Category : C Source Code
Archive   : CEPHES22.ZIP
Filename : INCBI.C

/* incbi()
*
* Inverse of imcomplete beta integral
*
*
*
* SYNOPSIS:
*
* double a, b, x, y, incbi();
*
* x = incbi( a, b, y );
*
*
*
* DESCRIPTION:
*
* Given y, the function finds x such that
*
* incbet( a, b, x ) = y.
*
* the routine performs up to 10 Newton iterations to find the
* root of incbet(a,b,x) - y = 0.
*
*
* ACCURACY:
*
* Relative error:
* x a,b
* arithmetic domain domain # trials peak rms
* DEC 0,1 0,100 2500 2.9e-13 5.9e-15
* IEEE 0,1 0,100 5000 8.3e-13 1.7e-14
*
* Larger errors were observed for a near zero and b large.
*/


/*
Cephes Math Library Release 2.2: July, 1992
Copyright 1984, 1987, 1992 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/

#include "mconf.h"

extern double MACHEP, MAXNUM, MAXLOG, MINLOG;

double incbi( aa, bb, yy0 )
double aa, bb, yy0;
{
double a, b, y0;
double d, y, x, x0, x1, lgm, yp, di;
int i, n, rflg;
double incbet();
double ndtri(), exp(), fabs(), log(), sqrt(), lgam();


if( yy0 <= 0 )
return(0.0);
if( yy0 >= 1.0 )
return(1.0);

/* approximation to inverse function */

yp = -ndtri(yy0);

if( yy0 > 0.5 )
{
rflg = 1;
a = bb;
b = aa;
y0 = 1.0 - yy0;
yp = -yp;
}
else
{
rflg = 0;
a = aa;
b = bb;
y0 = yy0;
}


if( (aa <= 1.0) || (bb <= 1.0) )
{
y = yp * yp / 2.0;
}
else
{
lgm = (yp * yp - 3.0)/6.0;
x0 = 2.0/( 1.0/(2.0*a-1.0) + 1.0/(2.0*b-1.0) );
y = yp * sqrt( x0 + lgm ) / x0
- ( 1.0/(2.0*b-1.0) - 1.0/(2.0*a-1.0) )
* (lgm + 5.0/6.0 - 2.0/(3.0*x0));
y = 2.0 * y;
if( y < MINLOG )
{
x0 = 1.0;
goto under;
}
}

x = a/( a + b * exp(y) );
y = incbet( a, b, x );
yp = (y - y0)/y0;
if( fabs(yp) < 1.0e-1 )
goto newt;

/* Resort to interval halving if not close enough */
x0 = 0.0;
x1 = 1.0;
di = 0.5;

for( i=0; i<20; i++ )
{
if( i != 0 )
{
x = di * x1 + (1.0-di) * x0;
y = incbet( a, b, x );
yp = (y - y0)/y0;
if( fabs(yp) < 1.0e-3 )
goto newt;
}

if( y < y0 )
{
x0 = x;
di = 0.5;
}
else
{
x1 = x;
di *= di;
if( di == 0.0 )
di = 0.5;
}
}

if( x0 == 0.0 )
{
under:
mtherr( "incbi", UNDERFLOW );
goto done;
}

newt:

x0 = x;
lgm = lgam(a+b) - lgam(a) - lgam(b);

for( i=0; i<10; i++ )
{
/* compute the function at this point */
if( i != 0 )
y = incbet(a,b,x0);
/* compute the derivative of the function at this point */
d = (a - 1.0) * log(x0) + (b - 1.0) * log(1.0-x0) + lgm;
if( d < MINLOG )
{
x0 = 0.0;
goto under;
}
d = exp(d);
/* compute the step to the next approximation of x */
d = (y - y0)/d;
x = x0;
x0 = x0 - d;
if( x0 <= 0.0 )
{
x0 = 0.0;
goto under;
}
if( x0 >= 1.0 )
{
x0 = 1.0;
goto under;
}
if( i < 2 )
continue;
if( fabs(d/x0) < 256.0 * MACHEP )
goto done;
}

done:
if( rflg )
x0 = 1.0 - x0;
return( x0 );
}