void acb_hypgeom_laguerre_l(acb_t res, const acb_t n, const acb_t m, const acb_t z, slong prec) { acb_t t, u, v; if (use_recurrence(n, m, prec)) { acb_hypgeom_laguerre_l_ui_recurrence(res, arf_get_si(arb_midref(acb_realref(n)), ARF_RND_DOWN), m, z, prec); return; } /* todo: should be a test of whether n contains any negative integer */ if (acb_contains_int(n) && !arb_is_nonnegative(acb_realref(n))) { acb_indeterminate(res); return; } acb_init(t); acb_init(u); acb_init(v); acb_neg(t, n); acb_add_ui(u, m, 1, prec); acb_hypgeom_m(t, t, u, z, 1, prec); acb_add_ui(u, n, 1, prec); acb_rising(u, u, m, prec); acb_mul(res, t, u, prec); acb_clear(t); acb_clear(u); acb_clear(v); }
/* todo: remove radii */ void acb_lambertw_halley_step(acb_t res, acb_t ew, const acb_t z, const acb_t w, slong prec) { acb_t t, u, v; acb_init(t); acb_init(u); acb_init(v); acb_exp(ew, w, prec); acb_add_ui(u, w, 2, prec); acb_add_ui(v, w, 1, prec); acb_mul_2exp_si(v, v, 1); acb_div(v, u, v, prec); acb_mul(t, ew, w, prec); acb_sub(u, t, z, prec); acb_mul(v, v, u, prec); acb_neg(v, v); acb_add(v, v, t, prec); acb_add(v, v, ew, prec); acb_div(t, u, v, prec); acb_sub(t, w, t, prec); acb_swap(res, t); acb_clear(t); acb_clear(u); acb_clear(v); }
void acb_log1p(acb_t r, const acb_t z, slong prec) { slong magz, magx, magy; if (acb_is_zero(z)) { acb_zero(r); return; } magx = arf_abs_bound_lt_2exp_si(arb_midref(acb_realref(z))); magy = arf_abs_bound_lt_2exp_si(arb_midref(acb_imagref(z))); magz = FLINT_MAX(magx, magy); if (magz < -prec) { acb_log1p_tiny(r, z, prec); } else { if (magz < 0) acb_add_ui(r, z, 1, prec + (-magz) + 4); else acb_add_ui(r, z, 1, prec + 4); acb_log(r, r, prec); } }
void acb_polygamma(acb_t res, const acb_t s, const acb_t z, long prec) { if (acb_is_zero(s)) { acb_digamma(res, z, prec); } else if (acb_is_int(s) && arb_is_positive(acb_realref(s))) { acb_t t, u; acb_init(t); acb_init(u); acb_add_ui(t, s, 1, prec); acb_gamma(u, t, prec); acb_hurwitz_zeta(t, t, z, prec); if (arf_is_int_2exp_si(arb_midref(acb_realref(s)), 1)) acb_neg(t, t); acb_mul(res, t, u, prec); acb_clear(t); acb_clear(u); } else { acb_t t, u; acb_struct v[2]; acb_init(t); acb_init(u); acb_init(v); acb_init(v + 1); /* u = psi(-s) + gamma */ acb_neg(t, s); acb_digamma(u, t, prec); arb_const_euler(acb_realref(v), prec); arb_add(acb_realref(u), acb_realref(u), acb_realref(v), prec); acb_add_ui(t, s, 1, prec); _acb_poly_zeta_cpx_series(v, t, z, 0, 2, prec); acb_addmul(v + 1, v, u, prec); acb_neg(t, s); acb_rgamma(u, t, prec); acb_mul(res, v + 1, u, prec); acb_clear(v); acb_clear(v + 1); acb_clear(t); acb_clear(u); } }
