static void acb_approx_mul(acb_t res, const acb_t x, const acb_t y, slong prec) { arf_complex_mul(arb_midref(acb_realref(res)), arb_midref(acb_imagref(res)), arb_midref(acb_realref(x)), arb_midref(acb_imagref(x)), arb_midref(acb_realref(y)), arb_midref(acb_imagref(y)), prec, ARB_RND); }
/* Extremely close to the branch point at -1/e, use the series expansion directly. */ int acb_lambertw_try_near_branch_point(acb_t res, const acb_t z, const acb_t ez1, const fmpz_t k, int flags, slong prec) { if (fmpz_is_zero(k) || (fmpz_is_one(k) && arb_is_negative(acb_imagref(z))) || (fmpz_equal_si(k, -1) && arb_is_nonnegative(acb_imagref(z)))) { if (acb_contains_zero(ez1) || (arf_cmpabs_2exp_si(arb_midref(acb_realref(ez1)), -prec / 4.5 - 6) < 0 && arf_cmpabs_2exp_si(arb_midref(acb_imagref(ez1)), -prec / 4.5 - 6) < 0)) { acb_t t; acb_init(t); acb_mul_2exp_si(t, ez1, 1); acb_sqrt(t, t, prec); if (!fmpz_is_zero(k)) acb_neg(t, t); acb_lambertw_branchpoint_series(res, t, 1, prec); acb_clear(t); return 1; } } return 0; }
int acb_cmp_pretty(const acb_t a, const acb_t b) { arb_t t, u, v; int res; arb_init(t); arb_init(u); arb_init(v); arb_abs(u, acb_imagref(a)); arb_abs(v, acb_imagref(b)); arb_sub(t, u, v, MAG_BITS); res = 0; if (arb_contains_zero(t)) { arb_sub(t, acb_realref(a), acb_realref(b), MAG_BITS); res = arb_is_positive(t) ? 1 : -1; } else { res = arb_is_positive(t) ? 1 : -1; } arb_clear(t); arb_clear(u); arb_clear(v); return res; }
void acb_get_mag(mag_t u, const acb_t z) { if (arb_is_zero(acb_imagref(z))) { arb_get_mag(u, acb_realref(z)); } else if (arb_is_zero(acb_realref(z))) { arb_get_mag(u, acb_imagref(z)); } else { mag_t v; mag_init(v); arb_get_mag(u, acb_realref(z)); arb_get_mag(v, acb_imagref(z)); mag_mul(u, u, u); mag_addmul(u, v, v); mag_sqrt(u, u); mag_clear(v); } }
static void acb_set_dddd(acb_t z, double a, double ar, double b, double br) { arf_set_d(arb_midref(acb_realref(z)), a); mag_set_d(arb_radref(acb_realref(z)), ar); arf_set_d(arb_midref(acb_imagref(z)), b); mag_set_d(arb_radref(acb_imagref(z)), br); }
void acb_hypgeom_erf(acb_t res, const acb_t z, slong prec) { double x, y, absz2, logz; slong prec2; if (!acb_is_finite(z)) { acb_indeterminate(res); return; } if (acb_is_zero(z)) { acb_zero(res); return; } if ((arf_cmpabs_2exp_si(arb_midref(acb_realref(z)), 0) < 0 && arf_cmpabs_2exp_si(arb_midref(acb_imagref(z)), 0) < 0)) { acb_hypgeom_erf_1f1a(res, z, prec); return; } if ((arf_cmpabs_2exp_si(arb_midref(acb_realref(z)), 64) > 0 || arf_cmpabs_2exp_si(arb_midref(acb_imagref(z)), 64) > 0)) { acb_hypgeom_erf_asymp(res, z, prec, prec); return; } x = arf_get_d(arb_midref(acb_realref(z)), ARF_RND_DOWN); y = arf_get_d(arb_midref(acb_imagref(z)), ARF_RND_DOWN); absz2 = x * x + y * y; logz = 0.5 * log(absz2); if (logz - absz2 < -(prec + 8) * 0.69314718055994530942) { /* If the asymptotic term is small, we can compute with reduced precision */ prec2 = FLINT_MIN(prec + 4 + (y*y - x*x - logz) * 1.4426950408889634074, (double) prec); prec2 = FLINT_MAX(8, prec2); prec2 = FLINT_MIN(prec2, prec); acb_hypgeom_erf_asymp(res, z, prec, prec2); } else if (arf_cmpabs(arb_midref(acb_imagref(z)), arb_midref(acb_realref(z))) > 0) { acb_hypgeom_erf_1f1a(res, z, prec); } else { acb_hypgeom_erf_1f1b(res, z, prec); } }
void acb_lambertw_principal_d(acb_t res, const acb_t z) { double za, zb, wa, wb, ewa, ewb, t, u, q, r; int k, maxk = 15; za = arf_get_d(arb_midref(acb_realref(z)), ARF_RND_DOWN); zb = arf_get_d(arb_midref(acb_imagref(z)), ARF_RND_DOWN); /* make sure we end up on the right branch */ if (za < -0.367 && zb > -1e-20 && zb <= 0.0 && arf_sgn(arb_midref(acb_imagref(z))) < 0) zb = -1e-20; wa = za; wb = zb; if (fabs(wa) > 2.0 || fabs(wb) > 2.0) { t = atan2(wb, wa); wa = 0.5 * log(wa * wa + wb * wb); wb = t; } else if (fabs(wa) > 0.25 || fabs(wb) > 0.25) { /* We have W(z) ~= -1 + (2(ez+1))^(1/2) near the branch point. Changing the exponent to 1/4 gives a much worse local guess which however does the job on a larger domain. */ wa *= 5.43656365691809; wb *= 5.43656365691809; wa += 2.0; t = atan2(wb, wa); r = pow(wa * wa + wb * wb, 0.125); wa = r * cos(0.25 * t); wb = r * sin(0.25 * t); wa -= 1.0; } for (k = 0; k < maxk; k++) { t = exp(wa); ewa = t * cos(wb); ewb = t * sin(wb); t = (ewa * wa - ewb * wb); q = t + ewa; t -= za; u = (ewb * wa + ewa * wb); r = u + ewb; u -= zb; ewa = q * t + r * u; ewb = q * u - r * t; r = 1.0 / (q * q + r * r); ewa *= r; ewb *= r; if ((ewa*ewa + ewb*ewb) < (wa*wa + wb*wb) * 1e-12) maxk = FLINT_MIN(maxk, k + 2); wa -= ewa; wb -= ewb; } acb_set_d_d(res, wa, wb); }
void acb_hypgeom_erf_propagated_error(mag_t re, mag_t im, const acb_t z) { mag_t x, y; mag_init(x); mag_init(y); /* |exp(-(x+y)^2)| = exp(y^2-x^2) */ arb_get_mag(y, acb_imagref(z)); mag_mul(y, y, y); arb_get_mag_lower(x, acb_realref(z)); mag_mul_lower(x, x, x); if (mag_cmp(y, x) >= 0) { mag_sub(re, y, x); mag_exp(re, re); } else { mag_sub_lower(re, x, y); mag_expinv(re, re); } /* Radius. */ mag_hypot(x, arb_radref(acb_realref(z)), arb_radref(acb_imagref(z))); mag_mul(re, re, x); /* 2/sqrt(pi) < 289/256 */ mag_mul_ui(re, re, 289); mag_mul_2exp_si(re, re, -8); if (arb_is_zero(acb_imagref(z))) { /* todo: could bound magnitude even for complex numbers */ mag_set_ui(y, 2); mag_min(re, re, y); mag_zero(im); } else if (arb_is_zero(acb_realref(z))) { mag_swap(im, re); mag_zero(re); } else { mag_set(im, re); } mag_clear(x); mag_clear(y); }
