/* the full definition of C_L(eta) for any valid L and eta
* [Abramowitz and Stegun 14.1.7]
* This depends on the complex gamma function. For large
* arguments the phase of the complex gamma function is not
* very accurately determined. However the modulus is, and that
* is all that we need to calculate C_L.
*
* This is not valid for L <= -3/2 or L = -1.
*/
static
int
CLeta(double L, double eta, gsl_sf_result * result)
{
gsl_sf_result ln1; /* log of numerator Gamma function */
gsl_sf_result ln2; /* log of denominator Gamma function */
double sgn = 1.0;
double arg_val, arg_err;
if(fabs(eta/(L+1.0)) < GSL_DBL_EPSILON) {
gsl_sf_lngamma_e(L+1.0, &ln1);
}
else {
gsl_sf_result p1; /* phase of numerator Gamma -- not used */
gsl_sf_lngamma_complex_e(L+1.0, eta, &ln1, &p1); /* should be ok */
}
gsl_sf_lngamma_e(2.0*(L+1.0), &ln2);
if(L < -1.0) sgn = -sgn;
arg_val = L*M_LN2 - 0.5*eta*M_PI + ln1.val - ln2.val;
arg_err = ln1.err + ln2.err;
arg_err += GSL_DBL_EPSILON * (fabs(L*M_LN2) + fabs(0.5*eta*M_PI));
return gsl_sf_exp_err_e(arg_val, arg_err, result);
}
/* The dominant part,
* D(a,x) := x^a e^(-x) / Gamma(a+1)
*/
static
int
gamma_inc_D(const double a, const double x, gsl_sf_result * result)
{
if(a < 10.0) {
double lnr;
gsl_sf_result lg;
gsl_sf_lngamma_e(a+1.0, &lg);
lnr = a * log(x) - x - lg.val;
result->val = exp(lnr);
result->err = 2.0 * GSL_DBL_EPSILON * (fabs(lnr) + 1.0) * fabs(result->val);
return GSL_SUCCESS;
}
else {
double mu = (x-a)/a;
double term1;
gsl_sf_result gstar;
gsl_sf_result ln_term;
gsl_sf_log_1plusx_mx_e(mu, &ln_term); /* log(1+mu) - mu */
gsl_sf_gammastar_e(a, &gstar);
term1 = exp(a*ln_term.val)/sqrt(2.0*M_PI*a);
result->val = term1/gstar.val;
result->err = 2.0 * GSL_DBL_EPSILON * (fabs(a*ln_term.val) + 1.0) * fabs(result->val);
result->err += gstar.err/fabs(gstar.val) * fabs(result->val);
return GSL_SUCCESS;
}
}
int
gsl_sf_bessel_lnKnu_e(const double nu, const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= 0.0 || nu < 0.0) {
DOMAIN_ERROR(result);
}
else if(nu == 0.0) {
gsl_sf_result K_scaled;
/* This cannot underflow, and
* it will not throw GSL_EDOM
* since that is already checked.
*/
gsl_sf_bessel_K0_scaled_e(x, &K_scaled);
result->val = -x + log(fabs(K_scaled.val));
result->err = GSL_DBL_EPSILON * fabs(x) + fabs(K_scaled.err/K_scaled.val);
result->err += GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x < 2.0 && nu > 1.0) {
/* Make use of the inequality
* Knu(x) <= 1/2 (2/x)^nu Gamma(nu),
* which follows from the integral representation
* [Abramowitz+Stegun, 9.6.23 (2)]. With this
* we decide whether or not there is an overflow
* problem because x is small.
*/
double ln_bound;
gsl_sf_result lg_nu;
gsl_sf_lngamma_e(nu, &lg_nu);
ln_bound = -M_LN2 - nu*log(0.5*x) + lg_nu.val;
if(ln_bound > GSL_LOG_DBL_MAX - 20.0) {
/* x must be very small or nu very large (or both).
*/
double xi = 0.25*x*x;
double sum = 1.0 - xi/(nu-1.0);
if(nu > 2.0) sum += (xi/(nu-1.0)) * (xi/(nu-2.0));
result->val = ln_bound + log(sum);
result->err = lg_nu.err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
/* can drop-through here */
}
{
/* We passed the above tests, so no problem.
* Evaluate as usual. Note the possible drop-through
* in the above code!
*/
gsl_sf_result K_scaled;
gsl_sf_bessel_Knu_scaled_e(nu, x, &K_scaled);
result->val = -x + log(fabs(K_scaled.val));
result->err = GSL_DBL_EPSILON * fabs(x) + fabs(K_scaled.err/K_scaled.val);
result->err += GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
}
/// Log lower Pochhammer symbol.