void acb_tan_pi(acb_t r, const acb_t z, slong prec) { if (arb_is_zero(acb_imagref(z))) { arb_tan_pi(acb_realref(r), acb_realref(z), prec); arb_zero(acb_imagref(r)); } else if (arb_is_zero(acb_realref(z))) { arb_t t; arb_init(t); arb_const_pi(t, prec + 4); arb_mul(t, acb_imagref(z), t, prec + 4); arb_tanh(acb_imagref(r), t, prec); arb_zero(acb_realref(r)); arb_clear(t); } else { acb_t t; acb_init(t); if (arf_cmpabs_2exp_si(arb_midref(acb_imagref(z)), 0) < 0) { acb_sin_cos_pi(r, t, z, prec + 4); acb_div(r, r, t, prec); } else { acb_mul_2exp_si(t, z, 1); if (arf_sgn(arb_midref(acb_imagref(z))) > 0) { acb_exp_pi_i(t, t, prec + 4); acb_add_ui(r, t, 1, prec + 4); acb_div(r, t, r, prec + 4); acb_mul_2exp_si(r, r, 1); acb_sub_ui(r, r, 1, prec); acb_div_onei(r, r); } else { acb_neg(t, t); acb_exp_pi_i(t, t, prec + 4); acb_add_ui(r, t, 1, prec + 4); acb_div(r, t, r, prec + 4); acb_mul_2exp_si(r, r, 1); acb_sub_ui(r, r, 1, prec); acb_mul_onei(r, r); } } acb_clear(t); } }
void acb_hypgeom_laguerre_l_ui_recurrence(acb_t res, ulong n, const acb_t m, const acb_t z, slong prec) { acb_t t, u, v; ulong k; if (n == 0) { acb_one(res); return; } if (n == 1) { acb_sub(res, m, z, prec); acb_add_ui(res, res, 1, prec); return; } acb_init(t); acb_init(u); acb_init(v); acb_one(t); acb_sub(u, m, z, prec); acb_add_ui(u, u, 1, prec); for (k = 2; k <= n; k++) { acb_add_ui(v, m, k - 1, prec); acb_mul(t, t, v, prec); acb_add_ui(v, m, 2 * k - 1, prec); acb_sub(v, v, z, prec); acb_mul(v, v, u, prec); acb_sub(t, v, t, prec); acb_div_ui(t, t, k, prec); acb_swap(t, u); } acb_set(res, u); acb_clear(t); acb_clear(u); acb_clear(v); }
/* Differential equation for F(a,b,c,y+z): (y+z)(y-1+z) F''(z) + ((y+z)(a+b+1) - c) F'(z) + a b F(z) = 0 Coefficients in the Taylor series are bounded by A * binomial(N+k, k) * nu^k using the Cauchy-Kovalevskaya majorant method. See J. van der Hoeven, "Fast evaluation of holonomic functions near and in regular singularities" */ static void bound(mag_t A, mag_t nu, mag_t N, const acb_t a, const acb_t b, const acb_t c, const acb_t y, const acb_t f0, const acb_t f1) { mag_t M0, M1, t, u; acb_t d; acb_init(d); mag_init(M0); mag_init(M1); mag_init(t); mag_init(u); /* nu = max(1/|y-1|, 1/|y|) = 1/min(|y-1|, |y|) */ acb_get_mag_lower(t, y); acb_sub_ui(d, y, 1, MAG_BITS); acb_get_mag_lower(u, d); mag_min(t, t, u); mag_one(u); mag_div(nu, u, t); /* M0 = 2 nu |ab| */ acb_get_mag(t, a); acb_get_mag(u, b); mag_mul(M0, t, u); mag_mul(M0, M0, nu); mag_mul_2exp_si(M0, M0, 1); /* M1 = 2 nu |(a+b+1)y-c| + 2|a+b+1| */ acb_add(d, a, b, MAG_BITS); acb_add_ui(d, d, 1, MAG_BITS); acb_get_mag(t, d); acb_mul(d, d, y, MAG_BITS); acb_sub(d, d, c, MAG_BITS); acb_get_mag(u, d); mag_mul(u, u, nu); mag_add(M1, t, u); mag_mul_2exp_si(M1, M1, 1); /* N = max(sqrt(2 M0), 2 M1) / nu */ mag_mul_2exp_si(M0, M0, 1); mag_sqrt(M0, M0); mag_mul_2exp_si(M1, M1, 1); mag_max(N, M0, M1); mag_div(N, N, nu); /* A = max(|f0|, |f1| / (nu (N+1)) */ acb_get_mag(t, f0); acb_get_mag(u, f1); mag_div(u, u, nu); mag_div(u, u, N); /* upper bound for dividing by N+1 */ mag_max(A, t, u); acb_clear(d); mag_clear(M0); mag_clear(M1); mag_clear(t); mag_clear(u); }
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_lgamma(acb_t y, const acb_t x, long prec) { int reflect; long r, n, wp; acb_t t, u; wp = prec + FLINT_BIT_COUNT(prec); acb_gamma_stirling_choose_param(&reflect, &r, &n, x, 0, 0, wp); /* log(gamma(x)) = log(gamma(x+r)) - log(rf(x,r)) */ acb_init(t); acb_init(u); acb_add_ui(t, x, r, wp); acb_gamma_stirling_eval(u, t, n, 0, wp); acb_rising_ui_rec(t, x, r, prec); acb_log(t, t, prec); _acb_log_rising_correct_branch(t, t, x, r, wp); acb_sub(y, u, t, prec); acb_clear(t); acb_clear(u); }
