void _acb_poly_lgamma_series(acb_ptr res, acb_srcptr h, slong hlen, slong len, slong prec) { int reflect; slong i, r, n, wp; acb_t zr; acb_ptr t, u; hlen = FLINT_MIN(hlen, len); if (hlen == 1) { acb_lgamma(res, h, prec); if (acb_is_finite(res)) _acb_vec_zero(res + 1, len - 1); else _acb_vec_indeterminate(res + 1, len - 1); return; } if (len == 2) { acb_t v; acb_init(v); acb_set(v, h + 1); acb_digamma(res + 1, h, prec); acb_lgamma(res, h, prec); acb_mul(res + 1, res + 1, v, prec); acb_clear(v); return; } /* use real code for real input and output */ if (_acb_vec_is_real(h, hlen) && arb_is_positive(acb_realref(h))) { arb_ptr tmp = _arb_vec_init(len); for (i = 0; i < hlen; i++) arb_set(tmp + i, acb_realref(h + i)); _arb_poly_lgamma_series(tmp, tmp, hlen, len, prec); for (i = 0; i < len; i++) acb_set_arb(res + i, tmp + i); _arb_vec_clear(tmp, len); return; } wp = prec + FLINT_BIT_COUNT(prec); t = _acb_vec_init(len); u = _acb_vec_init(len); acb_init(zr); /* use Stirling series */ acb_gamma_stirling_choose_param(&reflect, &r, &n, h, 1, 0, wp); if (reflect) { /* log gamma(h+x) = log rf(1-(h+x), r) - log gamma(1-(h+x)+r) - log sin(pi (h+x)) + log(pi) */ if (r != 0) /* otherwise t = 0 */ { acb_sub_ui(u, h, 1, wp); acb_neg(u, u); _log_rising_ui_series(t, u, r, len, wp); for (i = 1; i < len; i += 2) acb_neg(t + i, t + i); } acb_sub_ui(u, h, 1, wp); acb_neg(u, u); acb_add_ui(zr, u, r, wp); _acb_poly_gamma_stirling_eval(u, zr, n, len, wp); for (i = 1; i < len; i += 2) acb_neg(u + i, u + i); _acb_vec_sub(t, t, u, len, wp); /* log(sin) is unstable with large imaginary parts; cot_pi is implemented in a numerically stable way */ acb_set(u, h); acb_one(u + 1); _acb_poly_cot_pi_series(u, u, 2, len - 1, wp); _acb_poly_integral(u, u, len, wp); acb_const_pi(u, wp); _acb_vec_scalar_mul(u + 1, u + 1, len - 1, u, wp); acb_log_sin_pi(u, h, wp); _acb_vec_sub(u, t, u, len, wp); acb_const_pi(t, wp); /* todo: constant for log pi */ acb_log(t, t, wp); acb_add(u, u, t, wp); } else { /* log gamma(x) = log gamma(x+r) - log rf(x,r) */ acb_add_ui(zr, h, r, wp); _acb_poly_gamma_stirling_eval(u, zr, n, len, wp); if (r != 0) { _log_rising_ui_series(t, h, r, len, wp); _acb_vec_sub(u, u, t, len, wp); } } /* compose with nonconstant part */ acb_zero(t); _acb_vec_set(t + 1, h + 1, hlen - 1); _acb_poly_compose_series(res, u, len, t, hlen, len, prec); acb_clear(zr); _acb_vec_clear(t, len); _acb_vec_clear(u, len); }
int main() { long iter; flint_rand_t state; printf("lgamma...."); fflush(stdout); flint_randinit(state); for (iter = 0; iter < 10000; iter++) { acb_t a, b, c; long prec1, prec2; prec1 = 2 + n_randint(state, 500); prec2 = prec1 + 30; acb_init(a); acb_init(b); acb_init(c); arb_randtest_precise(acb_realref(a), state, 1 + n_randint(state, 1000), 1 + n_randint(state, 10)); arb_randtest_precise(acb_imagref(a), state, 1 + n_randint(state, 1000), 1 + n_randint(state, 10)); acb_lgamma(b, a, prec1); if (n_randint(state, 4) == 0) { acb_randtest(c, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 10)); acb_add(a, a, c, prec1); acb_sub(a, a, c, prec1); } acb_lgamma(c, a, prec2); if (!acb_overlaps(b, c)) { printf("FAIL: overlap\n\n"); printf("a = "); acb_print(a); printf("\n\n"); printf("b = "); acb_print(b); printf("\n\n"); printf("c = "); acb_print(c); printf("\n\n"); abort(); } /* check lgamma(z+1) = lgamma(z) + log(z) */ acb_log(c, a, prec1); acb_add(b, b, c, prec1); acb_add_ui(c, a, 1, prec1); acb_lgamma(c, c, prec1); if (!acb_overlaps(b, c)) { printf("FAIL: functional equation\n\n"); printf("a = "); acb_print(a); printf("\n\n"); printf("b = "); acb_print(b); printf("\n\n"); printf("c = "); acb_print(c); printf("\n\n"); abort(); } acb_clear(a); acb_clear(b); acb_clear(c); } flint_randclear(state); flint_cleanup(); printf("PASS\n"); return EXIT_SUCCESS; }