double
lpochhammer_l(double a, double x)
{
if (a == x)
return std::numeric_limits<double>::infinity();
gsl_sf_result result_num;
int stat_num = gsl_sf_lngamma_e(std::abs(a - x), &result_num);
gsl_sf_result result_den;
int stat_den = gsl_sf_lngamma_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;
}
int
gsl_sf_hyperg_2F1_conj_renorm_e(const double aR, const double aI, const double c,
const double x,
gsl_sf_result * result
)
{
const double rintc = floor(c + 0.5);
const double rinta = floor(aR + 0.5);
const int a_neg_integer = ( aR < 0.0 && fabs(aR-rinta) < locEPS && aI == 0.0);
const int c_neg_integer = ( c < 0.0 && fabs(c - rintc) < locEPS );
if(c_neg_integer) {
if(a_neg_integer && aR > c+0.1) {
/* 2F1 terminates early */
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else {
/* 2F1 does not terminate early enough, so something survives */
/* [Abramowitz+Stegun, 15.1.2] */
gsl_sf_result g1, g2;
gsl_sf_result g3;
gsl_sf_result a1, a2;
int stat = 0;
stat += gsl_sf_lngamma_complex_e(aR-c+1, aI, &g1, &a1);
stat += gsl_sf_lngamma_complex_e(aR, aI, &g2, &a2);
stat += gsl_sf_lngamma_e(-c+2.0, &g3);
if(stat != 0) {
DOMAIN_ERROR(result);
}
else {
gsl_sf_result F;
int stat_F = gsl_sf_hyperg_2F1_conj_e(aR-c+1, aI, -c+2, x, &F);
double ln_pre_val = 2.0*(g1.val - g2.val) - g3.val;
double ln_pre_err = 2.0 * (g1.err + g2.err) + g3.err;
int stat_e = gsl_sf_exp_mult_err_e(ln_pre_val, ln_pre_err,
F.val, F.err,
result);
return GSL_ERROR_SELECT_2(stat_e, stat_F);
}
}
}
else {
/* generic c */
gsl_sf_result F;
gsl_sf_result lng;
double sgn;
int stat_g = gsl_sf_lngamma_sgn_e(c, &lng, &sgn);
int stat_F = gsl_sf_hyperg_2F1_conj_e(aR, aI, c, x, &F);
int stat_e = gsl_sf_exp_mult_err_e(-lng.val, lng.err,
sgn*F.val, F.err,
result);
return GSL_ERROR_SELECT_3(stat_e, stat_F, stat_g);
}
}
/* asymptotic expansion
* j + 2.0 > 0.0
*/
static
int
fd_asymp(const double j, const double x, gsl_sf_result * result)
{
const int j_integer = ( fabs(j - floor(j+0.5)) < 100.0*GSL_DBL_EPSILON );
const int itmax = 200;
gsl_sf_result lg;
int stat_lg = gsl_sf_lngamma_e(j + 2.0, &lg);
double seqn_val = 0.5;
double seqn_err = 0.0;
double xm2 = (1.0/x)/x;
double xgam = 1.0;
double add = GSL_DBL_MAX;
double cos_term;
double ln_x;
double ex_term_1;
double ex_term_2;
gsl_sf_result fneg;
gsl_sf_result ex_arg;
gsl_sf_result ex;
int stat_fneg;
int stat_e;
int n;
for(n=1; n<=itmax; n++) {
double add_previous = add;
gsl_sf_result eta;
gsl_sf_eta_int_e(2*n, &eta);
xgam = xgam * xm2 * (j + 1.0 - (2*n-2)) * (j + 1.0 - (2*n-1));
add = eta.val * xgam;
if(!j_integer && fabs(add) > fabs(add_previous)) break;
if(fabs(add/seqn_val) < GSL_DBL_EPSILON) break;
seqn_val += add;
seqn_err += 2.0 * GSL_DBL_EPSILON * fabs(add);
}
seqn_err += fabs(add);
stat_fneg = fd_neg(j, -x, &fneg);
ln_x = log(x);
ex_term_1 = (j+1.0)*ln_x;
ex_term_2 = lg.val;
ex_arg.val = ex_term_1 - ex_term_2; /*(j+1.0)*ln_x - lg.val; */
ex_arg.err = GSL_DBL_EPSILON*(fabs(ex_term_1) + fabs(ex_term_2)) + lg.err;
stat_e = gsl_sf_exp_err_e(ex_arg.val, ex_arg.err, &ex);
cos_term = cos(j*M_PI);
result->val = cos_term * fneg.val + 2.0 * seqn_val * ex.val;
result->err = fabs(2.0 * ex.err * seqn_val);
result->err += fabs(2.0 * ex.val * seqn_err);
result->err += fabs(cos_term) * fneg.err;
result->err += 4.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_3(stat_e, stat_fneg, stat_lg);
}
/* Calculate series for small eta*lambda.
* Assumes eta > 0, lambda != 0.
*
* This is just the defining hypergeometric for the Legendre function.
