int gsl_sf_eta_e(const double s, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(s > 100.0) {
result->val = 1.0;
result->err = GSL_DBL_EPSILON;
return GSL_SUCCESS;
}
else if(fabs(s-1.0) < 10.0*GSL_ROOT5_DBL_EPSILON) {
double del = s-1.0;
double c0 = M_LN2;
double c1 = M_LN2 * (M_EULER - 0.5*M_LN2);
double c2 = -0.0326862962794492996;
double c3 = 0.0015689917054155150;
double c4 = 0.00074987242112047532;
result->val = c0 + del * (c1 + del * (c2 + del * (c3 + del * c4)));
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
gsl_sf_result z;
gsl_sf_result p;
int stat_z = gsl_sf_zeta_e(s, &z);
int stat_p = gsl_sf_exp_e((1.0-s)*M_LN2, &p);
int stat_m = gsl_sf_multiply_e(1.0-p.val, z.val, result);
result->err = fabs(p.err * (M_LN2*(1.0-s)) * z.val) + z.err * fabs(p.val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_3(stat_m, stat_p, stat_z);
}
}
int
gsl_sf_hyperg_2F1_renorm_e(const double a, const double b, const double c,
const double x,
gsl_sf_result * result
)
{
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 );
if(c_neg_integer) {
if((a_neg_integer && a > c+0.1) || (b_neg_integer && b > 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, g3, g4, g5;
double s1, s2, s3, s4, s5;
int stat = 0;
stat += gsl_sf_lngamma_sgn_e(a-c+1, &g1, &s1);
stat += gsl_sf_lngamma_sgn_e(b-c+1, &g2, &s2);
stat += gsl_sf_lngamma_sgn_e(a, &g3, &s3);
stat += gsl_sf_lngamma_sgn_e(b, &g4, &s4);
stat += gsl_sf_lngamma_sgn_e(-c+2, &g5, &s5);
if(stat != 0) {
DOMAIN_ERROR(result);
}
else {
gsl_sf_result F;
int stat_F = gsl_sf_hyperg_2F1_e(a-c+1, b-c+1, -c+2, x, &F);
double ln_pre_val = g1.val + g2.val - g3.val - g4.val - g5.val;
double ln_pre_err = g1.err + g2.err + g3.err + g4.err + g5.err;
double sg = s1 * s2 * s3 * s4 * s5;
int stat_e = gsl_sf_exp_mult_err_e(ln_pre_val, ln_pre_err,
sg * 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_e(a, b, 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);
}
}
int gsl_sf_psi_n_e(const int n, const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(n == 0)
{
return gsl_sf_psi_e(x, result);
}
else if(n == 1)
{
return gsl_sf_psi_1_e(x, result);
}
else if(n < 0 || x <= 0.0) {
DOMAIN_ERROR(result);
}
else {
gsl_sf_result ln_nf;
gsl_sf_result hzeta;
int stat_hz = gsl_sf_hzeta_e(n+1.0, x, &hzeta);
int stat_nf = gsl_sf_lnfact_e((unsigned int) n, &ln_nf);
int stat_e = gsl_sf_exp_mult_err_e(ln_nf.val, ln_nf.err,
hzeta.val, hzeta.err,
result);
if(GSL_IS_EVEN(n)) result->val = -result->val;
return GSL_ERROR_SELECT_3(stat_e, stat_nf, stat_hz);
}
}
int gsl_sf_bessel_J0_e(const double x, gsl_sf_result * result)
{
double y = fabs(x);
/* CHECK_POINTER(result) */
if(y < 2.0*GSL_SQRT_DBL_EPSILON) {
result->val = 1.0;
result->err = y*y;
return GSL_SUCCESS;
}
else if(y <= 4.0) {
return cheb_eval_e(&bj0_cs, 0.125*y*y - 1.0, result);
}
else {
const double z = 32.0/(y*y) - 1.0;
gsl_sf_result ca;
gsl_sf_result ct;
gsl_sf_result cp;
