source: sasview/sansmodels/src/cephes/stdtr.c @ 476977b

/*                                                      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 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*/

#include "mconf.h"

extern double PI, MACHEP, MAXNUM;
#ifdef ANSIPROT
extern double sqrt ( double );
extern double atan ( double );
extern double incbet ( double, double, double );
extern double incbi ( double, double, double );
extern double fabs ( double );
#else
double sqrt(), atan(), incbet(), incbi(), fabs();
#endif

double stdtr( k, t )
int k;
double t;
{
double x, rk, z, f, tz, p, xsqk;
int j;

if( k <= 0 )
        {
        mtherr( "stdtr", DOMAIN );
        return(0.0);
        }

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/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( k, p )
int k;
double p;
{
double t, rk, z;
int rflg;

if( k <= 0 || p <= 0.0 || p >= 1.0 )
        {
        mtherr( "stdtri", DOMAIN );
        return(0.0);
        }

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( MAXNUM * z < rk )
        return(rflg* MAXNUM);
t = sqrt( rk/z - rk );
return( rflg * t );
}