*
* P^{mu}_{-1/2 + I lam}(z) = 1/Gamma(l+3/2) ((z+1)/(z-1)^(mu/2)
* 2F1(1/2 - I lam, 1/2 + I lam; l+3/2; (1-z)/2)
* We use
* z = cosh(eta)
* (z-1)/2 = sinh^2(eta/2)
*
* And recall
* H3d = sqrt(Pi Norm /(2 lam^2 sinh(eta))) P^{-l-1/2}_{-1/2 + I lam}(cosh(eta))
*/
static
int
legendre_H3d_series(const int ell, const double lambda, const double eta,
gsl_sf_result * result)
{
const int nmax = 5000;
const double shheta = sinh(0.5*eta);
const double ln_zp1 = M_LN2 + log(1.0 + shheta*shheta);
const double ln_zm1 = M_LN2 + 2.0*log(shheta);
const double zeta = -shheta*shheta;
gsl_sf_result lg_lp32;
double term = 1.0;
double sum = 1.0;
double sum_err = 0.0;
gsl_sf_result lnsheta;
double lnN;
double lnpre_val, lnpre_err, lnprepow;
int stat_e;
int n;
gsl_sf_lngamma_e(ell + 3.0/2.0, &lg_lp32);
gsl_sf_lnsinh_e(eta, &lnsheta);
legendre_H3d_lnnorm(ell, lambda, &lnN);
lnprepow = 0.5*(ell + 0.5) * (ln_zm1 - ln_zp1);
lnpre_val = lnprepow + 0.5*(lnN + M_LNPI - M_LN2 - lnsheta.val) - lg_lp32.val - log(fabs(lambda));
lnpre_err = lnsheta.err + lg_lp32.err + GSL_DBL_EPSILON * fabs(lnpre_val);
lnpre_err += 2.0*GSL_DBL_EPSILON * (fabs(lnN) + M_LNPI + M_LN2);
lnpre_err += 2.0*GSL_DBL_EPSILON * (0.5*(ell + 0.5) * (fabs(ln_zm1) + fabs(ln_zp1)));
for(n=1; n<nmax; n++) {
double aR = n - 0.5;
term *= (aR*aR + lambda*lambda)*zeta/(ell + n + 0.5)/n;
sum += term;
sum_err += 2.0*GSL_DBL_EPSILON*fabs(term);
if(fabs(term/sum) < 2.0 * GSL_DBL_EPSILON) break;
}
stat_e = gsl_sf_exp_mult_err_e(lnpre_val, lnpre_err, sum, fabs(term)+sum_err, result);
return GSL_ERROR_SELECT_2(stat_e, (n==nmax ? GSL_EMAXITER : GSL_SUCCESS));
}
int
gsl_sf_bessel_IJ_taylor_e(const double nu, const double x,
const int sign,
const int kmax,
const double threshold,
gsl_sf_result * result
)
{
/* CHECK_POINTER(result) */
if(nu < 0.0 || x < 0.0) {
DOMAIN_ERROR(result);
}
else if(x == 0.0) {
if(nu == 0.0) {
result->val = 1.0;
result->err = 0.0;
}
else {
result->val = 0.0;
result->err = 0.0;
}
return GSL_SUCCESS;
}
else {
gsl_sf_result prefactor; /* (x/2)^nu / Gamma(nu+1) */
gsl_sf_result sum;
int stat_pre;
int stat_sum;
int stat_mul;
if(nu == 0.0) {
prefactor.val = 1.0;
prefactor.err = 0.0;
stat_pre = GSL_SUCCESS;
}
else if(nu < INT_MAX-1) {
/* Separate the integer part and use
* y^nu / Gamma(nu+1) = y^N /N! y^f / (N+1)_f,
* to control the error.
*/
const int N = (int)floor(nu + 0.5);
const double f = nu - N;
gsl_sf_result poch_factor;
gsl_sf_result tc_factor;
const int stat_poch = gsl_sf_poch_e(N+1.0, f, &poch_factor);
const int stat_tc = gsl_sf_taylorcoeff_e(N, 0.5*x, &tc_factor);
const double p = pow(0.5*x,f);
prefactor.val = tc_factor.val * p / poch_factor.val;
prefactor.err = tc_factor.err * p / poch_factor.val;
prefactor.err += fabs(prefactor.val) / poch_factor.val * poch_factor.err;
prefactor.err += 2.0 * GSL_DBL_EPSILON * fabs(prefactor.val);
stat_pre = GSL_ERROR_SELECT_2(stat_tc, stat_poch);
}
else {
gsl_sf_result lg;
const int stat_lg = gsl_sf_lngamma_e(nu+1.0, &lg);
const double term1 = nu*log(0.5*x);
const double term2 = lg.val;
const double ln_pre = term1 - term2;
const double ln_pre_err = GSL_DBL_EPSILON * (fabs(term1)+fabs(term2)) + lg.err;
const int stat_ex = gsl_sf_exp_err_e(ln_pre, ln_pre_err, &prefactor);
stat_pre = GSL_ERROR_SELECT_2(stat_ex, stat_lg);
}
/* Evaluate the sum.
* [Abramowitz+Stegun, 9.1.10]
* [Abramowitz+Stegun, 9.6.7]
*/
{
const double y = sign * 0.25 * x*x;
double sumk = 1.0;
double term = 1.0;
int k;
for(k=1; k<=kmax; k++) {
term *= y/((nu+k)*k);
sumk += term;
if(fabs(term/sumk) < threshold) break;
}
sum.val = sumk;
sum.err = threshold * fabs(sumk);
stat_sum = ( k >= kmax ? GSL_EMAXITER : GSL_SUCCESS );
}
stat_mul = gsl_sf_multiply_err_e(prefactor.val, prefactor.err,
sum.val, sum.err,
result);
return GSL_ERROR_SELECT_3(stat_mul, stat_pre, stat_sum);
}
}
/* series of hypergeometric functions for integer j > 0, x > 0
* [Goano (7)]
*/
static
int
fd_UMseries_int(const int j, const double x, gsl_sf_result * result)
{
const int nmax = 2000;
double pre;
double lnpre_val;
double lnpre_err;
double sum_even_val = 1.0;
double sum_even_err = 0.0;
double sum_odd_val = 0.0;
double sum_odd_err = 0.0;
int stat_sum;
int stat_e;
int stat_h = GSL_SUCCESS;
int n;
if(x < 500.0 && j < 80) {
double p = gsl_sf_pow_int(x, j+1);
gsl_sf_result g;
gsl_sf_fact_e(j+1, &g); /* Gamma(j+2) */
lnpre_val = 0.0;
lnpre_err = 0.0;
pre = p/g.val;
}
else {
double lnx = log(x);
gsl_sf_result lg;
gsl_sf_lngamma_e(j + 2.0, &lg);
lnpre_val = (j+1.0)*lnx - lg.val;
lnpre_err = 2.0 * GSL_DBL_EPSILON * fabs((j+1.0)*lnx) + lg.err;
pre = 1.0;
}
/* Add up the odd terms of the sum.