void acb_poly_add_si(acb_poly_t res, const acb_poly_t x, long y, long prec) { long len = x->length; if (len == 0) { acb_poly_set_si(res, y); } else { acb_poly_fit_length(res, len); if (y >= 0) acb_add_ui(res->coeffs, x->coeffs, y, prec); else acb_sub_ui(res->coeffs, x->coeffs, -y, prec); if (res != x) _acb_vec_set(res->coeffs + 1, x->coeffs + 1, len - 1); _acb_poly_set_length(res, len); _acb_poly_normalise(res); } }
void acb_hypgeom_u(acb_t res, const acb_t a, const acb_t b, const acb_t z, long prec) { acb_t t; acb_init(t); acb_sub(t, a, b, prec); acb_add_ui(t, t, 1, prec); if ((acb_is_int(a) && arf_sgn(arb_midref(acb_realref(a))) <= 0) || (acb_is_int(t) && arf_sgn(arb_midref(acb_realref(t))) <= 0) || acb_hypgeom_u_use_asymp(z, prec)) { acb_neg(t, a); acb_pow(t, z, t, prec); acb_hypgeom_u_asymp(res, a, b, z, -1, prec); acb_mul(res, res, t, prec); } else { acb_hypgeom_u_1f1(res, a, b, z, prec); } acb_clear(t); }
static void _acb_gamma(acb_t y, const acb_t x, long prec, int inverse) { int reflect; long r, n, wp; acb_t t, u, v; wp = prec + FLINT_BIT_COUNT(prec); acb_gamma_stirling_choose_param(&reflect, &r, &n, x, 1, 0, wp); acb_init(t); acb_init(u); acb_init(v); if (reflect) { /* gamma(x) = (rf(1-x, r) * pi) / (gamma(1-x+r) sin(pi x)) */ acb_sub_ui(t, x, 1, wp); acb_neg(t, t); acb_rising_ui_rec(u, t, r, wp); arb_const_pi(acb_realref(v), wp); acb_mul_arb(u, u, acb_realref(v), wp); acb_add_ui(t, t, r, wp); acb_gamma_stirling_eval(v, t, n, 0, wp); acb_exp(v, v, wp); acb_sin_pi(t, x, wp); acb_mul(v, v, t, wp); } else { /* gamma(x) = gamma(x+r) / rf(x,r) */ acb_add_ui(t, x, r, wp); acb_gamma_stirling_eval(u, t, n, 0, wp); acb_exp(u, u, prec); acb_rising_ui_rec(v, x, r, wp); } if (inverse) acb_div(y, v, u, prec); else acb_div(y, u, v, prec); acb_clear(t); acb_clear(u); acb_clear(v); }
void acb_hypgeom_jacobi_p_ui_direct(acb_t res, ulong n, const acb_t a, const acb_t b, const acb_t z, slong prec) { acb_ptr terms; acb_t t, u, v; slong k; terms = _acb_vec_init(n + 1); acb_init(t); acb_init(u); acb_init(v); acb_one(terms); acb_add_ui(u, z, 1, prec); for (k = 1; k <= n; k++) { acb_add_ui(t, a, n + 1 - k, prec); acb_mul(t, t, u, prec); acb_div_ui(t, t, 2 * k, prec); acb_mul(terms + k, terms + k - 1, t, prec); } acb_sub_ui(u, z, 1, prec); acb_one(v); for (k = 1; k <= n; k++) { acb_add_ui(t, b, n + 1 - k, prec); acb_mul(t, t, u, prec); acb_div_ui(t, t, 2 * k, prec); acb_mul(v, v, t, prec); acb_mul(terms + n - k, terms + n - k, v, prec); } acb_set(res, terms); for (k = 1; k <= n; k++) acb_add(res, res, terms + k, prec); _acb_vec_clear(terms, n + 1); acb_clear(t); acb_clear(u); acb_clear(v); }