int main() { flint_printf("spherical_y...."); fflush(stdout); { slong i, n, m; acb_t z, w, x, y; acb_init(z); acb_init(w); acb_init(x); acb_init(y); i = 0; arb_set_str(acb_realref(x), "0.2", 64); arb_set_str(acb_imagref(x), "0.3", 64); arb_set_str(acb_realref(y), "0.3", 64); arb_set_str(acb_imagref(y), "0.4", 64); for (n = -4; n <= 4; n++) { for (m = -4; m <= 4; m++) { acb_hypgeom_spherical_y(z, n, m, x, y, 64); acb_set_d_d(w, testdata[2 * i], testdata[2 * i + 1]); mag_set_d(arb_radref(acb_realref(w)), 1e-13); mag_set_d(arb_radref(acb_imagref(w)), 1e-13); if (!acb_overlaps(z, w)) { flint_printf("FAIL: value\n\n"); flint_printf("n = %wd, m = %wd\n", n, m); flint_printf("z = "); acb_printd(z, 20); flint_printf("\n\n"); flint_printf("w = "); acb_printd(w, 20); flint_printf("\n\n"); abort(); } i++; } } acb_clear(z); acb_clear(w); acb_clear(x); acb_clear(y); } flint_cleanup(); flint_printf("PASS\n"); return EXIT_SUCCESS; }
void acb_hypgeom_bessel_jy(acb_t res1, acb_t res2, const acb_t nu, const acb_t z, slong prec) { acb_t jnu, t, u, v; acb_init(jnu); acb_init(t); acb_init(u); acb_init(v); acb_hypgeom_bessel_j(jnu, nu, z, prec); if (acb_is_int(nu)) { int is_real = acb_is_real(nu) && acb_is_real(z) && arb_is_positive(acb_realref(z)); acb_mul_onei(t, z); acb_hypgeom_bessel_k(t, nu, t, prec); acb_onei(u); acb_pow(u, u, nu, prec); acb_mul(t, t, u, prec); acb_const_pi(u, prec); acb_div(t, t, u, prec); acb_mul_2exp_si(t, t, 1); acb_neg(t, t); phase(v, acb_realref(z), acb_imagref(z)); acb_mul(u, jnu, v, prec); acb_mul_onei(u, u); acb_sub(res2, t, u, prec); if (is_real) arb_zero(acb_imagref(res2)); } else { acb_sin_cos_pi(t, u, nu, prec); acb_mul(v, jnu, u, prec); acb_neg(u, nu); acb_hypgeom_bessel_j(u, u, z, prec); acb_sub(v, v, u, prec); acb_div(res2, v, t, prec); } if (res1 != NULL) acb_set(res1, jnu); acb_clear(jnu); acb_clear(t); acb_clear(u); acb_clear(v); }
/* note: the tmp variable t should have zero radius */ static void acb_approx_div(acb_t z, const acb_t x, const acb_t y, acb_t t, slong prec) { arf_set(arb_midref(acb_realref(t)), arb_midref(acb_realref(y))); arf_set(arb_midref(acb_imagref(t)), arb_midref(acb_imagref(y))); acb_inv(t, t, prec); mag_zero(arb_radref(acb_realref(t))); mag_zero(arb_radref(acb_imagref(t))); acb_approx_mul(z, x, t, prec); }
static void bsplit(arb_poly_t pol, const arb_t sqrtD, const slong * qbf, slong a, slong b, slong prec) { if (b - a == 0) { arb_poly_one(pol); } else if (b - a == 1) { acb_t z; acb_init(z); /* j((-b+sqrt(-D))/(2a)) */ arb_set_si(acb_realref(z), -FLINT_ABS(qbf[3 * a + 1])); arb_set(acb_imagref(z), sqrtD); acb_div_si(z, z, 2 * qbf[3 * a], prec); acb_modular_j(z, z, prec); if (qbf[3 * a + 1] < 0) { /* (x^2 - 2re(j) x + |j|^2) */ arb_poly_fit_length(pol, 3); arb_mul(pol->coeffs, acb_realref(z), acb_realref(z), prec); arb_addmul(pol->coeffs, acb_imagref(z), acb_imagref(z), prec); arb_mul_2exp_si(pol->coeffs + 1, acb_realref(z), 1); arb_neg(pol->coeffs + 1, pol->coeffs + 1); arb_one(pol->coeffs + 2); _arb_poly_set_length(pol, 3); } else { /* (x-j) */ arb_poly_fit_length(pol, 2); arb_neg(pol->coeffs, acb_realref(z)); arb_one(pol->coeffs + 1); _arb_poly_set_length(pol, 2); } acb_clear(z); } else { arb_poly_t tmp; arb_poly_init(tmp); bsplit(pol, sqrtD, qbf, a, a + (b - a) / 2, prec); bsplit(tmp, sqrtD, qbf, a + (b - a) / 2, b, prec); arb_poly_mul(pol, pol, tmp, prec); arb_poly_clear(tmp); } }
void acb_dirichlet_gauss_sum_order2(acb_t res, const dirichlet_group_t G, const dirichlet_char_t chi, slong prec) { if (dirichlet_parity_char(G, chi)) { arb_zero(acb_realref(res)); arb_sqrt_ui(acb_imagref(res), G->q, prec); } else { arb_zero(acb_imagref(res)); arb_sqrt_ui(acb_realref(res), G->q, prec); } }
void acb_hypgeom_erf_asymp(acb_t res, const acb_t z, slong prec, slong prec2) { acb_t a, t, u; acb_init(a); acb_init(t); acb_init(u); acb_one(a); acb_mul_2exp_si(a, a, -1); acb_mul(t, z, z, prec2); acb_hypgeom_u_asymp(u, a, a, t, -1, prec2); acb_neg(t, t); acb_exp(t, t, prec2); acb_mul(u, u, t, prec2); acb_const_pi(t, prec2); acb_sqrt(t, t, prec2); acb_mul(t, t, z, prec2); acb_div(u, u, t, prec2); /* branch cut term: -1 or 1 */ if (arb_contains_zero(acb_realref(z))) { arb_zero(acb_imagref(t)); arf_zero(arb_midref(acb_realref(t))); mag_one(arb_radref(acb_realref(t))); } else { acb_set_si(t, arf_sgn(arb_midref(acb_realref(z)))); } acb_sub(t, t, u, prec); if (arb_is_zero(acb_imagref(z))) arb_zero(acb_imagref(t)); else if (arb_is_zero(acb_realref(z))) arb_zero(acb_realref(t)); acb_set(res, t); acb_clear(a); acb_clear(t); acb_clear(u); }
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); } }
int bessel(acb_ptr out, const acb_t inp, void * params, long order, long prec) { acb_ptr t; acb_t z; ulong n; t = _acb_vec_init(order); acb_init(z); acb_set(t, inp); if (order > 1) acb_one(t + 1); n = 10; arb_set_si(acb_realref(z), 20); arb_set_si(acb_imagref(z), 10); /* z sin(t) */ _acb_poly_sin_series(out, t, FLINT_MIN(2, order), order, prec); _acb_vec_scalar_mul(out, out, order, z, prec); /* t n */ _acb_vec_scalar_mul_ui(t, t, FLINT_MIN(2, order), n, prec); _acb_poly_sub(out, t, FLINT_MIN(2, order), out, order, prec); _acb_poly_cos_series(out, out, order, order, prec); _acb_vec_clear(t, order); acb_clear(z); return 0; }
void _acb_poly_zeta_cpx_series(acb_ptr z, const acb_t s, const acb_t a, int deflate, long d, long prec) { ulong M, N; long i; arf_t bound; arb_ptr vb; if (d < 1) return; if (!acb_is_finite(s) || !acb_is_finite(a)) { _acb_vec_indeterminate(z, d); return; } arf_init(bound); vb = _arb_vec_init(d); _acb_poly_zeta_em_choose_param(bound, &N, &M, s, a, FLINT_MIN(d, 2), prec, MAG_BITS); _acb_poly_zeta_em_bound(vb, s, a, N, M, d, MAG_BITS); _acb_poly_zeta_em_sum(z, s, a, deflate, N, M, d, prec); for (i = 0; i < d; i++) { arb_get_abs_ubound_arf(bound, vb + i, MAG_BITS); arb_add_error_arf(acb_realref(z + i), bound); arb_add_error_arf(acb_imagref(z + i), bound); } arf_clear(bound); _arb_vec_clear(vb, d); }
