int gsl_sf_gamma_inc_e(const double a, const double x, gsl_sf_result * result) { if(x < 0.0) { DOMAIN_ERROR(result); } else if(x == 0.0) { return gsl_sf_gamma_e(a, result); } else if(a == 0.0) { return GAMMA_INC_A_0(x, result); } else if(a > 0.0) { return gamma_inc_a_gt_0(a, x, result); } else if(x > 0.25) { /* continued fraction seems to fail for x too small; otherwise it is ok, independent of the value of |x/a|, because of the non-oscillation in the expansion, i.e. the CF is un-conditionally convergent for a < 0 and x > 0 */ return gamma_inc_CF(a, x, result); } else if(fabs(a) < 0.5) { return gamma_inc_series(a, x, result); } else { /* a = fa + da; da >= 0 */ const double fa = floor(a); const double da = a - fa; gsl_sf_result g_da; const int stat_g_da = ( da > 0.0 ? gamma_inc_a_gt_0(da, x, &g_da) : GAMMA_INC_A_0(x, &g_da)); double alpha = da; double gax = g_da.val; /* Gamma(alpha-1,x) = 1/(alpha-1) (Gamma(a,x) - x^(alpha-1) e^-x) */ do { const double shift = exp(-x + (alpha-1.0)*log(x)); gax = (gax - shift) / (alpha - 1.0); alpha -= 1.0; } while(alpha > a); result->val = gax; result->err = 2.0*(1.0 + fabs(a))*GSL_DBL_EPSILON*fabs(gax); return stat_g_da; } }
/// Lower Pochhammer symbol. double pochhammer_l(double a, double x) { if (a == x) return std::numeric_limits<double>::infinity(); if (std::fmod(std::abs(a - x), M_PI) < 1.0e-12) return std::numeric_limits<double>::infinity(); if (std::fmod(std::abs(a), M_PI) < 1.0e-12) return std::numeric_limits<double>::infinity(); gsl_sf_result result_num; int stat_num = gsl_sf_gamma_e(std::abs(a - x), &result_num); gsl_sf_result result_den; int stat_den = gsl_sf_gamma_e(std::abs(a), &result_den); if (stat_num != GSL_SUCCESS && stat_den != GSL_SUCCESS) { std::ostringstream msg("Error in lpochhammer_l:"); msg << " a=" << a << " x=" << x; throw std::runtime_error(msg.str()); } else return result_num.val / result_den.val; }
/* series for small a and x, but not defined for a == 0 */ static int gamma_inc_series(double a, double x, gsl_sf_result * result) { gsl_sf_result Q; gsl_sf_result G; const int stat_Q = gamma_inc_Q_series(a, x, &Q); const int stat_G = gsl_sf_gamma_e(a, &G); result->val = Q.val * G.val; result->err = fabs(Q.val * G.err) + fabs(Q.err * G.val); result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); return GSL_ERROR_SELECT_2(stat_Q, stat_G); }
int gsl_sf_beta_e(const double x, const double y, gsl_sf_result * result) { if((x > 0 && y > 0) && x < 50.0 && y < 50.0) { /* Handle the easy case */ gsl_sf_result gx, gy, gxy; gsl_sf_gamma_e(x, &gx); gsl_sf_gamma_e(y, &gy); gsl_sf_gamma_e(x+y, &gxy); result->val = (gx.val*gy.val)/gxy.val; result->err = gx.err * fabs(gy.val/gxy.val); result->err += gy.err * fabs(gx.val/gxy.val); result->err += fabs((gx.val*gy.val)/(gxy.val*gxy.val)) * gxy.err; result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); return GSL_SUCCESS; } else if (isnegint(x) || isnegint(y)) { DOMAIN_ERROR(result); } else if (isnegint(x+y)) { /* infinity in the denominator */ result->val = 0.0; result->err = 0.0; return GSL_SUCCESS; } else { gsl_sf_result lb; double sgn; int stat_lb = gsl_sf_lnbeta_sgn_e(x, y, &lb, &sgn); if(stat_lb == GSL_SUCCESS) { int status = gsl_sf_exp_err_e(lb.val, lb.err, result); result->val *= sgn; return status; } else { result->val = 0.0; result->err = 0.0; return stat_lb; } } }
/// Non-normalized lower incomplete gamma functions. double gamma_l(double a, double x) { gsl_sf_result result; int stat = gsl_sf_gamma_e(a, &result); if (stat != GSL_SUCCESS) { std::ostringstream msg("Error in gamma_q:"); msg << " a=" << a << " x=" << x; throw std::runtime_error(msg.str()); } else return result.val - gamma_u(a, x); }