static void bsplit(acb_t p, acb_t q, const acb_t x, ulong a, ulong b, slong prec) { if (b - a < 8) { ulong k; acb_t t; acb_one(p); acb_add_ui(q, x, a, prec); acb_init(t); for (k = a + 1; k < b; k++) { acb_add_ui(t, x, k, prec); acb_mul(p, p, t, prec); acb_add(p, p, q, prec); acb_mul(q, q, t, prec); } acb_clear(t); } else { acb_t r, s; ulong m; acb_init(r); acb_init(s); m = a + (b - a) / 2; bsplit(p, q, x, a, m, prec); bsplit(r, s, x, m, b, prec); acb_mul(p, p, s, prec); acb_mul(r, r, q, prec); acb_add(p, p, r, prec); acb_mul(q, q, s, prec); acb_clear(r); acb_clear(s); } }
void acb_digamma(acb_t y, const acb_t x, long prec) { int reflect; long r, n, wp; acb_t t, u, v; wp = prec + FLINT_BIT_COUNT(prec); acb_gamma_stirling_choose_param(&reflect, &r, &n, x, 1, 1, wp); acb_init(t); acb_init(u); acb_init(v); /* psi(x) = psi((1-x)+r) - h(1-x,r) - pi*cot(pi*x) */ if (reflect) { acb_sub_ui(t, x, 1, wp); acb_neg(t, t); acb_cot_pi(v, x, wp); arb_const_pi(acb_realref(u), wp); acb_mul_arb(v, v, acb_realref(u), wp); acb_rising2_ui(y, u, t, r, wp); acb_div(u, u, y, wp); acb_add(v, v, u, wp); acb_add_ui(t, t, r, wp); acb_gamma_stirling_eval(u, t, n, 1, wp); acb_sub(y, u, v, wp); } else { acb_add_ui(t, x, r, wp); acb_gamma_stirling_eval(u, t, n, 1, wp); acb_rising2_ui(y, t, x, r, wp); acb_div(t, t, y, wp); acb_sub(y, u, t, prec); } acb_clear(t); acb_clear(u); acb_clear(v); }
void acb_hypgeom_jacobi_p(acb_t res, const acb_t n, const acb_t a, const acb_t b, const acb_t z, slong prec) { acb_t t, u, v, w; if (use_recurrence(n, a, b, prec)) { acb_hypgeom_jacobi_p_ui_direct(res, arf_get_si(arb_midref(acb_realref(n)), ARF_RND_DOWN), a, b, z, prec); return; } acb_init(t); acb_init(u); acb_init(v); acb_init(w); acb_neg(t, n); acb_add_ui(v, a, 1, prec); acb_add(u, n, v, prec); acb_add(u, u, b, prec); acb_sub_ui(w, z, 1, prec); acb_mul_2exp_si(w, w, -1); acb_neg(w, w); acb_hypgeom_2f1(w, t, u, v, w, 0, prec); acb_rising(t, v, n, prec); acb_mul(w, w, t, prec); acb_add_ui(t, n, 1, prec); acb_rgamma(t, t, prec); acb_mul(w, w, t, prec); acb_set(res, w); acb_clear(t); acb_clear(u); acb_clear(v); acb_clear(w); }
/* f(z) = 1/(1+z^2) */ int f_atanderiv(acb_ptr res, const acb_t z, void * param, slong order, slong prec) { if (order > 1) flint_abort(); /* Would be needed for Taylor method. */ acb_mul(res, z, z, prec); acb_add_ui(res, res, 1, prec); acb_inv(res, res, prec); return 0; }
void acb_lambertw(acb_t res, const acb_t z, const fmpz_t k, int flags, slong prec) { acb_t ez1; if (!acb_is_finite(z)) { acb_indeterminate(res); return; } if (flags == ACB_LAMBERTW_LEFT) { acb_lambertw_left(res, z, k, prec); return; } if (flags == ACB_LAMBERTW_MIDDLE) { acb_lambertw_middle(res, z, prec); return; } if (acb_contains_zero(z) && !fmpz_is_zero(k)) { acb_indeterminate(res); return; } acb_init(ez1); /* precompute z*e + 1 */ arb_const_e(acb_realref(ez1), prec); acb_mul(ez1, ez1, z, prec); acb_add_ui(ez1, ez1, 1, prec); /* Compute standard branches */ /* use real code when possible */ if (acb_is_real(z) && arb_is_positive(acb_realref(ez1)) && (fmpz_is_zero(k) || (fmpz_equal_si(k, -1) && arb_is_negative(acb_realref(z))))) { arb_lambertw(acb_realref(res), acb_realref(z), !fmpz_is_zero(k), prec); arb_zero(acb_imagref(res)); } else { _acb_lambertw(res, z, ez1, k, flags, prec); } acb_clear(ez1); }