*/
for(n=1; n<nmax; n+=2) {
double del_val;
double del_err;
gsl_sf_result U;
gsl_sf_result M;
int stat_h_U = gsl_sf_hyperg_U_int_e(1, j+2, n*x, &U);
int stat_h_F = gsl_sf_hyperg_1F1_int_e(1, j+2, -n*x, &M);
stat_h = GSL_ERROR_SELECT_3(stat_h, stat_h_U, stat_h_F);
del_val = ((j+1.0)*U.val - M.val);
del_err = (fabs(j+1.0)*U.err + M.err);
sum_odd_val += del_val;
sum_odd_err += del_err;
if(fabs(del_val/sum_odd_val) < GSL_DBL_EPSILON) break;
}
/* Add up the even terms of the sum.
*/
for(n=2; n<nmax; n+=2) {
double del_val;
double del_err;
gsl_sf_result U;
gsl_sf_result M;
int stat_h_U = gsl_sf_hyperg_U_int_e(1, j+2, n*x, &U);
int stat_h_F = gsl_sf_hyperg_1F1_int_e(1, j+2, -n*x, &M);
stat_h = GSL_ERROR_SELECT_3(stat_h, stat_h_U, stat_h_F);
del_val = ((j+1.0)*U.val - M.val);
del_err = (fabs(j+1.0)*U.err + M.err);
sum_even_val -= del_val;
sum_even_err += del_err;
if(fabs(del_val/sum_even_val) < GSL_DBL_EPSILON) break;
}
stat_sum = ( n >= nmax ? GSL_EMAXITER : GSL_SUCCESS );
stat_e = gsl_sf_exp_mult_err_e(lnpre_val, lnpre_err,
pre*(sum_even_val + sum_odd_val),
pre*(sum_even_err + sum_odd_err),
result);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_3(stat_e, stat_h, stat_sum);
}
/* Assumes a>0 and a+x>0.
*/
static
int
lnpoch_pos(const double a, const double x, gsl_sf_result * result)
{
double absx = fabs(x);
if(absx > 0.1*a || absx*log(GSL_MAX_DBL(a,2.0)) > 0.1) {
if(a < GSL_SF_GAMMA_XMAX && a+x < GSL_SF_GAMMA_XMAX) {
/* If we can do it by calculating the gamma functions
* directly, then that will be more accurate than
* doing the subtraction of the logs.
*/
gsl_sf_result g1;
gsl_sf_result g2;
gsl_sf_gammainv_e(a, &g1);
gsl_sf_gammainv_e(a+x, &g2);
result->val = -log(g2.val/g1.val);
result->err = g1.err/fabs(g1.val) + g2.err/fabs(g2.val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
/* Otherwise we must do the subtraction.
*/
gsl_sf_result lg1;
gsl_sf_result lg2;
int stat_1 = gsl_sf_lngamma_e(a, &lg1);
int stat_2 = gsl_sf_lngamma_e(a+x, &lg2);
result->val = lg2.val - lg1.val;
result->err = lg2.err + lg1.err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_1, stat_2);
}
}
else if(absx < 0.1*a && a > 15.0) {
/* Be careful about the implied subtraction.
* Note that both a+x and and a must be
* large here since a is not small
* and x is not relatively large.
* So we calculate using Stirling for Log[Gamma(z)].
*
* Log[Gamma(a+x)/Gamma(a)] = x(Log[a]-1) + (x+a-1/2)Log[1+x/a]
* + (1/(1+eps) - 1) / (12 a)
* - (1/(1+eps)^3 - 1) / (360 a^3)
* + (1/(1+eps)^5 - 1) / (1260 a^5)
* - (1/(1+eps)^7 - 1) / (1680 a^7)
* + ...