const int stat_ca = cheb_eval_e(&_gsl_sf_bessel_amp_phase_bm0_cs, z, &ca);
const int stat_ct = cheb_eval_e(&_gsl_sf_bessel_amp_phase_bth0_cs, z, &ct);
const int stat_cp = gsl_sf_bessel_cos_pi4_e(y, ct.val/y, &cp);
const double sqrty = sqrt(y);
const double ampl = (0.75 + ca.val) / sqrty;
result->val = ampl * cp.val;
result->err = fabs(cp.val) * ca.err/sqrty + fabs(ampl) * cp.err;
result->err += GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_3(stat_ca, stat_ct, stat_cp);
}
}
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);
}
}
int gsl_sf_bessel_Y1_e(const double x, gsl_sf_result * result)
{
const double two_over_pi = 2.0/M_PI;
const double xmin = 1.571*GSL_DBL_MIN; /*exp ( amax1(alog(r1mach(1)), -alog(r1mach(2)))+.01) */
const double x_small = 2.0 * GSL_SQRT_DBL_EPSILON;
const double xmax = 1.0/GSL_DBL_EPSILON;
/* CHECK_POINTER(result) */
if(x <= 0.0) {
DOMAIN_ERROR(result);
}
else if(x < xmin) {
OVERFLOW_ERROR(result);
}
else if(x < x_small) {
const double lnterm = log(0.5*x);
gsl_sf_result J1;
gsl_sf_result c;
int status = gsl_sf_bessel_J1_e(x, &J1);
cheb_eval_e(&by1_cs, -1.0, &c);
result->val = two_over_pi * lnterm * J1.val + (0.5 + c.val)/x;
result->err = fabs(lnterm) * (fabs(GSL_DBL_EPSILON * J1.val) + J1.err) + c.err/x;
return status;
}
else if(x < 4.0) {
const double lnterm = log(0.5*x);
int status;
gsl_sf_result J1;
gsl_sf_result c;
cheb_eval_e(&by1_cs, 0.125*x*x-1.0, &c);
status = gsl_sf_bessel_J1_e(x, &J1);
result->val = two_over_pi * lnterm * J1.val + (0.5 + c.val)/x;
result->err = fabs(lnterm) * (fabs(GSL_DBL_EPSILON * J1.val) + J1.err) + c.err/x;
return status;
}
else if(x < xmax) {
const double z = 32.0/(x*x) - 1.0;
gsl_sf_result ca;
gsl_sf_result ct;
gsl_sf_result cp;
const int stat_ca = cheb_eval_e(&_gsl_sf_bessel_amp_phase_bm1_cs, z, &ca);
const int stat_ct = cheb_eval_e(&_gsl_sf_bessel_amp_phase_bth1_cs, z, &ct);
const int stat_cp = gsl_sf_bessel_cos_pi4_e(x, ct.val/x, &cp);
const double sqrtx = sqrt(x);
const double ampl = (0.75 + ca.val) / sqrtx;
result->val = -ampl * cp.val;
result->err = fabs(cp.val) * ca.err/sqrtx + fabs(ampl) * cp.err;
result->err += GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_3(stat_ca, stat_ct, stat_cp);
}
else {
UNDERFLOW_ERROR(result);
}
}
/* 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);
}
int gsl_sf_bessel_J1_e(const double x, gsl_sf_result * result)
{
double y = fabs(x);
/* CHECK_POINTER(result) */
if(y == 0.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(y < 2.0*GSL_DBL_MIN) {
UNDERFLOW_ERROR(result);
}
else if(y < ROOT_EIGHT * GSL_SQRT_DBL_EPSILON) {
result->val = 0.5*x;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(y < 4.0) {
gsl_sf_result c;
cheb_eval_e(&bj1_cs, 0.125*y*y-1.0, &c);
result->val = x * (0.25 + c.val);
result->err = fabs(x * c.err);
return GSL_SUCCESS;
}
else {
/* Because the leading term in the phase is y,
* which we assume is exactly known, the error
* in the cos() evaluation is bounded.
