* Special functions: log(gamma()), erfc(), P(a,x) (incomplete gamma)

 * Not using Numerical Recipe due to licensing issues

diff --git a/bcftools/kfunc.c b/bcftools/kfunc.c
new file mode 100644 (file)
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+++ b/bcftools/kfunc.c
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+#include <math.h>
+
+
+/* Log gamma function
+ * \log{\Gamma(z)}
+ * AS245, 2nd algorithm, http://lib.stat.cmu.edu/apstat/245
+ */
+double kf_lgamma(double z)
+{
+       double x = 0;
+       x += 0.1659470187408462e-06 / (z+7);
+       x += 0.9934937113930748e-05 / (z+6);
+       x -= 0.1385710331296526     / (z+5);
+       x += 12.50734324009056      / (z+4);
+       x -= 176.6150291498386      / (z+3);
+       x += 771.3234287757674      / (z+2);
+       x -= 1259.139216722289      / (z+1);
+       x += 676.5203681218835      / z;
+       x += 0.9999999999995183;
+       return log(x) - 5.58106146679532777 - z + (z-0.5) * log(z+6.5);
+}
+
+/* complementary error function
+ * \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2} dt
+ * AS66, 2nd algorithm, http://lib.stat.cmu.edu/apstat/66
+ */
+double kf_erfc(double x)
+{
+       const double p0 = 220.2068679123761;
+       const double p1 = 221.2135961699311;
+       const double p2 = 112.0792914978709;
+       const double p3 = 33.912866078383;
+       const double p4 = 6.37396220353165;
+       const double p5 = .7003830644436881;
+       const double p6 = .03526249659989109;
+       const double q0 = 440.4137358247522;
+       const double q1 = 793.8265125199484;
+       const double q2 = 637.3336333788311;
+       const double q3 = 296.5642487796737;
+       const double q4 = 86.78073220294608;
+       const double q5 = 16.06417757920695;
+       const double q6 = 1.755667163182642;
+       const double q7 = .08838834764831844;
+       double expntl, z, p;
+       z = fabs(x) * M_SQRT2;
+       if (z > 37.) return x > 0.? 0. : 2.;
+       expntl = exp(z * z * - .5);
+       if (z < 10. / M_SQRT2) // for small z
+           p = expntl * ((((((p6 * z + p5) * z + p4) * z + p3) * z + p2) * z + p1) * z + p0)
+                       / (((((((q7 * z + q6) * z + q5) * z + q4) * z + q3) * z + q2) * z + q1) * z + q0);
+       else p = expntl / 2.506628274631001 / (z + 1. / (z + 2. / (z + 3. / (z + 4. / (z + .65)))));
+       return x > 0.? 2. * p : 2. * (1. - p);
+}
+
+/* Regularized (incomplete lower) gamma function
+ * \frac{\gamma(p,x)}{\Gamma(p)}=\frac{1}{\Gamma(p)} \int_0^x t^{p-1}e^{-t} dt
+ * AS245, http://lib.stat.cmu.edu/apstat/245
+ */
+double kf_gammap(double p, double x)
+{
+    double ret_val;
+    double a, b, c, an, rn, pn1, pn2, pn3, pn4, pn5, pn6, arg;
+
+       if (x == 0.) return 0.;
+       // The following line is not thoroughly tested, so it is commented out.
+       if (p > 1e3) return .5 * kf_erfc(-M_SQRT1_2 * sqrt(p) * 3. * (pow(x / p, 1./3.) + 1. / (p * 9.) - 1.));
+       if (x > 1e8) return 1.;
+       if (x <= 1. || x < p) { // series expansion
+               c = 1.;
+               arg = p * log(x) - x - kf_lgamma(p + 1.);
+               ret_val = 1.;
+               a = p;
+               while (c > 1e-14) {
+                       a += 1.;
+                       c = c * x / a;
+                       ret_val += c;
+               }
+               arg += log(ret_val);
+               ret_val = 0.;
+               if (arg >= -88.) ret_val = exp(arg);
+       } else { // continued expansion
+               arg = p * log(x) - x - kf_lgamma(p);
+               a = 1. - p;
+               b = a + x + 1.;
+               c = 0.;
+               pn1 = 1.;
+               pn2 = x;
+               pn3 = x + 1.;
+               pn4 = x * b;
+               ret_val = pn3 / pn4;
+               while (1) {
+                       a += 1.; b += 2.; c += 1.;
+                       an = a * c;
+                       pn5 = b * pn3 - an * pn1;
+                       pn6 = b * pn4 - an * pn2;
+                       if (fabs(pn6) > 0.) {
+                               rn = pn5 / pn6;
+                               if (fabs(ret_val - rn) <= fmin(1e-14, rn * 1e-14)) break;
+                               ret_val = rn;
+                       }
+                       pn1 = pn3; pn2 = pn4; pn3 = pn5; pn4 = pn6;
+                       if (fabs(pn5) >= 1e37)
+                               pn1 /= 1e37, pn2 /= 1e37, pn3 /= 1e37, pn4 /= 1e37;
+               }
+               arg += log(ret_val);
+               ret_val = 1.;
+               if (arg >= -88.) ret_val = 1. - exp(arg);
+    }
+       return ret_val;
+}
+
+/* Numerical Recipe separates series expansion and continued
+ * expansion. This may potentially reduce underflow for some
+ * combinations of p and x. Nonetheless, the precision here is good
+ * enough for me. I will not spend more time for now.
+ */
+double kf_gammaq(double p, double x)
+{
+       return 1. - kf_gammap(p, x);
+}
+
+#ifdef KF_MAIN
+#include <stdio.h>
+int main(int argc, char *argv[])
+{
+       double x = 10, y = 2.5;
+       printf("erfc(%lg): %lg, %lg\n", x, erfc(x), kf_erfc(x));
+       printf("lower-gamma(%lg,%lg): %lg\n", x, y, (1.0-kf_gammap(y, x))*tgamma(y));
+       return 0;
+}
+#endif