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; }