stdtr.c

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/*                                                     stdtr.c
 *
 *     Student's t distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * double t, stdtr();
 * short k;
 *
 * y = stdtr( k, t );
 *
 *
 * DESCRIPTION:
 *
 * Computes the integral from minus infinity to t of the Student
 * t distribution with integer k > 0 degrees of freedom:
 *
 *                                      t
 *                                      -
 *                                     | |
 *              -                      |         2   -(k+1)/2
 *             | ( (k+1)/2 )           |  (     x   )
 *       ----------------------        |  ( 1 + --- )        dx
 *                     -               |  (      k  )
 *       sqrt( k pi ) | ( k/2 )        |
 *                                   | |
 *                                    -
 *                                   -inf.
 *
 * Relation to incomplete beta integral:
 *
 *        1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z )
 * where
 *        z = k/(k + t**2).
 *
 * For t < -2, this is the method of computation.  For higher t,
 * a direct method is derived from integration by parts.
 * Since the function is symmetric about t=0, the area under the
 * right tail of the density is found by calling the function
 * with -t instead of t.
 *
 * ACCURACY:
 *
 * Tested at random 1 <= k <= 25.  The "domain" refers to t.
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE     -100,-2      50000       5.9e-15     1.4e-15
 *    IEEE     -2,100      500000       2.7e-15     4.9e-17
 */

/*                                                     stdtri.c
 *
 *     Functional inverse of Student's t distribution
 *
 *
 *
 * SYNOPSIS:
 *
 * double p, t, stdtri();
 * int k;
 *
 * t = stdtri( k, p );
 *
 *
 * DESCRIPTION:
 *
 * Given probability p, finds the argument t such that stdtr(k,t)
 * is equal to p.
 *
 * ACCURACY:
 *
 * Tested at random 1 <= k <= 100.  The "domain" refers to p:
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE    .001,.999     25000       5.7e-15     8.0e-16
 *    IEEE    10^-6,.001    25000       2.0e-12     2.9e-14
 */

 
/*
 * Cephes Math Library Release 2.3:  March, 1995
 * Copyright 1984, 1987, 1995 by Stephen L. Moshier
 */
 
#include "mconf.h"
#include <float.h>
 
extern double MACHEP;
 
double stdtr(int k, double t)
{
    double x, rk, z, f, tz, p, xsqk;
    int j;
 
    if (k <= 0) {
        sf_error("stdtr", SF_ERROR_DOMAIN, NULL);
        return (NAN);
    }
 
    if (t == 0)
        return (0.5);
 
    if (t < -2.0) {
        rk = k;
        z = rk / (rk + t * t);
        p = 0.5 * incbet(0.5 * rk, 0.5, z);
        return (p);
    }
 
    /*     compute integral from -t to + t */
 
    if (t < 0)
        x = -t;
    else
        x = t;
 
    rk = k;                     /* degrees of freedom */
    z = 1.0 + (x * x) / rk;
 
    /* test if k is odd or even */
    if ((k & 1) != 0) {
 
        /*      computation for odd k   */
 
        xsqk = x / sqrt(rk);
        p = atan(xsqk);
        if (k > 1) {
            f = 1.0;
            tz = 1.0;
            j = 3;
            while ((j <= (k - 2)) && ((tz / f) > MACHEP)) {
                tz *= (j - 1) / (z * j);
                f += tz;
                j += 2;
            }
            p += f * xsqk / z;
        }
        p *= 2.0 / M_PI;
    }
 
 
    else {
 
        /*      computation for even k  */
 
        f = 1.0;
        tz = 1.0;
        j = 2;
 
        while ((j <= (k - 2)) && ((tz / f) > MACHEP)) {
            tz *= (j - 1) / (z * j);
            f += tz;
            j += 2;
        }
        p = f * x / sqrt(z * rk);
    }
 
    /*     common exit     */
 
 
    if (t < 0)
        p = -p;                 /* note destruction of relative accuracy */
 
    p = 0.5 + 0.5 * p;
    return (p);
}
 
double stdtri(int k, double p)
{
    double t, rk, z;
    int rflg;
 
    if (k <= 0 || p <= 0.0 || p >= 1.0) {
        sf_error("stdtri", SF_ERROR_DOMAIN, NULL);
        return (NAN);
    }
 
    rk = k;
 
    if (p > 0.25 && p < 0.75) {
        if (p == 0.5)
            return (0.0);
        z = 1.0 - 2.0 * p;
        z = incbi(0.5, 0.5 * rk, fabs(z));
        t = sqrt(rk * z / (1.0 - z));
        if (p < 0.5)
            t = -t;
        return (t);
    }
    rflg = -1;
    if (p >= 0.5) {
        p = 1.0 - p;
        rflg = 1;
    }
    z = incbi(0.5 * rk, 0.5, 2.0 * p);
 
    if (DBL_MAX * z < rk)
        return (rflg * INFINITY);
    t = sqrt(rk / z - rk);
    return (rflg * t);
}