slong hypgeom_estimate_terms(const mag_t z, int r, slong prec) { double y, t; t = mag_get_d(z); if (t == 0) return 1; if (r == 0) { if (t >= 1) { flint_printf("z must be smaller than 1\n"); abort(); } y = (log(1-t) - prec * LOG2) / log(t) + 1; } else { y = (prec * LOG2) / (d_root(t, r) * EXP1 * r); y = (prec * LOG2) / (r * d_lambertw(y)) + 1; } if (y >= WORD_MAX / 2) { flint_printf("error: series will converge too slowly\n"); abort(); } return y; }
slong _arb_mat_exp_choose_N(const mag_t norm, slong prec) { if (mag_is_special(norm) || mag_cmp_2exp_si(norm, 30) > 0 || mag_cmp_2exp_si(norm, -prec) < 0) { return 1; } else if (mag_cmp_2exp_si(norm, -300) < 0) { slong N = -MAG_EXP(norm); return (prec + N - 1) / N; } else { double c, t; c = mag_get_d(norm); t = d_lambertw(prec * LOG2_OVER_E / c); t = c * exp(t + 1.0); return FLINT_MIN((slong) (t + 1.0), 2 * 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); }