*/
const double eps = x/a;
const double den = 1.0 + eps;
const double d3 = den*den*den;
const double d5 = d3*den*den;
const double d7 = d5*den*den;
const double c1 = -eps/den;
const double c3 = -eps*(3.0+eps*(3.0+eps))/d3;
const double c5 = -eps*(5.0+eps*(10.0+eps*(10.0+eps*(5.0+eps))))/d5;
const double c7 = -eps*(7.0+eps*(21.0+eps*(35.0+eps*(35.0+eps*(21.0+eps*(7.0+eps))))))/d7;
const double p8 = gsl_sf_pow_int(1.0+eps,8);
const double c8 = 1.0/p8 - 1.0; /* these need not */
const double c9 = 1.0/(p8*(1.0+eps)) - 1.0; /* be very accurate */
const double a4 = a*a*a*a;
const double a6 = a4*a*a;
const double ser_1 = c1 + c3/(30.0*a*a) + c5/(105.0*a4) + c7/(140.0*a6);
const double ser_2 = c8/(99.0*a6*a*a) - 691.0/360360.0 * c9/(a6*a4);
const double ser = (ser_1 + ser_2)/ (12.0*a);
double term1 = x * log(a/M_E);
double term2;
gsl_sf_result ln_1peps;
gsl_sf_log_1plusx_e(eps, &ln_1peps); /* log(1 + x/a) */
term2 = (x + a - 0.5) * ln_1peps.val;
result->val = term1 + term2 + ser;
result->err = GSL_DBL_EPSILON*fabs(term1);
result->err += fabs((x + a - 0.5)*ln_1peps.err);
result->err += fabs(ln_1peps.val) * GSL_DBL_EPSILON * (fabs(x) + fabs(a) + 0.5);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
gsl_sf_result poch_rel;
int stat_p = pochrel_smallx(a, x, &poch_rel);
double eps = x*poch_rel.val;
int stat_e = gsl_sf_log_1plusx_e(eps, result);
result->err = 2.0 * fabs(x * poch_rel.err / (1.0 + eps));
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_e, stat_p);
}
}
int
gsl_sf_hyperg_U_large_b_e(const double a, const double b, const double x,
gsl_sf_result * result,
double * ln_multiplier
)
{
double N = floor(b); /* b = N + eps */
double eps = b - N;
if(fabs(eps) < GSL_SQRT_DBL_EPSILON) {
double lnpre_val;
double lnpre_err;
gsl_sf_result M;
if(b > 1.0) {
double tmp = (1.0-b)*log(x);
gsl_sf_result lg_bm1;
gsl_sf_result lg_a;
gsl_sf_lngamma_e(b-1.0, &lg_bm1);
gsl_sf_lngamma_e(a, &lg_a);
lnpre_val = tmp + x + lg_bm1.val - lg_a.val;
lnpre_err = lg_bm1.err + lg_a.err + GSL_DBL_EPSILON * (fabs(x) + fabs(tmp));
gsl_sf_hyperg_1F1_large_b_e(1.0-a, 2.0-b, -x, &M);
}
else {
gsl_sf_result lg_1mb;
gsl_sf_result lg_1pamb;
gsl_sf_lngamma_e(1.0-b, &lg_1mb);
gsl_sf_lngamma_e(1.0+a-b, &lg_1pamb);
lnpre_val = lg_1mb.val - lg_1pamb.val;
lnpre_err = lg_1mb.err + lg_1pamb.err;
gsl_sf_hyperg_1F1_large_b_e(a, b, x, &M);
}
if(lnpre_val > GSL_LOG_DBL_MAX-10.0) {
result->val = M.val;
result->err = M.err;
*ln_multiplier = lnpre_val;
GSL_ERROR ("overflow", GSL_EOVRFLW);
}
else {
gsl_sf_result epre;
int stat_e = gsl_sf_exp_err_e(lnpre_val, lnpre_err, &epre);
result->val = epre.val * M.val;
result->err = epre.val * M.err + epre.err * fabs(M.val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
*ln_multiplier = 0.0;
return stat_e;
}
}
else {
double omb_lnx = (1.0-b)*log(x);
gsl_sf_result lg_1mb;
double sgn_1mb;
gsl_sf_result lg_1pamb;
double sgn_1pamb;
gsl_sf_result lg_bm1;
double sgn_bm1;
gsl_sf_result lg_a;
double sgn_a;
gsl_sf_result M1, M2;
double lnpre1_val, lnpre2_val;
double lnpre1_err, lnpre2_err;
double sgpre1, sgpre2;
gsl_sf_hyperg_1F1_large_b_e( a, b, x, &M1);
gsl_sf_hyperg_1F1_large_b_e(1.0-a, 2.0-b, x, &M2);
gsl_sf_lngamma_sgn_e(1.0-b, &lg_1mb, &sgn_1mb);
gsl_sf_lngamma_sgn_e(1.0+a-b, &lg_1pamb, &sgn_1pamb);
gsl_sf_lngamma_sgn_e(b-1.0, &lg_bm1, &sgn_bm1);
gsl_sf_lngamma_sgn_e(a, &lg_a, &sgn_a);
lnpre1_val = lg_1mb.val - lg_1pamb.val;
lnpre1_err = lg_1mb.err + lg_1pamb.err;
lnpre2_val = lg_bm1.val - lg_a.val - omb_lnx - x;
lnpre2_err = lg_bm1.err + lg_a.err + GSL_DBL_EPSILON * (fabs(omb_lnx)+fabs(x));
sgpre1 = sgn_1mb * sgn_1pamb;
sgpre2 = sgn_bm1 * sgn_a;
if(lnpre1_val > GSL_LOG_DBL_MAX-10.0 || lnpre2_val > GSL_LOG_DBL_MAX-10.0) {
double max_lnpre_val = GSL_MAX(lnpre1_val,lnpre2_val);
double max_lnpre_err = GSL_MAX(lnpre1_err,lnpre2_err);
double lp1 = lnpre1_val - max_lnpre_val;
double lp2 = lnpre2_val - max_lnpre_val;
double t1 = sgpre1*exp(lp1);
double t2 = sgpre2*exp(lp2);
result->val = t1*M1.val + t2*M2.val;