void acb_hypgeom_bessel_j_0f1(acb_t res, const acb_t nu, const acb_t z, long prec) { acb_struct b[2]; acb_t w, c, t; if (acb_is_int(nu) && arb_is_negative(acb_realref(nu))) { acb_init(t); acb_neg(t, nu); acb_hypgeom_bessel_j_0f1(res, t, z, prec); acb_mul_2exp_si(t, t, -1); if (!acb_is_int(t)) acb_neg(res, res); acb_clear(t); return; } acb_init(b + 0); acb_init(b + 1); acb_init(w); acb_init(c); acb_init(t); acb_add_ui(b + 0, nu, 1, prec); acb_one(b + 1); /* (z/2)^nu / gamma(nu+1) */ acb_mul_2exp_si(c, z, -1); acb_pow(c, c, nu, prec); acb_rgamma(t, b + 0, prec); acb_mul(c, t, c, prec); /* -z^2/4 */ acb_mul(w, z, z, prec); acb_mul_2exp_si(w, w, -2); acb_neg(w, w); acb_hypgeom_pfq_direct(t, NULL, 0, b, 2, w, -1, prec); acb_mul(res, t, c, prec); acb_clear(b + 0); acb_clear(b + 1); acb_clear(w); acb_clear(c); acb_clear(t); }
void acb_hypgeom_beta_lower(acb_t res, const acb_t a, const acb_t b, const acb_t z, int regularized, slong prec) { acb_t t, u; if (acb_is_zero(z) && arb_is_positive(acb_realref(a))) { acb_zero(res); return; } if (acb_is_one(z) && arb_is_positive(acb_realref(b))) { if (regularized) acb_one(res); else acb_beta(res, a, b, prec); return; } acb_init(t); acb_init(u); acb_sub_ui(t, b, 1, prec); acb_neg(t, t); acb_add_ui(u, a, 1, prec); if (regularized) { acb_hypgeom_2f1(t, a, t, u, z, 1, prec); acb_add(u, a, b, prec); acb_gamma(u, u, prec); acb_mul(t, t, u, prec); acb_rgamma(u, b, prec); acb_mul(t, t, u, prec); } else { acb_hypgeom_2f1(t, a, t, u, z, 0, prec); acb_div(t, t, a, prec); } acb_pow(u, z, a, prec); acb_mul(t, t, u, prec); acb_set(res, t); acb_clear(t); acb_clear(u); }
/* f(z) = -log(z) / (1 + z) */ int f_log_div1p(acb_ptr res, const acb_t z, void * param, slong order, slong prec) { acb_t t; if (order > 1) flint_abort(); /* Would be needed for Taylor method. */ acb_init(t); acb_add_ui(t, z, 1, prec); acb_log(res, z, prec); acb_div(res, res, t, prec); acb_neg(res, res); acb_clear(t); return 0; }
void acb_rising_ui(acb_t y, const acb_t x, ulong n, long prec) { if (n < FLINT_MAX(prec, 100)) { acb_rising_ui_rec(y, x, n, prec); } else { acb_t t; acb_init(t); acb_add_ui(t, x, n, prec); acb_gamma(t, t, prec); acb_rgamma(y, x, prec); acb_mul(y, y, t, prec); acb_clear(t); } }
/* f(z) = sin(z) + exp(-200-z^2) */ int f_sin_plus_small(acb_ptr res, const acb_t z, void * param, slong order, slong prec) { acb_t t; if (order > 1) flint_abort(); /* Would be needed for Taylor method. */ acb_init(t); acb_mul(t, z, z, prec); acb_add_ui(t, t, 200, prec); acb_neg(t, t); acb_exp(t, t, prec); acb_sin(res, z, prec); acb_add(res, res, t, prec); acb_clear(t); return 0; }
