int main() { long iter; flint_rand_t state; printf("legendre_q...."); fflush(stdout); flint_randinit(state); for (iter = 0; iter < 1000; iter++) { acb_t n, m, z, res1, res2; long prec1, prec2, ebits; acb_init(n); acb_init(m); acb_init(z); acb_init(res1); acb_init(res2); prec1 = 2 + n_randint(state, 300); prec2 = 2 + n_randint(state, 300); ebits = 1 + n_randint(state, 10); if (n_randint(state, 2)) { acb_set_si(m, n_randint(state, 20) - 10); acb_set_si(n, n_randint(state, 20) - 10); } else { acb_randtest_param(n, state, 1 + n_randint(state, 400), ebits); acb_randtest_param(m, state, 1 + n_randint(state, 400), ebits); } acb_randtest_param(z, state, 1 + n_randint(state, 400), ebits); _acb_hypgeom_legendre_q_single(res1, n, m, z, prec1); _acb_hypgeom_legendre_q_double(res2, n, m, z, prec2); if (!acb_overlaps(res1, res2)) { printf("FAIL: consistency 1\n\n"); printf("iter = %ld, prec1 = %ld, prec2 = %ld\n\n", iter, prec1, prec2); printf("m = "); acb_printd(m, 30); printf("\n\n"); printf("n = "); acb_printd(n, 30); printf("\n\n"); printf("z = "); acb_printd(z, 30); printf("\n\n"); printf("res1 = "); acb_printd(res1, 30); printf("\n\n"); printf("res2 = "); acb_printd(res2, 30); printf("\n\n"); abort(); } acb_clear(n); acb_clear(m); acb_clear(z); acb_clear(res1); acb_clear(res2); } for (iter = 0; iter < 2000; iter++) { acb_t n, m, z, res1, res2, t, u; long prec1, prec2, ebits; int type; acb_init(n); acb_init(m); acb_init(z); acb_init(res1); acb_init(res2); acb_init(t); acb_init(u); prec1 = 2 + n_randint(state, 300); prec2 = 2 + n_randint(state, 300); ebits = 1 + n_randint(state, 10); if (n_randint(state, 2)) { acb_set_si(m, n_randint(state, 20) - 10); acb_set_si(n, n_randint(state, 20) - 10); } else { acb_randtest_param(n, state, 1 + n_randint(state, 400), ebits); acb_randtest_param(m, state, 1 + n_randint(state, 400), ebits); } acb_randtest_param(z, state, 1 + n_randint(state, 400), ebits); type = n_randint(state, 2); acb_hypgeom_legendre_q(res1, n, m, z, type, prec1); acb_neg(t, m); acb_hypgeom_legendre_p(res2, n, t, z, type, prec2); acb_add(u, m, n, prec2); acb_add_ui(u, u, 1, prec2); acb_gamma(u, u, prec2); acb_mul(res2, res2, u, prec2); acb_sub(u, n, m, prec2); acb_add_ui(u, u, 1, prec2); acb_rgamma(u, u, prec2); acb_mul(res2, res2, u, prec2); acb_hypgeom_legendre_p(t, n, m, z, type, prec2); if (type == 0) { acb_cos_pi(u, m, prec2); acb_mul(t, t, u, prec2); } acb_sub(res2, t, res2, prec2); if (type == 1) { acb_exp_pi_i(t, m, prec2); acb_mul(res2, res2, t, prec2); } acb_sin_pi(t, m, prec2); if (acb_contains_zero(t)) acb_indeterminate(res2); else acb_div(res2, res2, t, prec2); acb_const_pi(t, prec2); acb_mul(res2, res2, t, prec2); acb_mul_2exp_si(res2, res2, -1); if (!acb_overlaps(res1, res2)) { printf("FAIL: consistency 2\n\n"); printf("iter = %ld, prec1 = %ld, prec2 = %ld\n\n", iter, prec1, prec2); printf("type = %d\n\n", type); printf("m = "); acb_printd(m, 30); printf("\n\n"); printf("n = "); acb_printd(n, 30); printf("\n\n"); printf("z = "); acb_printd(z, 30); printf("\n\n"); printf("res1 = "); acb_printd(res1, 30); printf("\n\n"); printf("res2 = "); acb_printd(res2, 30); printf("\n\n"); abort(); } acb_clear(n); acb_clear(m); acb_clear(z); acb_clear(res1); acb_clear(res2); acb_clear(t); acb_clear(u); } flint_randclear(state); flint_cleanup(); printf("PASS\n"); return EXIT_SUCCESS; }
