int main() { slong iter; flint_rand_t state; flint_printf("rsqrt...."); fflush(stdout); flint_randinit(state); /* check (a^(-1/2))^(-2) = a */ for (iter = 0; iter < 10000; iter++) { acb_t a, b, c; slong prec; acb_init(a); acb_init(b); acb_init(c); acb_randtest(a, state, 1 + n_randint(state, 2000), 10); acb_randtest(b, state, 1 + n_randint(state, 2000), 10); prec = 2 + n_randint(state, 2000); acb_rsqrt(b, a, prec); acb_inv(c, b, prec); acb_mul(c, c, c, prec); if (!acb_contains(c, a)) { flint_printf("FAIL: containment\n\n"); flint_printf("a = "); acb_print(a); flint_printf("\n\n"); flint_printf("b = "); acb_print(b); flint_printf("\n\n"); flint_printf("c = "); acb_print(c); flint_printf("\n\n"); abort(); } acb_rsqrt(a, a, prec); if (!acb_equal(a, b)) { flint_printf("FAIL: aliasing\n\n"); flint_printf("a = "); acb_print(a); flint_printf("\n\n"); flint_printf("b = "); acb_print(b); flint_printf("\n\n"); abort(); } acb_clear(a); acb_clear(b); acb_clear(c); } flint_randclear(state); flint_cleanup(); flint_printf("PASS\n"); return EXIT_SUCCESS; }
/* REAL: erf(x) = 2x/sqrt(pi) * exp(-x^2) 1F1(1, 3/2, x^2) */ void acb_hypgeom_erf_1f1b(acb_t res, const acb_t z, slong prec) { acb_t a, b, t, w; acb_init(a); acb_init(b); acb_init(t); acb_init(w); acb_set_ui(b, 3); acb_mul_2exp_si(b, b, -1); acb_mul(w, z, z, prec); acb_hypgeom_pfq_direct(t, a, 0, b, 1, w, -1, prec); acb_neg(w, w); acb_exp(w, w, prec); acb_mul(t, t, w, prec); acb_const_pi(w, prec); acb_rsqrt(w, w, prec); acb_mul(t, t, w, prec); acb_mul(t, t, z, prec); acb_mul_2exp_si(res, t, 1); acb_clear(a); acb_clear(b); acb_clear(t); acb_clear (w); }
void acb_hypgeom_bessel_j_asymp_prefactors(acb_t Ap, acb_t Am, acb_t C, const acb_t nu, const acb_t z, long prec) { if (arb_is_positive(acb_realref(z))) { acb_t t, u; acb_init(t); acb_init(u); /* -(2nu+1)/4 * pi + z */ acb_mul_2exp_si(t, nu, 1); acb_add_ui(t, t, 1, prec); acb_mul_2exp_si(t, t, -2); acb_neg(t, t); acb_const_pi(u, prec); acb_mul(t, t, u, prec); acb_add(t, t, z, prec); acb_mul_onei(t, t); acb_exp_invexp(Ap, Am, t, prec); /* (2 pi z)^(-1/2) */ acb_const_pi(C, prec); acb_mul_2exp_si(C, C, 1); acb_mul(C, C, z, prec); acb_rsqrt(C, C, prec); acb_clear(t); acb_clear(u); return; } acb_hypgeom_bessel_j_asymp_prefactors_fallback(Ap, Am, C, nu, z, prec); }
void acb_hypgeom_bessel_i_asymp_prefactors(acb_t A, acb_t B, acb_t C, const acb_t nu, const acb_t z, long prec) { acb_t t, u; acb_init(t); acb_init(u); /* C = (2 pi z)^(-1/2) */ acb_const_pi(C, prec); acb_mul_2exp_si(C, C, 1); acb_mul(C, C, z, prec); acb_rsqrt(C, C, prec); if (arb_is_positive(acb_imagref(z)) || (arb_is_zero(acb_imagref(z)) && arb_is_negative(acb_realref(z)))) { acb_exp_pi_i(t, nu, prec); acb_mul_onei(t, t); } else if (arb_is_negative(acb_imagref(z)) || (arb_is_zero(acb_imagref(z)) && arb_is_positive(acb_realref(z)))) { acb_neg(t, nu); acb_exp_pi_i(t, t, prec); acb_mul_onei(t, t); acb_neg(t, t); } else { acb_exp_pi_i(t, nu, prec); acb_mul_onei(t, t); acb_neg(u, nu); acb_exp_pi_i(u, u, prec); acb_mul_onei(u, u); acb_neg(u, u); arb_union(acb_realref(t), acb_realref(t), acb_realref(u), prec); arb_union(acb_imagref(t), acb_imagref(t), acb_imagref(u), prec); } acb_exp_invexp(B, A, z, prec); acb_mul(A, A, t, prec); acb_clear(t); acb_clear(u); }