*/
const double z = 32.0/(y*y) - 1.0;
gsl_sf_result ca;
gsl_sf_result ct;
gsl_sf_result sp;
const int stat_ca = cheb_eval_e(&_gsl_sf_bessel_amp_phase_bm1_cs, z, &ca);
const int stat_ct = cheb_eval_e(&_gsl_sf_bessel_amp_phase_bth1_cs, z, &ct);
const int stat_sp = gsl_sf_bessel_sin_pi4_e(y, ct.val/y, &sp);
const double sqrty = sqrt(y);
const double ampl = (0.75 + ca.val) / sqrty;
result->val = (x < 0.0 ? -ampl : ampl) * sp.val;
result->err = fabs(sp.val) * ca.err/sqrty + fabs(ampl) * sp.err;
result->err += GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_3(stat_ca, stat_ct, stat_sp);
}
}
/* normalization for hydrogenic wave functions */
static
int
R_norm(const int n, const int l, const double Z, gsl_sf_result * result)
{
double A = 2.0*Z/n;
double pre = sqrt(A*A*A /(2.0*n));
gsl_sf_result ln_a, ln_b;
gsl_sf_result ex;
int stat_a = gsl_sf_lnfact_e(n+l, &ln_a);
int stat_b = gsl_sf_lnfact_e(n-l-1, &ln_b);
double diff_val = 0.5*(ln_b.val - ln_a.val);
double diff_err = 0.5*(ln_b.err + ln_a.err) + GSL_DBL_EPSILON * fabs(diff_val);
int stat_e = gsl_sf_exp_err_e(diff_val, diff_err, &ex);
result->val = pre * ex.val;
result->err = pre * ex.err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_3(stat_e, stat_a, stat_b);
}
/* generic polygamma; assumes n >= 0 and x > 0
*/
static int
psi_n_xg0(const int n, const double x, gsl_sf_result * result)
{
if(n == 0) {
return gsl_sf_psi_e(x, result);
}
else {
/* Abramowitz + Stegun 6.4.10 */
gsl_sf_result ln_nf;
gsl_sf_result hzeta;
int stat_hz = gsl_sf_hzeta_e(n+1.0, x, &hzeta);
int stat_nf = gsl_sf_lnfact_e((unsigned int) n, &ln_nf);
int stat_e = gsl_sf_exp_mult_err_e(ln_nf.val, ln_nf.err,
hzeta.val, hzeta.err,
result);
if(GSL_IS_EVEN(n)) result->val = -result->val;
return GSL_ERROR_SELECT_3(stat_e, stat_nf, stat_hz);
}
}
int gsl_sf_bessel_Y0_e(const double x, gsl_sf_result * result)
{
const double two_over_pi = 2.0/M_PI;
const double xmax = 1.0/GSL_DBL_EPSILON;
/* CHECK_POINTER(result) */
if (x <= 0.0) {
DOMAIN_ERROR(result);
}
else if(x < 4.0) {
gsl_sf_result J0;
gsl_sf_result c;
int stat_J0 = gsl_sf_bessel_J0_e(x, &J0);
cheb_eval_e(&by0_cs, 0.125*x*x-1.0, &c);
result->val = two_over_pi*(-M_LN2 + log(x))*J0.val + 0.375 + c.val;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val) + c.err;
return stat_J0;
}
else if(x < xmax) {
/* Leading behaviour of phase is x, which is exact,
* so the error is bounded.
*/
const double z = 32.0/(x*x) - 1.0;
gsl_sf_result c1;
gsl_sf_result c2;
gsl_sf_result sp;
const int stat_c1 = cheb_eval_e(&_gsl_sf_bessel_amp_phase_bm0_cs, z, &c1);
const int stat_c2 = cheb_eval_e(&_gsl_sf_bessel_amp_phase_bth0_cs, z, &c2);
const int stat_sp = gsl_sf_bessel_sin_pi4_e(x, c2.val/x, &sp);
const double sqrtx = sqrt(x);
const double ampl = (0.75 + c1.val) / sqrtx;
result->val = ampl * sp.val;
result->err = fabs(sp.val) * c1.err/sqrtx + fabs(ampl) * sp.err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_3(stat_sp, stat_c1, stat_c2);
}
else {
UNDERFLOW_ERROR(result);
}
}
int gsl_sf_eta_int_e(int n, gsl_sf_result * result)
{
if(n > ETA_POS_TABLE_NMAX) {
result->val = 1.0;
result->err = GSL_DBL_EPSILON;
return GSL_SUCCESS;
}
else if(n >= 0) {
result->val = eta_pos_int_table[n];
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
/* n < 0 */
if(!GSL_IS_ODD(n)) {
/* exactly zero at even negative integers */
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(n > -ETA_NEG_TABLE_NMAX) {
result->val = eta_neg_int_table[-(n+1)/2];
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
gsl_sf_result z;
gsl_sf_result p;
int stat_z = gsl_sf_zeta_int_e(n, &z);
int stat_p = gsl_sf_exp_e((1.0-n)*M_LN2, &p);
int stat_m = gsl_sf_multiply_e(-p.val, z.val, result);
result->err = fabs(p.err * (M_LN2*(1.0-n)) * z.val) + z.err * fabs(p.val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_3(stat_m, stat_p, stat_z);
}
}
}
