1, 30); flint_printf("\n\n");
flint_printf("bi2 = "); acb_printd(bi2, 30); flint_printf("\n\n");
abort();
}
if (!acb_overlaps(bip1, bip2))
{
flint_printf("FAIL: consistency (Bi')\n\n");
flint_printf("z = "); acb_printd(z, 30); flint_printf("\n\n");
flint_printf("bip1 = "); acb_printd(bip1, 30); flint_printf("\n\n");
flint_printf("bip2 = "); acb_printd(bip2, 30); flint_printf("\n\n");
abort();
}
acb_mul(w, ai1, bip1, prec1);
acb_submul(w, bi1, aip1, prec1);
acb_const_pi(t, prec1);
acb_inv(t, t, prec1);
if (!acb_overlaps(w, t))
{
flint_printf("FAIL: wronskian\n\n");
flint_printf("z = "); acb_printd(z, 30); flint_printf("\n\n");
flint_printf("ai1 = "); acb_printd(ai1, 30); flint_printf("\n\n");
flint_printf("aip1 = "); acb_printd(aip1, 30); flint_printf("\n\n");
flint_printf("bi1 = "); acb_printd(bi1, 30); flint_printf("\n\n");
flint_printf("bip1 = "); acb_printd(bip1, 30); flint_printf("\n\n");
flint_printf("w = "); acb_printd(w, 30); flint_printf("\n\n");
abort();
}
mask = n_randlimb(state);
acb_hypgeom_airy((mask & 1) ? ai2 : NULL,
(mask & 2) ? aip2 : NULL,
(mask & 4) ? bi2 : NULL,
(mask & 8) ? bip2 : NULL, z, prec2);
if (!acb_overlaps(ai1, ai2) || !acb_overlaps(aip1, aip2) ||
!acb_overlaps(bi1, bi2) || !acb_overlaps(bip1, bip2))
{
flint_printf("FAIL: consistency (mask)\n\n");
flint_printf("mask = %u\n\n", mask);
flint_printf("z = "); acb_printd(z, 30); flint_printf("\n\n");
flint_printf("ai1 = "); acb_printd(ai1, 30); flint_printf("\n\n");
flint_printf("ai2 = "); acb_printd(ai2, 30); flint_printf("\n\n");
flint_printf("aip1 = "); acb_printd(aip1, 30); flint_printf("\n\n");
flint_printf("aip2 = "); acb_printd(aip2, 30); flint_printf("\n\n");
flint_printf("bi1 = "); acb_printd(bi1, 30); flint_printf("\n\n");
flint_printf("bi2 = "); acb_printd(bi2, 30); flint_printf("\n\n");
flint_printf("bip1 = "); acb_printd(bip1, 30); flint_printf("\n\n");
flint_printf("bip2 = "); acb_printd(bip2, 30); flint_printf("\n\n");
abort();
}
acb_clear(z); acb_clear(t); acb_clear(w);
acb_clear(ai1); acb_clear(aip1); acb_clear(bi1); acb_clear(bip1);
acb_clear(ai2); acb_clear(aip2); acb_clear(bi2); acb_clear(bip2);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
