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/* stdtrl.c
*
* Student's t distribution
*
*
*
* SYNOPSIS:
*
* long double p, t, stdtrl();
* int k;
*
* p = stdtrl( 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 < -1.6, 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 <= 100. The "domain" refers to t.
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -100,-1.6 10000 5.7e-18 9.8e-19
* IEEE -1.6,100 10000 3.8e-18 1.0e-19
*/
/* stdtril.c
*
* Functional inverse of Student's t distribution
*
*
*
* SYNOPSIS:
*
* long double p, t, stdtril();
* int k;
*
* t = stdtril( k, p );
*
*
* DESCRIPTION:
*
* Given probability p, finds the argument t such that stdtrl(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 0,1 3500 4.2e-17 4.1e-18
*/
/*
Cephes Math Library Release 2.3: January, 1995
Copyright 1984, 1995 by Stephen L. Moshier
*/
#include <math.h>
extern long double PIL, MACHEPL, MAXNUML;
#ifdef ANSIPROT
extern long double sqrtl ( long double );
extern long double atanl ( long double );
extern long double incbetl ( long double, long double, long double );
extern long double incbil ( long double, long double, long double );
extern long double fabsl ( long double );
#else
long double sqrtl(), atanl(), incbetl(), incbil(), fabsl();
#endif
long double stdtrl( k, t )
int k;
long double t;
{
long double x, rk, z, f, tz, p, xsqk;
int j;
if( k <= 0 )
{
mtherr( "stdtrl", DOMAIN );
return(0.0L);
}
if( t == 0.0L )
return( 0.5L );
if( t < -1.6L )
{
rk = k;
z = rk / (rk + t * t);
p = 0.5L * incbetl( 0.5L*rk, 0.5L, z );
return( p );
}
/* compute integral from -t to + t */
if( t < 0.0L )
x = -t;
else
x = t;
rk = k; /* degrees of freedom */
z = 1.0L + ( x * x )/rk;
/* test if k is odd or even */
if( (k & 1) != 0)
{
/* computation for odd k */
xsqk = x/sqrtl(rk);
p = atanl( xsqk );
if( k > 1 )
{
f = 1.0L;
tz = 1.0L;
j = 3;
while( (j<=(k-2)) && ( (tz/f) > MACHEPL ) )
{
tz *= (j-1)/( z * j );
f += tz;
j += 2;
}
p += f * xsqk/z;
}
p *= 2.0L/PIL;
}
else
{
/* computation for even k */
f = 1.0L;
tz = 1.0L;
j = 2;
while( ( j <= (k-2) ) && ( (tz/f) > MACHEPL ) )
{
tz *= (j - 1)/( z * j );
f += tz;
j += 2;
}
p = f * x/sqrtl(z*rk);
}
/* common exit */
if( t < 0.0L )
p = -p; /* note destruction of relative accuracy */
p = 0.5L + 0.5L * p;
return(p);
}
long double stdtril( k, p )
int k;
long double p;
{
long double t, rk, z;
int rflg;
if( k <= 0 || p <= 0.0L || p >= 1.0L )
{
mtherr( "stdtril", DOMAIN );
return(0.0L);
}
rk = k;
if( p > 0.25L && p < 0.75L )
{
if( p == 0.5L )
return( 0.0L );
z = 1.0L - 2.0L * p;
z = incbil( 0.5L, 0.5L*rk, fabsl(z) );
t = sqrtl( rk*z/(1.0L-z) );
if( p < 0.5L )
t = -t;
return( t );
}
rflg = -1;
if( p >= 0.5L)
{
p = 1.0L - p;
rflg = 1;
}
z = incbil( 0.5L*rk, 0.5L, 2.0L*p );
if( MAXNUML * z < rk )
return(rflg* MAXNUML);
t = sqrtl( rk/z - rk );
return( rflg * t );
}
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