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kfunc.c

#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);
}

/* The following computes regularized incomplete gamma functions.
 * Formulas are taken from Wiki, with additional input from Numerical
 * Recipes in C (for modified Lentz's algorithm) and AS245
 * (http://lib.stat.cmu.edu/apstat/245).
 *
 * A good online calculator is available at:
 *
 *   http://www.danielsoper.com/statcalc/calc23.aspx
 *
 * It calculates upper incomplete gamma function, which equals
 * kf_gammaq(s,z)*tgamma(s).
 */

#define KF_GAMMA_EPS 1e-14
#define KF_TINY 1e-290

// regularized lower incomplete gamma function, by series expansion
static double _kf_gammap(double s, double z)
{
      double sum, x;
      int k;
      for (k = 1, sum = x = 1.; k < 100; ++k) {
            sum += (x *= z / (s + k));
            if (x / sum < KF_GAMMA_EPS) break;
      }
      return exp(s * log(z) - z - kf_lgamma(s + 1.) + log(sum));
}
// regularized upper incomplete gamma function, by continued fraction
static double _kf_gammaq(double s, double z)
{
      int j;
      double C, D, f;
      f = 1. + z - s; C = f; D = 0.;
      // Modified Lentz's algorithm for computing continued fraction
      // See Numerical Recipes in C, 2nd edition, section 5.2
      for (j = 1; j < 100; ++j) {
            double a = j * (s - j), b = (j<<1) + 1 + z - s, d;
            D = b + a * D;
            if (D < KF_TINY) D = KF_TINY;
            C = b + a / C;
            if (C < KF_TINY) C = KF_TINY;
            D = 1. / D;
            d = C * D;
            f *= d;
            if (fabs(d - 1.) < KF_GAMMA_EPS) break;
      }
      return exp(s * log(z) - z - kf_lgamma(s) - log(f));
}

double kf_gammap(double s, double z)
{
      return z <= 1. || z < s? _kf_gammap(s, z) : 1. - _kf_gammaq(s, z);
}

double kf_gammaq(double s, double z)
{
      return z <= 1. || z < s? 1. - _kf_gammap(s, z) : _kf_gammaq(s, z);
}

/* Regularized incomplete beta function. The method is taken from
 * Numerical Recipe in C, 2nd edition, section 6.4. The following web
 * page calculates the incomplete beta function, which equals
 * kf_betai(a,b,x) * gamma(a) * gamma(b) / gamma(a+b):
 *
 *   http://www.danielsoper.com/statcalc/calc36.aspx
 */
static double kf_betai_aux(double a, double b, double x)
{
      double C, D, f;
      int j;
      if (x == 0.) return 0.;
      if (x == 1.) return 1.;
      f = 1.; C = f; D = 0.;
      // Modified Lentz's algorithm for computing continued fraction
      for (j = 1; j < 200; ++j) {
            double aa, d;
            int m = j>>1;
            aa = (j&1)? -(a + m) * (a + b + m) * x / ((a + 2*m) * (a + 2*m + 1))
                  : m * (b - m) * x / ((a + 2*m - 1) * (a + 2*m));
            D = 1. + aa * D;
            if (D < KF_TINY) D = KF_TINY;
            C = 1. + aa / C;
            if (C < KF_TINY) C = KF_TINY;
            D = 1. / D;
            d = C * D;
            f *= d;
            if (fabs(d - 1.) < KF_GAMMA_EPS) break;
      }
      return exp(kf_lgamma(a+b) - kf_lgamma(a) - kf_lgamma(b) + a * log(x) + b * log(1.-x)) / a / f;
}
double kf_betai(double a, double b, double x)
{
      return x < (a + 1.) / (a + b + 2.)? kf_betai_aux(a, b, x) : 1. - kf_betai_aux(b, a, 1. - x);
}

#ifdef KF_MAIN
#include <stdio.h>
int main(int argc, char *argv[])
{
      double x = 5.5, y = 3;
      double a, b;
      printf("erfc(%lg): %lg, %lg\n", x, erfc(x), kf_erfc(x));
      printf("upper-gamma(%lg,%lg): %lg\n", x, y, kf_gammaq(y, x)*tgamma(y));
      a = 2; b = 2; x = 0.5;
      printf("incomplete-beta(%lg,%lg,%lg): %lg\n", a, b, x, kf_betai(a, b, x) / exp(kf_lgamma(a+b) - kf_lgamma(a) - kf_lgamma(b)));
      return 0;
}
#endif

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