static int gamma_inc_a_gt_0(double a, double x, gsl_sf_result * result) { /* x > 0 and a > 0; use result for Q */ gsl_sf_result Q; gsl_sf_result G; const int stat_Q = gsl_sf_gamma_inc_Q_e(a, x, &Q); const int stat_G = gsl_sf_gamma_e(a, &G); result->val = G.val * Q.val; result->err = fabs(G.val * Q.err) + fabs(G.err * Q.val); result->err += 2.0*GSL_DBL_EPSILON * fabs(result->val); return GSL_ERROR_SELECT_2(stat_G, stat_Q); }
APLFLOAT PrimFnMonQuoteDotFisF (APLFLOAT aplFloatRht, LPPRIMSPEC lpPrimSpec) { int iRet; gsl_sf_result gsr = {0}; // Check for indeterminates: !N for integer N < 0 if (aplFloatRht < 0) { if (aplFloatRht EQ floor (aplFloatRht) || aplFloatRht EQ ceil (aplFloatRht)) return TranslateQuadICIndex (0, ICNDX_QDOTn, aplFloatRht, FALSE); } // End IF // Check for PosInfinity if (IsFltPosInfinity (aplFloatRht)) return fltPosInfinity; // Check for too large for GSL if ((aplFloatRht + 1) > GSL_SF_GAMMA_XMAX) RaiseException (EXCEPTION_DOMAIN_ERROR, 0, 0, NULL); // Use the GNU Scientific Library Gamma function iRet = gsl_sf_gamma_e (aplFloatRht + 1, &gsr); // Check the return code switch (iRet) { case GSL_SUCCESS: break; case GSL_FAILURE: case GSL_EDOM: case GSL_ERANGE: RaiseException (EXCEPTION_DOMAIN_ERROR, 0, 0, NULL); defstop break; } // End SWITCH return gsr.val; } // End PrimFnMonQuoteDotFisF
int gsl_sf_zeta_e(const double s, gsl_sf_result * result) { /* CHECK_POINTER(result) */ if(s == 1.0) { DOMAIN_ERROR(result); } else if(s >= 0.0) { return riemann_zeta_sgt0(s, result); } else { /* reflection formula, [Abramowitz+Stegun, 23.2.5] */ gsl_sf_result zeta_one_minus_s; const int stat_zoms = riemann_zeta1ms_slt0(s, &zeta_one_minus_s); const double sin_term = (fmod(s,2.0) == 0.0) ? 0.0 : sin(0.5*M_PI*fmod(s,4.0))/M_PI; if(sin_term == 0.0) { result->val = 0.0; result->err = 0.0; return GSL_SUCCESS; } else if(s > -170) { /* We have to be careful about losing digits * in calculating pow(2 Pi, s). The gamma * function is fine because we were careful * with that implementation. * We keep an array of (2 Pi)^(10 n). */ const double twopi_pow[18] = { 1.0, 9.589560061550901348e+007, 9.195966217409212684e+015, 8.818527036583869903e+023, 8.456579467173150313e+031, 8.109487671573504384e+039, 7.776641909496069036e+047, 7.457457466828644277e+055, 7.151373628461452286e+063, 6.857852693272229709e+071, 6.576379029540265771e+079, 6.306458169130020789e+087, 6.047615938853066678e+095, 5.799397627482402614e+103, 5.561367186955830005e+111, 5.333106466365131227e+119, 5.114214477385391780e+127, 4.904306689854036836e+135 }; const int n = floor((-s)/10.0); const double fs = s + 10.0*n; const double p = pow(2.0*M_PI, fs) / twopi_pow[n]; gsl_sf_result g; const int stat_g = gsl_sf_gamma_e(1.0-s, &g); result->val = p * g.val * sin_term * zeta_one_minus_s.val; result->err = fabs(p * g.val * sin_term) * zeta_one_minus_s.err; result->err += fabs(p * sin_term * zeta_one_minus_s.val) * g.err; result->err += GSL_DBL_EPSILON * (fabs(s)+2.0) * fabs(result->val); return GSL_ERROR_SELECT_2(stat_g, stat_zoms); } else { /* The actual zeta function may or may not * overflow here. But we have no easy way * to calculate it when the prefactor(s) * overflow. Trying to use log's and exp * is no good because we loose a couple * digits to the exp error amplification. * When we gather a little more patience, * we can implement something here. Until * then just give up. */ OVERFLOW_ERROR(result); } } }