/* 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;
}
}
int
gsl_sf_lnbeta_sgn_e(const double x, const double y, gsl_sf_result * result, double * sgn)
{
/* CHECK_POINTER(result) */
if(x == 0.0 || y == 0.0) {
*sgn = 0.0;
DOMAIN_ERROR(result);
} else if (isnegint(x) || isnegint(y)) {
*sgn = 0.0;
DOMAIN_ERROR(result); /* not defined for negative integers */
}
/* See if we can handle the postive case with min/max < 0.2 */
if (x > 0 && y > 0) {
const double max = GSL_MAX(x,y);
const double min = GSL_MIN(x,y);
const double rat = min/max;
if(rat < 0.2) {
/* min << max, so be careful
* with the subtraction
*/
double lnpre_val;
double lnpre_err;
double lnpow_val;
double lnpow_err;
double t1, t2, t3;
gsl_sf_result lnopr;
gsl_sf_result gsx, gsy, gsxy;
gsl_sf_gammastar_e(x, &gsx);
gsl_sf_gammastar_e(y, &gsy);
gsl_sf_gammastar_e(x+y, &gsxy);
gsl_sf_log_1plusx_e(rat, &lnopr);
lnpre_val = log(gsx.val*gsy.val/gsxy.val * M_SQRT2*M_SQRTPI);
lnpre_err = gsx.err/gsx.val + gsy.err/gsy.val + gsxy.err/gsxy.val;
t1 = min*log(rat);
t2 = 0.5*log(min);
t3 = (x+y-0.5)*lnopr.val;
lnpow_val = t1 - t2 - t3;
lnpow_err = GSL_DBL_EPSILON * (fabs(t1) + fabs(t2) + fabs(t3));
lnpow_err += fabs(x+y-0.5) * lnopr.err;
result->val = lnpre_val + lnpow_val;
result->err = lnpre_err + lnpow_err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
*sgn = 1.0;
return GSL_SUCCESS;
}
}
/* General case - Fallback */
{
gsl_sf_result lgx, lgy, lgxy;
double sgx, sgy, sgxy, xy = x+y;
int stat_gx = gsl_sf_lngamma_sgn_e(x, &lgx, &sgx);
int stat_gy = gsl_sf_lngamma_sgn_e(y, &lgy, &sgy);
int stat_gxy = gsl_sf_lngamma_sgn_e(xy, &lgxy, &sgxy);
*sgn = sgx * sgy * sgxy;
result->val = lgx.val + lgy.val - lgxy.val;
result->err = lgx.err + lgy.err + lgxy.err;
result->err += 2.0 * GSL_DBL_EPSILON * (fabs(lgx.val) + fabs(lgy.val) + fabs(lgxy.val));
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_3(stat_gx, stat_gy, stat_gxy);
}
}
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);
}
int
gsl_sf_beta_inc_e(
const double a,
const double b,
const double x,
gsl_sf_result * result
)
{
if(a <= 0.0 || b <= 0.0 || x < 0.0 || x > 1.0) {
DOMAIN_ERROR(result);
}
else if(x == 0.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(x == 1.0) {
result->val = 1.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else {
gsl_sf_result ln_beta;
gsl_sf_result ln_x;
gsl_sf_result ln_1mx;
gsl_sf_result prefactor;
const int stat_ln_beta = gsl_sf_lnbeta_e(a, b, &ln_beta);
const int stat_ln_1mx = gsl_sf_log_1plusx_e(-x, &ln_1mx);
const int stat_ln_x = gsl_sf_log_e(x, &ln_x);
const int stat_ln = GSL_ERROR_SELECT_3(stat_ln_beta, stat_ln_1mx, stat_ln_x);
const double ln_pre_val = -ln_beta.val + a * ln_x.val + b * ln_1mx.val;
const double ln_pre_err = ln_beta.err + fabs(a*ln_x.err) + fabs(b*ln_1mx.err);
const int stat_exp = gsl_sf_exp_err_e(ln_pre_val, ln_pre_err, &prefactor);
if(stat_ln != GSL_SUCCESS) {
result->val = 0.0;
result->err = 0.0;
GSL_ERROR ("error", GSL_ESANITY);
}
if(x < (a + 1.0)/(a+b+2.0)) {
/* Apply continued fraction directly. */
gsl_sf_result cf;
const int stat_cf = beta_cont_frac(a, b, x, &cf);
int stat;
result->val = prefactor.val * cf.val / a;
result->err = (fabs(prefactor.err * cf.val) + fabs(prefactor.val * cf.err))/a;
stat = GSL_ERROR_SELECT_2(stat_exp, stat_cf);
if(stat == GSL_SUCCESS) {
CHECK_UNDERFLOW(result);
}
return stat;
}
else {
/* Apply continued fraction after hypergeometric transformation. */
gsl_sf_result cf;
const int stat_cf = beta_cont_frac(b, a, 1.0-x, &cf);
int stat;
const double term = prefactor.val * cf.val / b;
result->val = 1.0 - term;
result->err = fabs(prefactor.err * cf.val)/b;
result->err += fabs(prefactor.val * cf.err)/b;
result->err += 2.0 * GSL_DBL_EPSILON * (1.0 + fabs(term));
stat = GSL_ERROR_SELECT_2(stat_exp, stat_cf);
if(stat == GSL_SUCCESS) {
CHECK_UNDERFLOW(result);
}
return stat;
}
}
}