/* (+/- iz)^(-1/2-v) * z^v * exp(+/- iz) */ void acb_hypgeom_bessel_j_asymp_prefactors_fallback(acb_t Ap, acb_t Am, acb_t C, const acb_t nu, const acb_t z, long prec) { acb_t t, u, v; acb_init(t); acb_init(u); acb_init(v); /* v = -1/2-nu */ acb_one(v); acb_mul_2exp_si(v, v, -1); acb_add(v, v, nu, prec); acb_neg(v, v); acb_mul_onei(t, z); /* t = iz */ acb_neg(u, t); /* u = -iz */ /* Ap, Am = (+/- iz)^(-1/2-nu) */ acb_pow(Ap, t, v, prec); acb_pow(Am, u, v, prec); /* Ap, Am *= exp(+/- iz) */ acb_exp_invexp(u, v, t, prec); acb_mul(Ap, Ap, u, prec); acb_mul(Am, Am, v, prec); /* z^nu */ acb_pow(t, z, nu, prec); acb_mul(Ap, Ap, t, prec); acb_mul(Am, Am, t, prec); /* (2 pi)^(-1/2) */ acb_const_pi(C, prec); acb_mul_2exp_si(C, C, 1); acb_rsqrt(C, C, prec); acb_clear(t); acb_clear(u); acb_clear(v); }
void acb_hypgeom_bessel_k_asymp(acb_t res, const acb_t nu, const acb_t z, slong prec) { acb_t t, a, b, w; acb_init(t); acb_init(a); acb_init(b); acb_init(w); acb_one(a); acb_mul_2exp_si(a, a, -1); acb_add(a, a, nu, prec); acb_mul_2exp_si(b, nu, 1); acb_add_ui(b, b, 1, prec); acb_mul_2exp_si(w, z, 1); acb_hypgeom_u_asymp(t, a, b, w, -1, prec); acb_neg(w, z); acb_exp(w, w, prec); acb_mul(t, t, w, prec); acb_mul_2exp_si(w, z, 1); acb_rsqrt(w, w, prec); acb_mul(res, t, w, prec); arb_const_sqrt_pi(acb_realref(w), prec); acb_mul_arb(res, res, acb_realref(w), prec); acb_clear(t); acb_clear(a); acb_clear(b); acb_clear(w); }
/* IMAG: erf(z) = 2z/sqrt(pi) * 1F1(1/2, 3/2, -z^2) */ void acb_hypgeom_erf_1f1a(acb_t res, const acb_t z, slong prec) { acb_t a, t, w; acb_struct b[2]; acb_init(a); acb_init(b); acb_init(b + 1); acb_init(t); acb_init(w); acb_one(a); acb_mul_2exp_si(a, a, -1); acb_set_ui(b, 3); acb_mul_2exp_si(b, b, -1); acb_one(b + 1); acb_mul(w, z, z, prec); acb_neg(w, w); acb_hypgeom_pfq_direct(t, a, 1, b, 2, w, -1, prec); acb_const_pi(w, prec); acb_rsqrt(w, w, prec); acb_mul(t, t, w, prec); acb_mul(t, t, z, prec); acb_mul_2exp_si(res, t, 1); acb_clear(a); acb_clear(b); acb_clear(b + 1); acb_clear(t); acb_clear(w); }
void _acb_hypgeom_legendre_q_double(acb_t res, const acb_t n, const acb_t m, const acb_t z, slong prec) { acb_t t, u, v; acb_init(t); acb_init(u); acb_init(v); if (acb_is_int(m)) { acb_sub_ui(t, z, 1, prec); acb_mul_2exp_si(u, m, -1); acb_pow(v, t, u, prec); acb_neg(t, t); acb_neg(u, u); acb_pow(t, t, u, prec); acb_mul(t, t, v, prec); acb_hypgeom_legendre_q(u, n, m, z, 0, prec); acb_mul(t, t, u, prec); acb_mul_2exp_si(u, m, -1); if (!acb_is_int(u)) acb_neg(t, t); acb_sub_ui(u, z, 1, prec); acb_sqrt(u, u, prec); acb_sub_ui(v, z, 1, prec); acb_neg(v, v); acb_rsqrt(v, v, prec); acb_mul(u, u, v, prec); acb_hypgeom_legendre_p(v, n, m, z, 1, prec); acb_mul(u, u, v, prec); acb_const_pi(v, prec); acb_mul(u, u, v, prec); acb_mul_2exp_si(u, u, -1); acb_sub(res, t, u, prec); } else { acb_sub(t, n, m, prec); acb_add_ui(t, t, 1, prec); acb_mul_2exp_si(u, m, 1); acb_rising(t, t, u, prec); acb_neg(u, m); acb_hypgeom_legendre_p(u, n, u, z, 1, prec); acb_mul(t, t, u, prec); acb_hypgeom_legendre_p(u, n, m, z, 1, prec); acb_sub(t, u, t, prec); acb_exp_pi_i(u, m, prec); acb_mul(t, t, u, prec); acb_sin_pi(u, m, prec); acb_div(t, t, u, prec); acb_const_pi(u, prec); acb_mul(t, t, u, prec); acb_mul_2exp_si(t, t, -1); acb_set(res, t); } acb_clear(t); acb_clear(u); acb_clear(v); }