void acb_hypgeom_expint(acb_t res, const acb_t s, const acb_t z, slong prec) { acb_t t; acb_init(t); acb_sub_ui(t, s, 1, prec); acb_neg(t, t); acb_hypgeom_gamma_upper(res, t, z, 2, prec); acb_clear(t); }
void arb_hypgeom_gamma_upper(arb_t res, const arb_t s, const arb_t z, int regularized, slong prec) { acb_t t, u; acb_init(t); acb_init(u); arb_set(acb_realref(t), s); arb_set(acb_realref(u), z); acb_hypgeom_gamma_upper(t, t, u, regularized, prec); if (acb_is_finite(t) && acb_is_real(t)) arb_swap(res, acb_realref(t)); else arb_indeterminate(res); acb_clear(t); acb_clear(u); }
int main() { long iter; flint_rand_t state; printf("gamma_upper...."); fflush(stdout); flint_randinit(state); for (iter = 0; iter < 2000; iter++) { acb_t a0, a1, b, z, w0, w1, t, u; long prec0, prec1; int modified; acb_init(a0); acb_init(a1); acb_init(b); acb_init(z); acb_init(w0); acb_init(w1); acb_init(t); acb_init(u); modified = n_randint(state, 2); prec0 = 2 + n_randint(state, 1000); prec1 = 2 + n_randint(state, 1000); acb_randtest_maybe_half_int(a0, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100)); acb_randtest(z, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100)); acb_randtest(w0, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100)); acb_randtest(w1, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100)); acb_add_ui(a1, a0, 1, prec0); switch (n_randint(state, 4)) { case 0: acb_hypgeom_gamma_upper_asymp(w0, a0, z, modified, prec0); break; case 1: acb_hypgeom_gamma_upper_1f1a(w0, a0, z, modified, prec0); break; case 2: acb_hypgeom_gamma_upper_1f1b(w0, a0, z, modified, prec0); break; default: acb_hypgeom_gamma_upper(w0, a0, z, modified, prec0); } switch (n_randint(state, 4)) { case 0: acb_hypgeom_gamma_upper_asymp(w1, a0, z, modified, prec1); break; case 1: acb_hypgeom_gamma_upper_1f1a(w1, a0, z, modified, prec1); break; case 2: acb_hypgeom_gamma_upper_1f1b(w1, a0, z, modified, prec1); break; default: acb_hypgeom_gamma_upper(w1, a0, z, modified, prec1); } if (!acb_overlaps(w0, w1)) { printf("FAIL: consistency\n\n"); printf("nu = "); acb_printd(a0, 30); printf("\n\n"); printf("z = "); acb_printd(z, 30); printf("\n\n"); printf("w0 = "); acb_printd(w0, 30); printf("\n\n"); printf("w1 = "); acb_printd(w1, 30); printf("\n\n"); abort(); } switch (n_randint(state, 4)) { case 0: acb_hypgeom_gamma_upper_asymp(w1, a1, z, modified, prec1); break; case 1: acb_hypgeom_gamma_upper_1f1a(w1, a1, z, modified, prec1); break; case 2: acb_hypgeom_gamma_upper_1f1b(w1, a1, z, modified, prec1); break; default: acb_hypgeom_gamma_upper(w1, a1, z, modified, prec1); } /* a Gamma(a,z) + exp(-z) z^a = Gamma(a+1,z) */ if (modified) acb_one(t); else acb_pow(t, z, a0, prec0); acb_neg(u, z); acb_exp(u, u, prec0); acb_mul(t, t, u, prec0); if (modified) { acb_mul(b, w1, z, prec0); acb_addmul(t, a0, w0, prec0); acb_sub(t, t, b, prec0); } else { acb_addmul(t, a0, w0, prec0); acb_sub(t, t, w1, prec0); } if (!acb_contains_zero(t)) { printf("FAIL: contiguous relation\n\n"); printf("nu = "); acb_printd(a0, 30); printf("\n\n"); printf("z = "); acb_printd(z, 30); printf("\n\n"); printf("w0 = "); acb_printd(w0, 30); printf("\n\n"); printf("w1 = "); acb_printd(w1, 30); printf("\n\n"); printf("t = "); acb_printd(t, 30); printf("\n\n"); abort(); } acb_clear(a0); acb_clear(a1); acb_clear(b); acb_clear(z); acb_clear(w0); acb_clear(w1); acb_clear(t); acb_clear(u); } flint_randclear(state); flint_cleanup(); printf("PASS\n"); return EXIT_SUCCESS; }