result->err = fabs(t1)*M1.err + fabs(t2)*M2.err;
result->err += GSL_DBL_EPSILON * exp(max_lnpre_err) * (fabs(t1*M1.val) + fabs(t2*M2.val));
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
*ln_multiplier = max_lnpre_val;
GSL_ERROR ("overflow", GSL_EOVRFLW);
}
else {
double t1 = sgpre1*exp(lnpre1_val);
double t2 = sgpre2*exp(lnpre2_val);
result->val = t1*M1.val + t2*M2.val;
result->err = fabs(t1) * M1.err + fabs(t2)*M2.err;
result->err += GSL_DBL_EPSILON * (exp(lnpre1_err)*fabs(t1*M1.val) + exp(lnpre2_err)*fabs(t2*M2.val));
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
*ln_multiplier = 0.0;
return GSL_SUCCESS;
}
}
}
int
gsl_sf_hyperg_2F1_e(double a, double b, const double c,
const double x,
gsl_sf_result * result)
{
const double d = c - a - b;
const double rinta = floor(a + 0.5);
const double rintb = floor(b + 0.5);
const double rintc = floor(c + 0.5);
const int a_neg_integer = ( a < 0.0 && fabs(a - rinta) < locEPS );
const int b_neg_integer = ( b < 0.0 && fabs(b - rintb) < locEPS );
const int c_neg_integer = ( c < 0.0 && fabs(c - rintc) < locEPS );
result->val = 0.0;
result->err = 0.0;
/* Handle x == 1.0 RJM */
if (fabs (x - 1.0) < locEPS && (c - a - b) > 0 && c != 0 && !c_neg_integer) {
gsl_sf_result lngamc, lngamcab, lngamca, lngamcb;
double lngamc_sgn, lngamca_sgn, lngamcb_sgn;
int status;
int stat1 = gsl_sf_lngamma_sgn_e (c, &lngamc, &lngamc_sgn);
int stat2 = gsl_sf_lngamma_e (c - a - b, &lngamcab);
int stat3 = gsl_sf_lngamma_sgn_e (c - a, &lngamca, &lngamca_sgn);
int stat4 = gsl_sf_lngamma_sgn_e (c - b, &lngamcb, &lngamcb_sgn);
if (stat1 != GSL_SUCCESS || stat2 != GSL_SUCCESS
|| stat3 != GSL_SUCCESS || stat4 != GSL_SUCCESS)
{
DOMAIN_ERROR (result);
}
status =
gsl_sf_exp_err_e (lngamc.val + lngamcab.val - lngamca.val - lngamcb.val,
lngamc.err + lngamcab.err + lngamca.err + lngamcb.err,
result);
result->val *= lngamc_sgn / (lngamca_sgn * lngamcb_sgn);
return status;
}
if(x < -1.0 || 1.0 <= x) {
DOMAIN_ERROR(result);
}
if(c_neg_integer) {
/* If c is a negative integer, then either a or b must be a
negative integer of smaller magnitude than c to ensure
cancellation of the series. */
if(! (a_neg_integer && a > c + 0.1) && ! (b_neg_integer && b > c + 0.1)) {
DOMAIN_ERROR(result);
}
}
if(fabs(c-b) < locEPS || fabs(c-a) < locEPS) {
return pow_omx(x, d, result); /* (1-x)^(c-a-b) */
}
if(a >= 0.0 && b >= 0.0 && c >=0.0 && x >= 0.0 && x < 0.995) {
/* Series has all positive definite
* terms and x is not close to 1.
*/
return hyperg_2F1_series(a, b, c, x, result);
}
if(fabs(a) < 10.0 && fabs(b) < 10.0) {
/* a and b are not too large, so we attempt
* variations on the series summation.
*/
if(a_neg_integer) {
return hyperg_2F1_series(rinta, b, c, x, result);
}
if(b_neg_integer) {
return hyperg_2F1_series(a, rintb, c, x, result);
}
if(x < -0.25) {
return hyperg_2F1_luke(a, b, c, x, result);
}
else if(x < 0.5) {
return hyperg_2F1_series(a, b, c, x, result);
}
else {
if(fabs(c) > 10.0) {
return hyperg_2F1_series(a, b, c, x, result);
}
else {
return hyperg_2F1_reflect(a, b, c, x, result);
}
}
}
else {
/* Either a or b or both large.
* Introduce some new variables ap,bp so that bp is
* the larger in magnitude.
*/
double ap, bp;
if(fabs(a) > fabs(b)) {
bp = a;
ap = b;
}
else {
bp = b;
ap = a;
}
if(x < 0.0) {
/* What the hell, maybe Luke will converge.
*/
return hyperg_2F1_luke(a, b, c, x, result);
}
if(GSL_MAX_DBL(fabs(a),1.0)*fabs(bp)*fabs(x) < 2.0*fabs(c)) {
/* If c is large enough or x is small enough,
* we can attempt the series anyway.
*/
return hyperg_2F1_series(a, b, c, x, result);
}
if(fabs(bp*bp*x*x) < 0.001*fabs(bp) && fabs(a) < 10.0) {
/* The famous but nearly worthless "large b" asymptotic.
*/
int stat = gsl_sf_hyperg_1F1_e(a, c, bp*x, result);
result->err = 0.001 * fabs(result->val);
return stat;
}
/* We give up. */
result->val = 0.0;
result->err = 0.0;
GSL_ERROR ("error", GSL_EUNIMPL);
}
}
/* Do the reflection described in [Moshier, p. 334].
* Assumes a,b,c != neg integer.