void acb_lambertw_cleared_cut(acb_t res, const acb_t z, const fmpz_t k, int flags, slong prec) { acb_t ez1; acb_init(ez1); /* compute e*z + 1 */ arb_const_e(acb_realref(ez1), prec); acb_mul(ez1, ez1, z, prec); acb_add_ui(ez1, ez1, 1, prec); if (acb_is_exact(z)) { acb_lambertw_main(res, z, ez1, k, flags, prec); } else { acb_t zz; mag_t err, rad; mag_init(err); mag_init(rad); acb_init(zz); acb_lambertw_bound_deriv(err, z, ez1, k); mag_hypot(rad, arb_radref(acb_realref(z)), arb_radref(acb_imagref(z))); mag_mul(err, err, rad); acb_set(zz, z); mag_zero(arb_radref(acb_realref(zz))); mag_zero(arb_radref(acb_imagref(zz))); /* todo: recompute ez1? */ acb_lambertw_main(res, zz, ez1, k, flags, prec); acb_add_error_mag(res, err); mag_clear(err); mag_clear(rad); acb_clear(zz); } acb_clear(ez1); }
/* f(z) = z sin(z) / (1 + cos(z)^2) */ int f_sin_cos_frac(acb_ptr res, const acb_t z, void * param, slong order, slong prec) { acb_t s, c; if (order > 1) flint_abort(); /* Would be needed for Taylor method. */ acb_init(s); acb_init(c); acb_sin_cos(s, c, z, prec); acb_mul(c, c, c, prec); acb_add_ui(c, c, 1, prec); acb_mul(s, s, z, prec); acb_div(res, s, c, prec); acb_clear(s); acb_clear(c); return 0; }
void _acb_poly_atan_series(acb_ptr g, acb_srcptr h, slong hlen, slong n, slong prec) { acb_t c; acb_init(c); acb_atan(c, h, prec); hlen = FLINT_MIN(hlen, n); if (hlen == 1) { _acb_vec_zero(g + 1, n - 1); } else { acb_ptr t, u; slong ulen; t = _acb_vec_init(n); u = _acb_vec_init(n); /* atan(h(x)) = integral(h'(x)/(1+h(x)^2)) */ ulen = FLINT_MIN(n, 2 * hlen - 1); _acb_poly_mullow(u, h, hlen, h, hlen, ulen, prec); acb_add_ui(u, u, 1, prec); _acb_poly_derivative(t, h, hlen, prec); _acb_poly_div_series(g, t, hlen - 1, u, ulen, n, prec); _acb_poly_integral(g, g, n, prec); _acb_vec_clear(t, n); _acb_vec_clear(u, n); } acb_swap(g, c); acb_clear(c); }
void acb_acosh(acb_t res, const acb_t z, slong prec) { if (acb_is_one(z)) { acb_zero(res); } else { acb_t t, u; acb_init(t); acb_init(u); acb_add_ui(t, z, 1, prec); acb_sub_ui(u, z, 1, prec); acb_sqrt(t, t, prec); acb_sqrt(u, u, prec); acb_mul(t, t, u, prec); acb_add(t, t, z, prec); if (!arb_is_zero(acb_imagref(z))) { acb_log(res, t, prec); } else { /* pure imaginary on (-1,1) */ arb_abs(acb_realref(u), acb_realref(z)); arb_one(acb_imagref(u)); acb_log(res, t, prec); if (arb_lt(acb_realref(u), acb_imagref(u))) arb_zero(acb_realref(res)); } acb_clear(t); acb_clear(u); } }
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); }
int main() { long iter; flint_rand_t state; printf("u...."); fflush(stdout); flint_randinit(state); for (iter = 0; iter < 2000; iter++) { acb_t a0, a1, a2, b, z, w0, w1, w2, t, u; long prec0, prec1, prec2; acb_init(a0); acb_init(a1); acb_init(a2); acb_init(b); acb_init(z); acb_init(w0); acb_init(w1); acb_init(w2); acb_init(t); acb_init(u); prec0 = 2 + n_randint(state, 700); prec1 = 2 + n_randint(state, 700); prec2 = 2 + n_randint(state, 700); acb_randtest_maybe_half_int(a0, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100)); acb_randtest_maybe_half_int(b, 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_randtest(w2, state, 1 + n_randint(state, 1000), 1 + n_randint(state, 100)); acb_add_ui(a1, a0, 1, prec0); acb_add_ui(a2, a0, 2, prec0); switch (n_randint(state, 3)) { case 0: acb_hypgeom_u_asymp_proper(w0, a0, b, z, prec0); break; case 