int main() { slong iter; flint_rand_t state; flint_printf("gamma_lower...."); fflush(stdout); flint_randinit(state); for (iter = 0; iter < 2000 * arb_test_multiplier(); iter++) { acb_t a0, a1, b, z, w0, w1, t, u, enz; slong prec0, prec1; int regularized; acb_init(a0); acb_init(a1); acb_init(b); acb_init(z); acb_init(w0); acb_init(w1); acb_init(t); acb_init(u); acb_init(enz); regularized = n_randint(state, 3); prec0 = 2 + n_randint(state, 1000); prec1 = 2 + n_randint(state, 1000); acb_randtest_param(a0, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100)); acb_randtest(z, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100)); acb_randtest(w0, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100)); acb_randtest(w1, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100)); acb_add_ui(a1, a0, 1, prec0); acb_hypgeom_gamma_lower(w0, a0, z, regularized, prec0); acb_hypgeom_gamma_lower(w1, a1, z, regularized, prec1); acb_neg(enz, z); acb_exp(enz, enz, prec0); /* recurrence relations */ if (regularized == 2) { /* gamma^{*}(a,z) - exp(-z)/Gamma(a+1) - z gamma^{*}(a+1,z) = 0 */ /* http://dlmf.nist.gov/8.8.E4 */ acb_set(t, w0); acb_rgamma(u, a1, prec0); acb_submul(t, enz, u, prec0); acb_submul(t, z, w1, prec0); } else if (regularized == 1) { /* P(a,z) - exp(-z) z^a / Gamma(a+1) - P(a+1,z) = 0 */ /* http://dlmf.nist.gov/8.8.E5 */ acb_pow(u, z, a0, prec0); acb_rgamma(b, a1, prec0); acb_mul(u, u, b, prec0); acb_sub(t, w0, w1, prec0); acb_submul(t, enz, u, prec0); } else { /* a gamma(a,z) - exp(-z) z^a - gamma(a+1,z) = 0 */ /* http://dlmf.nist.gov/8.8.E1 */ acb_pow(u, z, a0, prec0); acb_mul(t, a0, w0, prec0); acb_submul(t, enz, u, prec0); acb_sub(t, t, w1, prec0); } if (!acb_contains_zero(t)) { flint_printf("FAIL: recurrence relation\n\n"); flint_printf("regularized = %d\n\n", regularized); flint_printf("a0 = "); acb_printd(a0, 30); flint_printf("\n\n"); flint_printf("z = "); acb_printd(z, 30); flint_printf("\n\n"); flint_printf("w0 = "); acb_printd(w0, 30); flint_printf("\n\n"); flint_printf("w1 = "); acb_printd(w1, 30); flint_printf("\n\n"); flint_printf("t = "); acb_printd(t, 30); flint_printf("\n\n"); abort(); } /* identities relating lower and upper incomplete gamma functions */ if (regularized == 0 || regularized == 1) { acb_t u0; acb_init(u0); acb_hypgeom_gamma_upper(u0, a0, z, regularized, prec0); acb_zero(t); if (regularized == 1) { /* P(s,z) + Q(s,z) - 1 = 0 */ /* http://dlmf.nist.gov/8.2.E5 */ acb_add(t, w0, u0, prec0); acb_sub_ui(t, t, 1, prec0); } else { /* gamma(s,z) + Gamma(s,z) - Gamma(s) = 0 */ /* excludes non-positive integer values of s */ /* http://dlmf.nist.gov/8.2.E3 */ if (!acb_is_int(a0) || arb_is_positive(acb_realref(a0))) { acb_gamma(b, a0, prec0); acb_add(t, w0, u0, prec0); acb_sub(t, t, b, prec0); } } if (!acb_contains_zero(t)) { flint_printf("FAIL: lower plus upper\n\n"); flint_printf("regularized = %d\n\n", regularized); flint_printf("a0 = "); acb_printd(a0, 30); flint_printf("\n\n"); flint_printf("z = "); acb_printd(z, 30); flint_printf("\n\n"); flint_printf("w0 = "); acb_printd(w0, 30); flint_printf("\n\n"); flint_printf("w1 = "); acb_printd(w1, 30); flint_printf("\n\n"); flint_printf("t = "); acb_printd(t, 30); flint_printf("\n\n"); abort(); } acb_clear(u0); } acb_clear(a0); acb_clear(a1); acb_clear(b); acb_clear(z); acb_clear(w0); acb_clear(w1); acb_clear(t); acb_clear(u); acb_clear(enz); } flint_randclear(state); flint_cleanup(); flint_printf("PASS\n"); return EXIT_SUCCESS; }