void mag_exp_tail(mag_t z, const mag_t x, ulong N) { if (N == 0 || mag_is_inf(x)) { mag_exp(z, x); } else if (mag_is_zero(x)) { mag_zero(z); } else { mag_t t; mag_init(t); mag_set_ui_2exp_si(t, N, -1); /* bound by geometric series when N >= 2*x <=> N/2 >= x */ if (mag_cmp(t, x) >= 0) { /* 2 c^N / N! */ mag_pow_ui(t, x, N); mag_rfac_ui(z, N); mag_mul(z, z, t); mag_mul_2exp_si(z, z, 1); } else { mag_exp(z, x); } mag_clear(t); } }
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); }
void mag_root(mag_t y, const mag_t x, ulong n) { if (n == 0) { mag_inf(y); } else if (n == 1 || mag_is_special(x)) { mag_set(y, x); } else if (n == 2) { mag_sqrt(y, x); } else if (n == 4) { mag_sqrt(y, x); mag_sqrt(y, y); } else { fmpz_t e, f; fmpz_init_set_ui(e, MAG_BITS); fmpz_init(f); /* We evaluate exp(log(1+2^(kn)x)/n) 2^-k where k is chosen so that 2^(kn) x ~= 2^30. TODO: this rewriting is probably unnecessary with the new exp/log functions. */ fmpz_sub(e, e, MAG_EXPREF(x)); fmpz_cdiv_q_ui(e, e, n); fmpz_mul_ui(f, e, n); mag_mul_2exp_fmpz(y, x, f); mag_log1p(y, y); mag_div_ui(y, y, n); mag_exp(y, y); fmpz_neg(e, e); mag_mul_2exp_fmpz(y, y, e); fmpz_clear(e); fmpz_clear(f); } }
/* derivatives: |8/sqrt(pi) sin(2z^2)|, |8/sqrt(pi) cos(2z^2)| <= 5 exp(4|xy|) */ void acb_hypgeom_fresnel_erf_error(acb_t res1, acb_t res2, const acb_t z, slong prec) { mag_t re; mag_t im; acb_t zmid; mag_init(re); mag_init(im); acb_init(zmid); if (arf_cmpabs_ui(arb_midref(acb_realref(z)), 1000) < 0 && arf_cmpabs_ui(arb_midref(acb_imagref(z)), 1000) < 0) { arb_get_mag(re, acb_realref(z)); arb_get_mag(im, acb_imagref(z)); mag_mul(re, re, im); mag_mul_2exp_si(re, re, 2); mag_exp(re, re); mag_mul_ui(re, re, 5); } else { arb_t t; arb_init(t); arb_mul(t, acb_realref(z), acb_imagref(z), prec); arb_abs(t, t); arb_mul_2exp_si(t, t, 2); arb_exp(t, t, prec); arb_get_mag(re, t); mag_mul_ui(re, re, 5); arb_clear(t); } mag_hypot(im, arb_radref(acb_realref(z)), arb_radref(acb_imagref(z))); mag_mul(re, re, im); if (arb_is_zero(acb_imagref(z))) { mag_set_ui(im, 8); /* For real x, |S(x)| < 4, |C(x)| < 4. */ mag_min(re, re, im); mag_zero(im); } else if (arb_is_zero(acb_realref(z))) { mag_set_ui(im, 8); mag_min(im, re, im); mag_zero(re); } else { mag_set(im, re); } arf_set(arb_midref(acb_realref(zmid)), arb_midref(acb_realref(z))); arf_set(arb_midref(acb_imagref(zmid)), arb_midref(acb_imagref(z))); acb_hypgeom_fresnel_erf(res1, res2, zmid, prec); if (res1 != NULL) { arb_add_error_mag(acb_realref(res1), re); arb_add_error_mag(acb_imagref(res1), im); } if (res2 != NULL) { arb_add_error_mag(acb_realref(res2), re); arb_add_error_mag(acb_imagref(res2), im); } mag_clear(re); mag_clear(im); acb_clear(zmid); }
void acb_hypgeom_u_asymp(acb_t res, const acb_t a, const acb_t b, const acb_t z, slong n, slong prec) { acb_struct aa[3]; acb_t s, t, w, winv; int R, p, q, is_real, is_terminating; slong n_terminating; if (!acb_is_finite(a) || !acb_is_finite(b) || !acb_is_finite(z)) { acb_indeterminate(res); return; } acb_init(aa); acb_init(aa + 1); acb_init(aa + 2); acb_init(s); acb_init(t); acb_init(w); acb_init(winv); is_terminating = 0; n_terminating = WORD_MAX; /* special case, for incomplete gamma [todo: also when they happen to be exact and with difference 1...] */ if (a == b) { acb_set(aa, a); p = 1; q = 0; } else { acb_set(aa, a); acb_sub(aa + 1, a, b, prec); acb_add_ui(aa + 1, aa + 1, 1, prec); acb_one(aa + 2); p = 2; q = 1; } if (acb_is_nonpositive_int(aa)) { is_terminating = 1; if (arf_cmpabs_ui(arb_midref(acb_realref(aa)), prec) < 0) n_terminating = 1 - arf_get_si(arb_midref(acb_realref(aa)), ARF_RND_DOWN); } if (p == 2 && acb_is_nonpositive_int(aa + 1)) { is_terminating = 1; if (arf_cmpabs_ui(arb_midref(acb_realref(aa + 1)), n_terminating) < 0) n_terminating = 1 - arf_get_si(arb_midref(acb_realref(aa + 1)), ARF_RND_DOWN); } acb_neg(w, z); acb_inv(w, w, prec); acb_neg(winv, z); /* low degree polynomial -- no need to try to terminate sooner */ if (is_terminating && n_terminating < 8) { acb_hypgeom_pfq_sum_invz(s, t, aa, p, aa + p, q, w, winv, n_terminating, prec); acb_set(res, s); } else { mag_t C1, Cn, alpha, nu, sigma, rho, zinv, tmp, err; mag_init(C1); mag_init(Cn); mag_init(alpha); mag_init(nu); mag_init(sigma); mag_init(rho); mag_init(zinv); mag_init(tmp); mag_init(err); acb_hypgeom_u_asymp_bound_factors(&R, alpha, nu, sigma, rho, zinv, a, b, z); is_real = acb_is_real(a) && acb_is_real(b) && acb_is_real(z) && (is_terminating || arb_is_positive(acb_realref(z))); if (R == 0) { /* if R == 0, the error bound is infinite unless terminating */ if (is_terminating && n_terminating < prec) { acb_hypgeom_pfq_sum_invz(s, t, aa, p, aa + p, q, w, winv, n_terminating, prec); acb_set(res, s); } else { acb_indeterminate(res); } } else { /* C1 */ acb_hypgeom_mag_Cn(C1, R, nu, sigma, 1); /* err = 2 * alpha * exp(...) */ mag_mul(tmp, C1, rho); mag_mul(tmp, tmp, alpha); mag_mul(tmp, tmp, zinv); mag_mul_2exp_si(tmp, tmp, 1); mag_exp(err, tmp); mag_mul(err, err, alpha); mag_mul_2exp_si(err, err, 1); /* choose n automatically */ if (n < 0) { slong moreprec; /* take err into account when finding truncation point */ /* we should take Cn into account as well, but this depends on n which is to be determined; it's easier to look only at exp(...) which should be larger anyway */ if (mag_cmp_2exp_si(err, 10 * prec) > 0) moreprec = 10 * prec; else if (mag_cmp_2exp_si(err, 0) < 0) moreprec = 0; else moreprec = MAG_EXP(err); n = acb_hypgeom_pfq_choose_n_max(aa, p, aa + p, q, w, prec + moreprec, FLINT_MIN(WORD_MAX / 2, 50 + 10.0 * prec)); } acb_hypgeom_pfq_sum_invz(s, t, aa, p, aa + p, q, w, winv, n, prec); /* add error bound, if not terminating */ if (!(is_terminating && n == n_terminating)) { acb_hypgeom_mag_Cn(Cn, R, nu, sigma, n); mag_mul(err, err, Cn); /* nth term * factor */ acb_get_mag(tmp, t); mag_mul(err, err, tmp); if (is_real) arb_add_error_mag(acb_realref(s), err); else acb_add_error_mag(s, err); } acb_set(res, s); } mag_clear(C1); mag_clear(Cn); mag_clear(alpha); mag_clear(nu); mag_clear(sigma); mag_clear(rho); mag_clear(zinv); mag_clear(tmp); mag_clear(err); } acb_clear(aa); acb_clear(aa + 1); acb_clear(aa + 2); acb_clear(s); acb_clear(t); acb_clear(w); acb_clear(winv); }