*/
static
int
hyperg_2F1_reflect(const double a, const double b, const double c,
const double x, gsl_sf_result * result)
{
const double d = c - a - b;
const int intd = floor(d+0.5);
const int d_integer = ( fabs(d - intd) < locEPS );
if(d_integer) {
const double ln_omx = log(1.0 - x);
const double ad = fabs(d);
int stat_F2 = GSL_SUCCESS;
double sgn_2;
gsl_sf_result F1;
gsl_sf_result F2;
double d1, d2;
gsl_sf_result lng_c;
gsl_sf_result lng_ad2;
gsl_sf_result lng_bd2;
int stat_c;
int stat_ad2;
int stat_bd2;
if(d >= 0.0) {
d1 = d;
d2 = 0.0;
}
else {
d1 = 0.0;
d2 = d;
}
stat_ad2 = gsl_sf_lngamma_e(a+d2, &lng_ad2);
stat_bd2 = gsl_sf_lngamma_e(b+d2, &lng_bd2);
stat_c = gsl_sf_lngamma_e(c, &lng_c);
/* Evaluate F1.
*/
if(ad < GSL_DBL_EPSILON) {
/* d = 0 */
F1.val = 0.0;
F1.err = 0.0;
}
else {
gsl_sf_result lng_ad;
gsl_sf_result lng_ad1;
gsl_sf_result lng_bd1;
int stat_ad = gsl_sf_lngamma_e(ad, &lng_ad);
int stat_ad1 = gsl_sf_lngamma_e(a+d1, &lng_ad1);
int stat_bd1 = gsl_sf_lngamma_e(b+d1, &lng_bd1);
if(stat_ad1 == GSL_SUCCESS && stat_bd1 == GSL_SUCCESS && stat_ad == GSL_SUCCESS) {
/* Gamma functions in the denominator are ok.
* Proceed with evaluation.
*/
int i;
double sum1 = 1.0;
double term = 1.0;
double ln_pre1_val = lng_ad.val + lng_c.val + d2*ln_omx - lng_ad1.val - lng_bd1.val;
double ln_pre1_err = lng_ad.err + lng_c.err + lng_ad1.err + lng_bd1.err + GSL_DBL_EPSILON * fabs(ln_pre1_val);
int stat_e;
/* Do F1 sum.
*/
for(i=1; i<ad; i++) {
int j = i-1;
term *= (a + d2 + j) * (b + d2 + j) / (1.0 + d2 + j) / i * (1.0-x);
sum1 += term;
}
stat_e = gsl_sf_exp_mult_err_e(ln_pre1_val, ln_pre1_err,
sum1, GSL_DBL_EPSILON*fabs(sum1),
&F1);
if(stat_e == GSL_EOVRFLW) {
OVERFLOW_ERROR(result);
}
}
else {
/* Gamma functions in the denominator were not ok.
* So the F1 term is zero.
*/
F1.val = 0.0;
F1.err = 0.0;
}
} /* end F1 evaluation */
/* Evaluate F2.
*/
if(stat_ad2 == GSL_SUCCESS && stat_bd2 == GSL_SUCCESS) {
/* Gamma functions in the denominator are ok.
* Proceed with evaluation.
*/
const int maxiter = 2000;
double psi_1 = -M_EULER;
gsl_sf_result psi_1pd;
gsl_sf_result psi_apd1;
gsl_sf_result psi_bpd1;
int stat_1pd = gsl_sf_psi_e(1.0 + ad, &psi_1pd);
int stat_apd1 = gsl_sf_psi_e(a + d1, &psi_apd1);
int stat_bpd1 = gsl_sf_psi_e(b + d1, &psi_bpd1);
int stat_dall = GSL_ERROR_SELECT_3(stat_1pd, stat_apd1, stat_bpd1);
double psi_val = psi_1 + psi_1pd.val - psi_apd1.val - psi_bpd1.val - ln_omx;
double psi_err = psi_1pd.err + psi_apd1.err + psi_bpd1.err + GSL_DBL_EPSILON*fabs(psi_val);
double fact = 1.0;
double sum2_val = psi_val;
double sum2_err = psi_err;
double ln_pre2_val = lng_c.val + d1*ln_omx - lng_ad2.val - lng_bd2.val;
double ln_pre2_err = lng_c.err + lng_ad2.err + lng_bd2.err + GSL_DBL_EPSILON * fabs(ln_pre2_val);
int stat_e;
int j;
/* Do F2 sum.
*/
for(j=1; j<maxiter; j++) {
/* values for psi functions use recurrence; Abramowitz+Stegun 6.3.5 */
double term1 = 1.0/(double)j + 1.0/(ad+j);
double term2 = 1.0/(a+d1+j-1.0) + 1.0/(b+d1+j-1.0);
double delta = 0.0;
psi_val += term1 - term2;
psi_err += GSL_DBL_EPSILON * (fabs(term1) + fabs(term2));
fact *= (a+d1+j-1.0)*(b+d1+j-1.0)/((ad+j)*j) * (1.0-x);
delta = fact * psi_val;
sum2_val += delta;
sum2_err += fabs(fact * psi_err) + GSL_DBL_EPSILON*fabs(delta);
if(fabs(delta) < GSL_DBL_EPSILON * fabs(sum2_val)) break;
}
if(j == maxiter) stat_F2 = GSL_EMAXITER;
if(sum2_val == 0.0) {
F2.val = 0.0;
F2.err = 0.0;
}
else {
stat_e = gsl_sf_exp_mult_err_e(ln_pre2_val, ln_pre2_err,
sum2_val, sum2_err,
&F2);
if(stat_e == GSL_EOVRFLW) {
result->val = 0.0;
result->err = 0.0;
GSL_ERROR ("error", GSL_EOVRFLW);
}
}
stat_F2 = GSL_ERROR_SELECT_2(stat_F2, stat_dall);
}
else {
/* Gamma functions in the denominator not ok.