void acb_dirichlet_theta_arb(acb_t res, const dirichlet_group_t G, const dirichlet_char_t chi, const arb_t t, slong prec) { slong len; ulong order; arb_t xt; mag_t e; len = acb_dirichlet_theta_length(G->q, t, prec); arb_init(xt); _acb_dirichlet_theta_argument_at_arb(xt, G->q, t, prec); mag_init(e); mag_tail_kexpk2_arb(e, xt, len); arb_neg(xt, xt); arb_exp(xt, xt, prec); /* TODO: tune this limit */ order = dirichlet_order_char(G, chi); if (order < 30) _acb_dirichlet_theta_arb_smallorder(res, G, chi, xt, len, prec); else _acb_dirichlet_theta_arb_naive(res, G, chi, xt, len, prec); arb_add_error_mag(acb_realref(res), e); arb_add_error_mag(acb_imagref(res), e); mag_clear(e); acb_mul_2exp_si(res, res, 1); arb_clear(xt); }
void acb_lambertw_cleared_cut_fix_small(acb_t res, const acb_t z, const acb_t ez1, const fmpz_t k, int flags, slong prec) { acb_t zz, zmid, zmide1; arf_t eps; acb_init(zz); acb_init(zmid); acb_init(zmide1); arf_init(eps); arf_mul_2exp_si(eps, arb_midref(acb_realref(z)), -prec); acb_set(zz, z); if (arf_sgn(arb_midref(acb_realref(zz))) < 0 && (!fmpz_is_zero(k) || arf_sgn(arb_midref(acb_realref(ez1))) < 0) && arf_cmpabs(arb_midref(acb_imagref(zz)), eps) < 0) { /* now the value must be in [0,2eps] */ arf_get_mag(arb_radref(acb_imagref(zz)), eps); arf_set_mag(arb_midref(acb_imagref(zz)), arb_radref(acb_imagref(zz))); if (arf_sgn(arb_midref(acb_imagref(z))) >= 0) { acb_lambertw_cleared_cut(res, zz, k, flags, prec); } else { fmpz_t kk; fmpz_init(kk); fmpz_neg(kk, k); acb_lambertw_cleared_cut(res, zz, kk, flags, prec); acb_conj(res, res); fmpz_clear(kk); } } else { acb_lambertw_cleared_cut(res, zz, k, flags, prec); } acb_clear(zz); acb_clear(zmid); acb_clear(zmide1); arf_clear(eps); }
void acb_hypgeom_m_asymp(acb_t res, const acb_t a, const acb_t b, const acb_t z, int regularized, slong prec) { acb_t t, u, v, c; acb_init(t); acb_init(u); acb_init(v); acb_init(c); acb_sub(c, b, a, prec); acb_neg(v, z); acb_hypgeom_u_asymp(t, a, b, z, -1, prec); acb_hypgeom_u_asymp(u, c, b, v, -1, prec); /* gamma(b-a) */ acb_rgamma(v, c, prec); acb_mul(t, t, v, prec); /* z^(a-b) */ acb_neg(c, c); acb_pow(v, z, c, prec); acb_mul(u, u, v, prec); /* gamma(a) */ acb_rgamma(v, a, prec); acb_mul(u, u, v, prec); /* exp(z) */ acb_exp(v, z, prec); acb_mul(u, u, v, prec); /* (-z)^(-a) */ acb_neg(c, a); acb_neg(v, z); acb_pow(v, v, c, prec); acb_mul(t, t, v, prec); acb_add(t, t, u, prec); if (!regularized) { acb_gamma(v, b, prec); acb_mul(t, t, v, prec); } if (acb_is_real(a) && acb_is_real(b) && acb_is_real(z)) { arb_zero(acb_imagref(t)); } acb_swap(res, t); acb_clear(t); acb_clear(u); acb_clear(v); acb_clear(c); }