1: acb_hypgeom_u_1f1(w0, a0, b, z, prec0); break; default: acb_hypgeom_u(w0, a0, b, z, prec0); } switch (n_randint(state, 3)) { case 0: acb_hypgeom_u_asymp_proper(w1, a0, b, z, prec1); break; case 1: acb_hypgeom_u_1f1(w1, a0, b, z, prec1); break; default: acb_hypgeom_u(w1, a0, b, z, prec1); } if (!acb_overlaps(w0, w1)) { printf("FAIL: consistency\n\n"); printf("a = "); acb_printd(a0, 30); printf("\n\n"); printf("b = "); acb_printd(b, 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, 3)) { case 0: acb_hypgeom_u_asymp_proper(w1, a1, b, z, prec1); break; case 1: acb_hypgeom_u_1f1(w1, a1, b, z, prec1); break; default: acb_hypgeom_u(w1, a1, b, z, prec1); } switch (n_randint(state, 3)) { case 0: acb_hypgeom_u_asymp_proper(w2, a2, b, z, prec2); break; case 1: acb_hypgeom_u_1f1(w2, a2, b, z, prec2); break; default: acb_hypgeom_u(w2, a2, b, z, prec2); } acb_set(t, w0); acb_mul_2exp_si(u, a0, 1); acb_sub(u, u, b, prec0); acb_add(u, u, z, prec0); acb_add_ui(u, u, 2, prec0); acb_submul(t, w1, u, prec0); acb_sub(u, a2, b, prec0); acb_mul(u, u, a1, prec0); acb_addmul(t, w2, u, prec0); if (!acb_contains_zero(t)) { printf("FAIL: contiguous relation\n\n"); printf("a = "); acb_printd(a0, 30); printf("\n\n"); printf("b = "); acb_printd(b, 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("w2 = "); acb_printd(w2, 30); printf("\n\n"); printf("t = "); acb_printd(t, 30); printf("\n\n"); abort(); } acb_clear(a0); acb_clear(a1); acb_clear(a2); acb_clear(b); acb_clear(z); acb_clear(w0); acb_clear(w1); acb_clear(w2); acb_clear(t); acb_clear(u); } flint_randclear(state); flint_cleanup(); printf("PASS\n"); return EXIT_SUCCESS; }
void acb_hypgeom_bessel_k_0f1(acb_t res, const acb_t nu, const acb_t z, slong prec) { if (acb_is_int(nu)) { acb_poly_t nux, zx, rx; acb_poly_init(nux); acb_poly_init(zx); acb_poly_init(rx); acb_poly_set_coeff_acb(nux, 0, nu); acb_poly_set_coeff_si(nux, 1, 1); acb_poly_set_acb(zx, z); acb_hypgeom_bessel_k_0f1_series(rx, nux, zx, 1, prec); acb_poly_get_coeff_acb(res, rx, 0); acb_poly_clear(nux); acb_poly_clear(zx); acb_poly_clear(rx); } else { acb_t t, u, v, w; acb_struct b[2]; acb_init(t); acb_init(u); acb_init(v); acb_init(w); acb_init(b + 0); acb_init(b + 1); /* u = 0F1(1+nu), v = 0F1(1-nu) */ acb_mul(t, z, z, prec); acb_mul_2exp_si(t, t, -2); acb_add_ui(b, nu, 1, prec); acb_one(b + 1); acb_hypgeom_pfq_direct(u, NULL, 0, b, 2, t, -1, prec); acb_sub_ui(b, nu, 1, prec); acb_neg(b, b); acb_hypgeom_pfq_direct(v, NULL, 0, b, 2, t, -1, prec); /* v = v * gamma(nu) / (z/2)^nu */ acb_mul_2exp_si(t, z, -1); acb_pow(t, t, nu, prec); acb_gamma(w, nu, prec); acb_mul(v, v, w, prec); acb_div(v, v, t, prec); /* u = u * t * pi / (gamma(nu) * nu * sin(pi nu)) */ acb_mul(u, u, t, prec); acb_const_pi(t, prec); acb_mul(u, u, t, prec); acb_sin_pi(t, nu, prec); acb_mul(t, t, w, prec); acb_mul(t, t, nu, prec); acb_div(u, u, t, prec); acb_sub(res, v, u, prec); acb_mul_2exp_si(res, res, -1); acb_clear(t); acb_clear(u); acb_clear(v); acb_clear(w); acb_clear(b + 0); acb_clear(b + 1); } }