* So the F2 term is zero.
*/
F2.val = 0.0;
F2.err = 0.0;
} /* end F2 evaluation */
sgn_2 = ( GSL_IS_ODD(intd) ? -1.0 : 1.0 );
result->val = F1.val + sgn_2 * F2.val;
result->err = F1.err + F2. err;
result->err += 2.0 * GSL_DBL_EPSILON * (fabs(F1.val) + fabs(F2.val));
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return stat_F2;
}
else {
/* d not an integer */
gsl_sf_result pre1, pre2;
double sgn1, sgn2;
gsl_sf_result F1, F2;
int status_F1, status_F2;
/* These gamma functions appear in the denominator, so we
* catch their harmless domain errors and set the terms to zero.
*/
gsl_sf_result ln_g1ca, ln_g1cb, ln_g2a, ln_g2b;
double sgn_g1ca, sgn_g1cb, sgn_g2a, sgn_g2b;
int stat_1ca = gsl_sf_lngamma_sgn_e(c-a, &ln_g1ca, &sgn_g1ca);
int stat_1cb = gsl_sf_lngamma_sgn_e(c-b, &ln_g1cb, &sgn_g1cb);
int stat_2a = gsl_sf_lngamma_sgn_e(a, &ln_g2a, &sgn_g2a);
int stat_2b = gsl_sf_lngamma_sgn_e(b, &ln_g2b, &sgn_g2b);
int ok1 = (stat_1ca == GSL_SUCCESS && stat_1cb == GSL_SUCCESS);
int ok2 = (stat_2a == GSL_SUCCESS && stat_2b == GSL_SUCCESS);
gsl_sf_result ln_gc, ln_gd, ln_gmd;
double sgn_gc, sgn_gd, sgn_gmd;
gsl_sf_lngamma_sgn_e( c, &ln_gc, &sgn_gc);
gsl_sf_lngamma_sgn_e( d, &ln_gd, &sgn_gd);
gsl_sf_lngamma_sgn_e(-d, &ln_gmd, &sgn_gmd);
sgn1 = sgn_gc * sgn_gd * sgn_g1ca * sgn_g1cb;
sgn2 = sgn_gc * sgn_gmd * sgn_g2a * sgn_g2b;
if(ok1 && ok2) {
double ln_pre1_val = ln_gc.val + ln_gd.val - ln_g1ca.val - ln_g1cb.val;
double ln_pre2_val = ln_gc.val + ln_gmd.val - ln_g2a.val - ln_g2b.val + d*log(1.0-x);
double ln_pre1_err = ln_gc.err + ln_gd.err + ln_g1ca.err + ln_g1cb.err;
double ln_pre2_err = ln_gc.err + ln_gmd.err + ln_g2a.err + ln_g2b.err;
if(ln_pre1_val < GSL_LOG_DBL_MAX && ln_pre2_val < GSL_LOG_DBL_MAX) {
gsl_sf_exp_err_e(ln_pre1_val, ln_pre1_err, &pre1);
gsl_sf_exp_err_e(ln_pre2_val, ln_pre2_err, &pre2);
pre1.val *= sgn1;
pre2.val *= sgn2;
}
else {
OVERFLOW_ERROR(result);
}
}
else if(ok1 && !ok2) {
double ln_pre1_val = ln_gc.val + ln_gd.val - ln_g1ca.val - ln_g1cb.val;
double ln_pre1_err = ln_gc.err + ln_gd.err + ln_g1ca.err + ln_g1cb.err;
if(ln_pre1_val < GSL_LOG_DBL_MAX) {
gsl_sf_exp_err_e(ln_pre1_val, ln_pre1_err, &pre1);
pre1.val *= sgn1;
pre2.val = 0.0;
pre2.err = 0.0;
}
else {
OVERFLOW_ERROR(result);
}
}
else if(!ok1 && ok2) {
double ln_pre2_val = ln_gc.val + ln_gmd.val - ln_g2a.val - ln_g2b.val + d*log(1.0-x);
double ln_pre2_err = ln_gc.err + ln_gmd.err + ln_g2a.err + ln_g2b.err;
if(ln_pre2_val < GSL_LOG_DBL_MAX) {
pre1.val = 0.0;
pre1.err = 0.0;
gsl_sf_exp_err_e(ln_pre2_val, ln_pre2_err, &pre2);
pre2.val *= sgn2;
}
else {
OVERFLOW_ERROR(result);
}
}
else {
pre1.val = 0.0;
pre2.val = 0.0;
UNDERFLOW_ERROR(result);
}
status_F1 = hyperg_2F1_series( a, b, 1.0-d, 1.0-x, &F1);
status_F2 = hyperg_2F1_series(c-a, c-b, 1.0+d, 1.0-x, &F2);
result->val = pre1.val*F1.val + pre2.val*F2.val;
result->err = fabs(pre1.val*F1.err) + fabs(pre2.val*F2.err);
result->err += fabs(pre1.err*F1.val) + fabs(pre2.err*F2.val);
result->err += 2.0 * GSL_DBL_EPSILON * (fabs(pre1.val*F1.val) + fabs(pre2.val*F2.val));
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
}