int acb_contains_int(const acb_t x) { if (!arb_contains_zero(acb_imagref(x))) return 0; return arb_contains_int(acb_realref(x)); }
void acb_log_sin_pi(acb_t res, const acb_t z, slong prec) { if (!acb_is_finite(z)) { acb_indeterminate(res); return; } if (arb_is_positive(acb_imagref(z)) || (arb_is_zero(acb_imagref(z)) && arb_is_negative(acb_realref(z)))) { acb_log_sin_pi_half(res, z, prec, 1); } else if (arb_is_negative(acb_imagref(z)) || (arb_is_zero(acb_imagref(z)) && arb_is_positive(acb_realref(z)))) { acb_log_sin_pi_half(res, z, prec, 0); } else { acb_t t; acb_init(t); acb_log_sin_pi_half(t, z, prec, 1); acb_log_sin_pi_half(res, z, prec, 0); arb_union(acb_realref(res), acb_realref(res), acb_realref(t), prec); arb_union(acb_imagref(res), acb_imagref(res), acb_imagref(t), prec); acb_clear(t); } }
static void _acb_unit_root(acb_t res, ulong order, slong prec) { fmpq_t t; fmpq_init(t); fmpq_set_si(t, 2, order); arb_sin_cos_pi_fmpq(acb_imagref(res), acb_realref(res), t, prec); fmpq_clear(t); }
/* invalid in (-1,0) */ int _acb_hypgeom_legendre_q_single_valid(const acb_t z) { arb_t t; int ok; if (!arb_contains_zero(acb_imagref(z))) return 1; if (arb_is_positive(acb_imagref(z))) return 1; arb_init(t); arb_one(t); arb_neg(t, t); ok = arb_lt(acb_realref(z), t); arb_clear(t); return ok; }
void acb_cot(acb_t r, const acb_t z, slong prec) { if (arb_is_zero(acb_imagref(z))) { arb_cot(acb_realref(r), acb_realref(z), prec); arb_zero(acb_imagref(r)); } else if (arb_is_zero(acb_realref(z))) { arb_coth(acb_imagref(r), acb_imagref(z), prec); arb_neg(acb_imagref(r), acb_imagref(r)); arb_zero(acb_realref(r)); } else { acb_t t; acb_init(t); if (arf_cmpabs_2exp_si(arb_midref(acb_imagref(z)), 0) < 0) { acb_sin_cos(r, t, z, prec + 4); acb_div(r, t, r, prec); } else { acb_mul_2exp_si(t, z, 1); if (arf_sgn(arb_midref(acb_imagref(z))) > 0) { acb_mul_onei(t, t); acb_exp(t, t, prec + 4); acb_sub_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); } else { acb_div_onei(t, t); acb_exp(t, t, prec + 4); acb_sub_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); } } acb_clear(t); } }
slong _acb_get_mid_mag(const acb_t z) { slong rm, im; rm = arf_abs_bound_lt_2exp_si(arb_midref(acb_realref(z))); im = arf_abs_bound_lt_2exp_si(arb_midref(acb_imagref(z))); return FLINT_MAX(rm, im); }