void acb_hypgeom_legendre_q(acb_t res, const acb_t n, const acb_t m, const acb_t z, int type, slong prec) { if (type == 0) { /* http://functions.wolfram.com/07.11.26.0033.01 */ /* todo: simplify the gamma quotients and the sqrt pi factor... */ acb_t a, b, c, z2, mn, nm, t, u; acb_init(a); acb_init(b); acb_init(c); acb_init(z2); acb_init(mn); acb_init(nm); acb_init(t); acb_init(u); acb_add(mn, m, n, prec); /* mn = m + n */ acb_sub(nm, n, m, prec); /* nm = n - m */ acb_mul(z2, z, z, prec); /* z2 = z^2 */ /* t = 2F1((1-m-n)/2, (n-m)/2+1, 3/2, z^2) */ acb_sub_ui(a, mn, 1, prec); acb_neg(a, a); acb_mul_2exp_si(a, a, -1); acb_mul_2exp_si(b, nm, -1); acb_add_ui(b, b, 1, prec); acb_set_ui(c, 3); acb_mul_2exp_si(c, c, -1); acb_hypgeom_2f1(t, a, b, c, z2, 0, prec); /* u = 2F1(-(m+n)/2, (n-m+1)/2, 1/2, z^2) */ acb_neg(a, mn); acb_mul_2exp_si(a, a, -1); acb_add_ui(b, nm, 1, prec); acb_mul_2exp_si(b, b, -1); acb_one(c); acb_mul_2exp_si(c, c, -1); acb_hypgeom_2f1(u, a, b, c, z2, 0, prec); /* a = cospi((m+n)/2) gamma((m+n)/2+1) rgamma((n-m+1)/2) z */ /* b = sinpi((m+n)/2) gamma((m+n+1)/2) rgamma((n-m)/2+1) / 2 */ acb_mul_2exp_si(a, mn, -1); acb_sin_cos_pi(b, a, a, prec); acb_mul_2exp_si(c, mn, -1); acb_add_ui(c, c, 1, prec); acb_gamma(c, c, prec); acb_mul(a, a, c, prec); acb_add_ui(c, nm, 1, prec); acb_mul_2exp_si(c, c, -1); acb_rgamma(c, c, prec); acb_mul(a, a, c, prec); acb_mul(a, a, z, prec); acb_add_ui(c, mn, 1, prec); acb_mul_2exp_si(c, c, -1); acb_gamma(c, c, prec); acb_mul(b, b, c, prec); acb_mul_2exp_si(c, nm, -1); acb_add_ui(c, c, 1, prec); acb_rgamma(c, c, prec); acb_mul(b, b, c, prec); acb_mul_2exp_si(b, b, -1); /* at - bu */ acb_mul(t, t, a, prec); acb_mul(u, u, b, prec); acb_sub(t, t, u, prec); /* prefactor sqrt(pi) 2^m (1-z^2)^(-m/2) */ if (!acb_is_zero(m)) { acb_sub_ui(u, z2, 1, prec); acb_neg(u, u); acb_neg(c, m); acb_mul_2exp_si(c, c, -1); acb_pow(u, u, c, prec); acb_set_ui(c, 2); acb_pow(c, c, m, prec); acb_mul(u, u, c, prec); acb_mul(t, t, u, prec); } arb_const_sqrt_pi(acb_realref(u), prec); acb_mul_arb(t, t, acb_realref(u), prec); acb_set(res, t); acb_clear(a); acb_clear(b); acb_clear(c); acb_clear(z2); acb_clear(mn); acb_clear(nm); acb_clear(t); acb_clear(u); } else if (type == 1) { if ((arf_cmpabs_2exp_si(arb_midref(acb_realref(z)), -2) < 0 && arf_cmpabs_2exp_si(arb_midref(acb_imagref(z)), -2) < 0) || !_acb_hypgeom_legendre_q_single_valid(z)) { _acb_hypgeom_legendre_q_double(res, n, m, z, prec); } else { _acb_hypgeom_legendre_q_single(res, n, m, z, prec); } } else { flint_printf("unsupported 'type' %d for legendre q\n", type); abort(); } }