void _arb_bell_sum_taylor(arb_t res, const fmpz_t n, const fmpz_t a, const fmpz_t b, const fmpz_t mmag, long tol) { fmpz_t m, r, R, tmp; mag_t B, C, D, bound; arb_t t, u; long wp, k, N; if (_fmpz_sub_small(b, a) < 5) { arb_bell_sum_bsplit(res, n, a, b, mmag, tol); return; } fmpz_init(m); fmpz_init(r); fmpz_init(R); fmpz_init(tmp); /* r = max(m - a, b - m) */ /* m = a + (b - a) / 2 */ fmpz_sub(r, b, a); fmpz_cdiv_q_2exp(r, r, 1); fmpz_add(m, a, r); fmpz_mul_2exp(R, r, RADIUS_BITS); mag_init(B); mag_init(C); mag_init(D); mag_init(bound); arb_init(t); arb_init(u); if (fmpz_cmp(R, m) >= 0) { mag_inf(C); mag_inf(D); } else { /* C = exp(R * |F'(m)| + (1/2) R^2 * (n/(m-R)^2 + 1/(m-R))) */ /* C = exp(R * (|F'(m)| + (1/2) R * (n/(m-R) + 1)/(m-R))) */ /* D = (1/2) R * (n/(m-R) + 1)/(m-R) */ fmpz_sub(tmp, m, R); mag_set_fmpz(D, n); mag_div_fmpz(D, D, tmp); mag_one(C); mag_add(D, D, C); mag_div_fmpz(D, D, tmp); mag_mul_fmpz(D, D, R); mag_mul_2exp_si(D, D, -1); /* C = |F'(m)| */ wp = 20 + 1.05 * fmpz_bits(n); arb_set_fmpz(t, n); arb_div_fmpz(t, t, m, wp); fmpz_add_ui(tmp, m, 1); arb_set_fmpz(u, tmp); arb_digamma(u, u, wp); arb_sub(t, t, u, wp); arb_get_mag(C, t); /* C = exp(R * (C + D)) */ mag_add(C, C, D); mag_mul_fmpz(C, C, R); mag_exp(C, C); } if (mag_cmp_2exp_si(C, tol / 4 + 2) > 0) { _arb_bell_sum_taylor(res, n, a, m, mmag, tol); _arb_bell_sum_taylor(t, n, m, b, mmag, tol); arb_add(res, res, t, 2 * tol); } else { arb_ptr mx, ser1, ser2, ser3; /* D = T(m) */ wp = 20 + 1.05 * fmpz_bits(n); arb_set_fmpz(t, m); arb_pow_fmpz(t, t, n, wp); fmpz_add_ui(tmp, m, 1); arb_gamma_fmpz(u, tmp, wp); arb_div(t, t, u, wp); arb_get_mag(D, t); /* error bound: (b-a) * C * D * B^N / (1 - B), B = r/R */ /* ((b-a) * C * D * 2) * 2^(-N*RADIUS_BITS) */ /* ((b-a) * C * D * 2) */ mag_mul(bound, C, D); mag_mul_2exp_si(bound, bound, 1); fmpz_sub(tmp, b, a); mag_mul_fmpz(bound, bound, tmp); /* N = (tol + log2((b-a)*C*D*2) - mmag) / RADIUS_BITS */ if (mmag == NULL) { /* estimate D ~= 2^mmag */ fmpz_add_ui(tmp, MAG_EXPREF(C), tol); fmpz_cdiv_q_ui(tmp, tmp, RADIUS_BITS); } else { fmpz_sub(tmp, MAG_EXPREF(bound), mmag); fmpz_add_ui(tmp, tmp, tol); fmpz_cdiv_q_ui(tmp, tmp, RADIUS_BITS); } if (fmpz_cmp_ui(tmp, 5 * tol / 4) > 0) N = 5 * tol / 4; else if (fmpz_cmp_ui(tmp, 2) < 0) N = 2; else N = fmpz_get_ui(tmp); /* multiply by 2^(-N*RADIUS_BITS) */ mag_mul_2exp_si(bound, bound, -N * RADIUS_BITS); mx = _arb_vec_init(2); ser1 = _arb_vec_init(N); ser2 = _arb_vec_init(N); ser3 = _arb_vec_init(N); /* estimate (this should work for moderate n and tol) */ wp = 1.1 * tol + 1.05 * fmpz_bits(n) + 5; /* increase precision until convergence */ while (1) { /* (m+x)^n / gamma(m+1+x) */ arb_set_fmpz(mx, m); arb_one(mx + 1); _arb_poly_log_series(ser1, mx, 2, N, wp); for (k = 0; k < N; k++) arb_mul_fmpz(ser1 + k, ser1 + k, n, wp); arb_add_ui(mx, mx, 1, wp); _arb_poly_lgamma_series(ser2, mx, 2, N, wp); _arb_vec_sub(ser1, ser1, ser2, N, wp); _arb_poly_exp_series(ser3, ser1, N, N, wp); /* t = a - m, u = b - m */ arb_set_fmpz(t, a); arb_sub_fmpz(t, t, m, wp); arb_set_fmpz(u, b); arb_sub_fmpz(u, u, m, wp); arb_power_sum_vec(ser1, t, u, N, wp); arb_zero(res); for (k = 0; k < N; k++) arb_addmul(res, ser3 + k, ser1 + k, wp); if (mmag != NULL) { if (_fmpz_sub_small(MAG_EXPREF(arb_radref(res)), mmag) <= -tol) break; } else { if (arb_rel_accuracy_bits(res) >= tol) break; } wp = 2 * wp; } /* add the series truncation bound */ arb_add_error_mag(res, bound); _arb_vec_clear(mx, 2); _arb_vec_clear(ser1, N); _arb_vec_clear(ser2, N); _arb_vec_clear(ser3, N); } mag_clear(B); mag_clear(C); mag_clear(D); mag_clear(bound); arb_clear(t); arb_clear(u); fmpz_clear(m); fmpz_clear(r); fmpz_clear(R); fmpz_clear(tmp); }