int main() { slong iter; flint_rand_t state; flint_printf("get_mag...."); fflush(stdout); flint_randinit(state); for (iter = 0; iter < 10000 * arb_test_multiplier(); iter++) { acb_t a; arf_t m2, x, y, s; mag_t m; acb_init(a); mag_init(m); arf_init(m2); arf_init(x); arf_init(y); arf_init(s); acb_randtest_special(a, state, 200, 10); acb_get_mag(m, a); MAG_CHECK_BITS(m) /* check m^2 >= x^2 + y^2 */ arf_set_mag(m2, m); arf_mul(m2, m2, m2, ARF_PREC_EXACT, ARF_RND_DOWN); arb_get_abs_ubound_arf(x, acb_realref(a), ARF_PREC_EXACT); arb_get_abs_ubound_arf(y, acb_imagref(a), ARF_PREC_EXACT); arf_sosq(s, x, y, ARF_PREC_EXACT, ARF_RND_DOWN); if (arf_cmp(m2, s) < 0) { flint_printf("FAIL:\n\n"); flint_printf("a = "); acb_print(a); flint_printf("\n\n"); flint_printf("m = "); mag_print(m); flint_printf("\n\n"); flint_abort(); } acb_clear(a); mag_clear(m); arf_clear(m2); arf_clear(x); arf_clear(y); arf_clear(s); } flint_randclear(state); flint_cleanup(); flint_printf("PASS\n"); return EXIT_SUCCESS; }
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); }
void acb_sin_cos_pi(acb_t s, acb_t c, const acb_t z, slong prec) { #define a acb_realref(z) #define b acb_imagref(z) if (arb_is_zero(b)) { arb_sin_cos_pi(acb_realref(s), acb_realref(c), a, prec); arb_zero(acb_imagref(s)); arb_zero(acb_imagref(c)); } else if (arb_is_zero(a)) { arb_t t; arb_init(t); arb_const_pi(t, prec); arb_mul(t, t, b, prec); arb_sinh_cosh(acb_imagref(s), acb_realref(c), t, prec); arb_zero(acb_realref(s)); arb_zero(acb_imagref(c)); arb_clear(t); } else { arb_t sa, ca, sb, cb; arb_init(sa); arb_init(ca); arb_init(sb); arb_init(cb); arb_const_pi(sb, prec); arb_mul(sb, sb, b, prec); arb_sin_cos_pi(sa, ca, a, prec); arb_sinh_cosh(sb, cb, sb, prec); arb_mul(acb_realref(s), sa, cb, prec); arb_mul(acb_imagref(s), sb, ca, prec); arb_mul(acb_realref(c), ca, cb, prec); arb_mul(acb_imagref(c), sa, sb, prec); arb_neg(acb_imagref(c), acb_imagref(c)); arb_clear(sa); arb_clear(ca); arb_clear(sb); arb_clear(cb); } #undef a #undef b }
void _acb_poly_zeta_cpx_series(acb_ptr z, const acb_t s, const acb_t a, int deflate, slong d, slong prec) { ulong M, N; slong i; mag_t bound; arb_ptr vb; int is_real, const_is_real; if (d < 1) return; if (!acb_is_finite(s) || !acb_is_finite(a)) { _acb_vec_indeterminate(z, d); return; } is_real = const_is_real = 0; if (acb_is_real(s) && acb_is_real(a)) { if (arb_is_positive(acb_realref(a))) { is_real = const_is_real = 1; } else if (arb_is_int(acb_realref(a)) && arb_is_int(acb_realref(s)) && arb_is_nonpositive(acb_realref(s))) { const_is_real = 1; } } mag_init(bound); vb = _arb_vec_init(d); _acb_poly_zeta_em_choose_param(bound, &N, &M, s, a, FLINT_MIN(d, 2), prec, MAG_BITS); _acb_poly_zeta_em_bound(vb, s, a, N, M, d, MAG_BITS); _acb_poly_zeta_em_sum(z, s, a, deflate, N, M, d, prec); for (i = 0; i < d; i++) { arb_get_mag(bound, vb + i); arb_add_error_mag(acb_realref(z + i), bound); if (!is_real && !(i == 0 && const_is_real)) arb_add_error_mag(acb_imagref(z + i), bound); } mag_clear(bound); _arb_vec_clear(vb, d); }