/* the L=0 normalization constant * [Abramowitz+Stegun 14.1.8] */ static double C0sq(double eta) { double twopieta = 2.0*M_PI*eta; if(fabs(eta) < GSL_DBL_EPSILON) { return 1.0; } else if(twopieta > GSL_LOG_DBL_MAX) { return 0.0; } else { gsl_sf_result scale; gsl_sf_expm1_e(twopieta, &scale); return twopieta/scale.val; } }
double gsl_sf_expm1(const double x) { EVAL_RESULT(gsl_sf_expm1_e(x, &result)); }
/* C When ABS(X) is so small that substantial cancellation will occur if C the straightforward formula is used, we use an expansion due C to Fields and discussed by Y. L. Luke, The Special Functions and Their C Approximations, Vol. 1, Academic Press, 1969, page 34. C C The ratio POCH(A,X) = GAMMA(A+X)/GAMMA(A) is written by Luke as C (A+(X-1)/2)**X * polynomial in (A+(X-1)/2)**(-2) . C In order to maintain significance in POCH1, we write for positive a C (A+(X-1)/2)**X = EXP(X*LOG(A+(X-1)/2)) = EXP(Q) C = 1.0 + Q*EXPREL(Q) . C Likewise the polynomial is written C POLY = 1.0 + X*POLY1(A,X) . C Thus, C POCH1(A,X) = (POCH(A,X) - 1) / X C = EXPREL(Q)*(Q/X + Q*POLY1(A,X)) + POLY1(A,X) C */ static int pochrel_smallx(const double a, const double x, gsl_sf_result * result) { /* SQTBIG = 1.0D0/SQRT(24.0D0*D1MACH(1)) ALNEPS = LOG(D1MACH(3)) */ const double SQTBIG = 1.0/(2.0*M_SQRT2*M_SQRT3*GSL_SQRT_DBL_MIN); const double ALNEPS = GSL_LOG_DBL_EPSILON - M_LN2; if(x == 0.0) { return gsl_sf_psi_e(a, result); } else { const double bp = ( (a < -0.5) ? 1.0-a-x : a ); const int incr = ( (bp < 10.0) ? 11.0-bp : 0 ); const double b = bp + incr; double dpoch1; gsl_sf_result dexprl; int stat_dexprl; int i; double var = b + 0.5*(x-1.0); double alnvar = log(var); double q = x*alnvar; double poly1 = 0.0; if(var < SQTBIG) { const int nterms = (int)(-0.5*ALNEPS/alnvar + 1.0); const double var2 = (1.0/var)/var; const double rho = 0.5 * (x + 1.0); double term = var2; double gbern[24]; int k, j; gbern[1] = 1.0; gbern[2] = -rho/12.0; poly1 = gbern[2] * term; if(nterms > 20) { /* NTERMS IS TOO BIG, MAYBE D1MACH(3) IS BAD */ /* nterms = 20; */ result->val = 0.0; result->err = 0.0; GSL_ERROR ("error", GSL_ESANITY); } for(k=2; k<=nterms; k++) { double gbk = 0.0; for(j=1; j<=k; j++) { gbk += bern[k-j+1]*gbern[j]; } gbern[k+1] = -rho*gbk/k; term *= (2*k-2-x)*(2*k-1-x)*var2; poly1 += gbern[k+1]*term; } } stat_dexprl = gsl_sf_expm1_e(q, &dexprl); if(stat_dexprl != GSL_SUCCESS) { result->val = 0.0; result->err = 0.0; return stat_dexprl; } dexprl.val = dexprl.val/q; poly1 *= (x - 1.0); dpoch1 = dexprl.val * (alnvar + q * poly1) + poly1; for(i=incr-1; i >= 0; i--) { /* C WE HAVE DPOCH1(B,X), BUT BP IS SMALL, SO WE USE BACKWARDS RECURSION C TO OBTAIN DPOCH1(BP,X). */ double binv = 1.0/(bp+i); dpoch1 = (dpoch1 - binv) / (1.0 + x*binv); } if(bp == a) { result->val = dpoch1; result->err = 2.0 * GSL_DBL_EPSILON * (fabs(incr) + 1.0) * fabs(result->val); return GSL_SUCCESS; } else { /* C WE HAVE DPOCH1(BP,X), BUT A IS LT -0.5. WE THEREFORE USE A C REFLECTION FORMULA TO OBTAIN DPOCH1(A,X). */ double sinpxx = sin(M_PI*x)/x; double sinpx2 = sin(0.5*M_PI*x); double t1 = sinpxx/tan(M_PI*b); double t2 = 2.0*sinpx2*(sinpx2/x); double trig = t1 - t2; result->val = dpoch1 * (1.0 + x*trig) + trig; result->err = (fabs(dpoch1*x) + 1.0) * GSL_DBL_EPSILON * (fabs(t1) + fabs(t2)); result->err += 2.0 * GSL_DBL_EPSILON * (fabs(incr) + 1.0) * fabs(result->val); return GSL_SUCCESS; } } }
/** * C++ version of gsl_sf_expm1_e(). * exp(x)-1 * @param x A real number * @param result The result as a @c gsl::sf::result object * @return GSL_SUCCESS or GSL_EOVRFLW */ inline int expm1_e( double const x, result& result ){ return gsl_sf_expm1_e( x, &result ); }