6 * AS245, 2nd algorithm, http://lib.stat.cmu.edu/apstat/245
8 double kf_lgamma(double z)
11 x += 0.1659470187408462e-06 / (z+7);
12 x += 0.9934937113930748e-05 / (z+6);
13 x -= 0.1385710331296526 / (z+5);
14 x += 12.50734324009056 / (z+4);
15 x -= 176.6150291498386 / (z+3);
16 x += 771.3234287757674 / (z+2);
17 x -= 1259.139216722289 / (z+1);
18 x += 676.5203681218835 / z;
19 x += 0.9999999999995183;
20 return log(x) - 5.58106146679532777 - z + (z-0.5) * log(z+6.5);
23 /* complementary error function
24 * \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2} dt
25 * AS66, 2nd algorithm, http://lib.stat.cmu.edu/apstat/66
27 double kf_erfc(double x)
29 const double p0 = 220.2068679123761;
30 const double p1 = 221.2135961699311;
31 const double p2 = 112.0792914978709;
32 const double p3 = 33.912866078383;
33 const double p4 = 6.37396220353165;
34 const double p5 = .7003830644436881;
35 const double p6 = .03526249659989109;
36 const double q0 = 440.4137358247522;
37 const double q1 = 793.8265125199484;
38 const double q2 = 637.3336333788311;
39 const double q3 = 296.5642487796737;
40 const double q4 = 86.78073220294608;
41 const double q5 = 16.06417757920695;
42 const double q6 = 1.755667163182642;
43 const double q7 = .08838834764831844;
45 z = fabs(x) * M_SQRT2;
46 if (z > 37.) return x > 0.? 0. : 2.;
47 expntl = exp(z * z * - .5);
48 if (z < 10. / M_SQRT2) // for small z
49 p = expntl * ((((((p6 * z + p5) * z + p4) * z + p3) * z + p2) * z + p1) * z + p0)
50 / (((((((q7 * z + q6) * z + q5) * z + q4) * z + q3) * z + q2) * z + q1) * z + q0);
51 else p = expntl / 2.506628274631001 / (z + 1. / (z + 2. / (z + 3. / (z + 4. / (z + .65)))));
52 return x > 0.? 2. * p : 2. * (1. - p);
55 /* Regularized (incomplete lower) gamma function
56 * \frac{\gamma(p,x)}{\Gamma(p)}=\frac{1}{\Gamma(p)} \int_0^x t^{p-1}e^{-t} dt
57 * AS245, http://lib.stat.cmu.edu/apstat/245
59 double kf_gammap(double p, double x)
62 double a, b, c, an, rn, pn1, pn2, pn3, pn4, pn5, pn6, arg;
64 if (x == 0.) return 0.;
65 // The following line is not thoroughly tested, so it is commented out.
66 if (p > 1e3) return .5 * kf_erfc(-M_SQRT1_2 * sqrt(p) * 3. * (pow(x / p, 1./3.) + 1. / (p * 9.) - 1.));
67 if (x > 1e8) return 1.;
68 if (x <= 1. || x < p) { // series expansion
70 arg = p * log(x) - x - kf_lgamma(p + 1.);
80 if (arg >= -88.) ret_val = exp(arg);
81 } else { // continued expansion
82 arg = p * log(x) - x - kf_lgamma(p);
92 a += 1.; b += 2.; c += 1.;
94 pn5 = b * pn3 - an * pn1;
95 pn6 = b * pn4 - an * pn2;
98 if (fabs(ret_val - rn) <= fmin(1e-14, rn * 1e-14)) break;
101 pn1 = pn3; pn2 = pn4; pn3 = pn5; pn4 = pn6;
102 if (fabs(pn5) >= 1e37)
103 pn1 /= 1e37, pn2 /= 1e37, pn3 /= 1e37, pn4 /= 1e37;
107 if (arg >= -88.) ret_val = 1. - exp(arg);
112 /* Numerical Recipe separates series expansion and continued
113 * expansion. This may potentially reduce underflow for some
114 * combinations of p and x. Nonetheless, the precision here is good
115 * enough for me. I will not spend more time for now.
117 double kf_gammaq(double p, double x)
119 return 1. - kf_gammap(p, x);
124 int main(int argc, char *argv[])
126 double x = 10, y = 2.5;
127 printf("erfc(%lg): %lg, %lg\n", x, erfc(x), kf_erfc(x));
128 printf("lower-gamma(%lg,%lg): %lg\n", x, y, (1.0-kf_gammap(y, x))*tgamma(y));