lib/studentbase.c

   1 #include "xmath.h"
   2
   3 #define TWOVRPI 0.636619772367581343
   4 #define HALF_PI 1.5707963268
   5 #define EPSILON .000001
   6
   7 extern double ppnd(), ppbeta();
   8
   9 /* CACM Algorithm 395, by G. W. Hill */
  10
  11 studentbase(x, df, cdf)
  12         double *x, *df, *cdf;
  13 {
  14   double t, y, b, a, z, j, n;
  15
  16   n = *df;
  17   z = 1.0;
  18   t = (*x) * (*x);
  19   y = t / n;
  20   b = 1.0 + y;
  21
  22   if (n > floor(n) || (n >= 20 && t < n) || (n > 20)) {
  23     if (n < 2.0 && n != 1.0) {
  24       /* beta integral aproximation for small df */
  25       double  da = 0.5, db = 0.5 * n, dx, dp;
  26       int ia = 0, ib = floor(db);
  27
  28       dx = db / (db + da * t);
  29       betabase(&dx, &db, &da, &ib, &ia, &dp);
  30       *cdf = (*x >= 0) ? 1.0 - .5 * dp : .5 * dp;
  31     }
  32     else {
  33       /* asymptotic series for large or non-integer df */
  34       if(y > EPSILON) y = log(b);
  35       a = n - 0.5;
  36       b = 48.0 * a * a;
  37       y = a * y;
  38       y = (((((-0.4 * y - 3.3) * y - 24.0) * y - 85.5)
  39             / (0.8 * y * y + 100.0 + b) + y + 3.0) / b + 1.0) * sqrt(y);
  40
  41       y = -1.0 * y;
  42       normbase(&y, cdf);
  43       if (*x > 0.0) *cdf = 1.0 - *cdf;
  44     }
  45   }
  46   else {
  47     /* nested summation of cosine series */
  48     if (n < 20 && t < 4.0) {
  49       a = y = sqrt(y);
  50       if(n == 1.0) a = 0.0;
  51     }
  52     else {
  53       a = sqrt(b);
  54       y = a * n;
  55       for(j = 2; fabs(a - z) > EPSILON; j += 2.0) {
  56         z = a;
  57         y = (y * (j - 1)) / (b * j);
  58         a = a + y / (n + j);
  59       }
  60       n += 2.0;
  61       z = y = 0.0;
  62       a = -a;
  63     }
  64     for(n = n - 2.0; n > 1.0; n -= 2.0) a = ((n - 1.0) / (b * n)) * a + y;
  65     a = (fabs(n) < EPSILON) ? a/sqrt(b) : TWOVRPI * (atan(y) + a / b);
  66     *cdf = z - a;
  67     if(*x > 0.0) *cdf = 1.0 - 0.5 * *cdf;
  68     else *cdf = 0.5 * *cdf;
  69   }
  70 }
  71
  72 /* CACM Algorithm 396, by G. W. Hill */
  73
  74 double ppstudent(pp, n, ifault)
  75      double pp, n;
  76      int *ifault;
  77 {
  78   double sq, p, a, b, c, d, x, y;
  79
  80   /* convert to double upper tailed probability */
  81   p = (pp < 0.5) ? 2.0 * pp : 2.0 * (1.0 - pp);
  82
  83   if (n <= 3.0) {
  84     if (n == 1) sq = tan(HALF_PI * (1.0 - p));
  85     else if (n == 2.0) sq = sqrt(2.0 / (p * (2.0 - p)) - 2.0);
  86     else {
  87       sq = ppbeta(p, 0.5 * n, 0.5, ifault);
  88       if (sq != 0.0) sq = sqrt((n / sq) - n);
  89     }
  90   }
  91   else {
  92     a = 1.0 / (n - 0.5);
  93     b = 48.0 / (a * a);
  94     c = ((20700.0 * a / b - 98.0) * a - 16) * a + 96.36;
  95     d = ((94.5 / (b + c) - 3.0) / b + 1.0) * sqrt(a * HALF_PI) * n;
  96     x = d * p;
  97     y = pow(x, 2.0 / n);
  98     if (y > 0.05 + a) {
  99       /* asymptotic inverse expansion about normal */
 100       x = ppnd(0.5 * p, ifault);
 101       y = x * x;
 102       if (n < 5) c = c + 0.3 * (n - 4.5) * (x + 0.6);
 103       c = (((0.05 * d * x - 5.0) * x - 7.0) * x - 2.0) * x + b + c;
 104       y = (((((0.4 * y + 6.3) * y + 36.0) * y + 94.5) / c - y - 3.0) / b
 105            + 1.0) * x;
 106       y = a * y * y;
 107       y = (y > .002) ? exp(y) - 1.0 : 0.5 * y * y + y;
 108     }
 109     else {
 110       y = ((1.0 / (((n + 6.0) / (n * y) - 0.089 * d - 0.822)
 111                    * (n + 2.0) * 3.0) + 0.5 / (n + 4.0)) * y - 1.0)
 112         * (n + 1.0) / (n + 2.0) + 1.0 / y;
 113     }
 114     sq = sqrt(n * y);
 115   }
 116
 117   /* decode based on tail */
 118   if (pp < 0.5) sq = -sq;
 119   return(sq);
 120 }