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/* incbif()
*
* Inverse of imcomplete beta integral
*
*
*
* SYNOPSIS:
*
* float a, b, x, y, incbif();
*
* x = incbif( 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
* IEEE 0,1 0,100 5000 2.8e-4 8.3e-6
*
* Overflow and larger errors may occur for one of a or b near zero
* and the other 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 <math.h>
extern float MACHEPF, MINLOGF;
#define fabsf(x) ( (x) < 0 ? -(x) : (x) )
#ifdef ANSIC
float incbetf(float, float, float);
float ndtrif(float), expf(float), logf(float), sqrtf(float), lgamf(float);
#else
float incbetf();
float ndtrif(), expf(), logf(), sqrtf(), lgamf();
#endif
float incbif( float aaa, float bbb, float yyy0 )
{
float aa, bb, yy0, a, b, y0;
float d, y, x, x0, x1, lgm, yp, di;
int i, rflg;
aa = aaa;
bb = bbb;
yy0 = yyy0;
if( yy0 <= 0 )
return(0.0);
if( yy0 >= 1.0 )
return(1.0);
/* approximation to inverse function */
yp = -ndtrif(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 = 0.5 * yp * yp;
}
else
{
lgm = (yp * yp - 3.0)* 0.16666666666666667;
x0 = 2.0/( 1.0/(2.0*a-1.0) + 1.0/(2.0*b-1.0) );
y = yp * sqrtf( x0 + lgm ) / x0
- ( 1.0/(2.0*b-1.0) - 1.0/(2.0*a-1.0) )
* (lgm + 0.833333333333333333 - 2.0/(3.0*x0));
y = 2.0 * y;
if( y < MINLOGF )
{
x0 = 1.0;
goto under;
}
}
x = a/( a + b * expf(y) );
y = incbetf( a, b, x );
yp = (y - y0)/y0;
if( fabsf(yp) < 0.1 )
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 = incbetf( a, b, x );
yp = (y - y0)/y0;
if( fabsf(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( "incbif", UNDERFLOW );
goto done;
}
newt:
x0 = x;
lgm = lgamf(a+b) - lgamf(a) - lgamf(b);
for( i=0; i<10; i++ )
{
/* compute the function at this point */
if( i != 0 )
y = incbetf(a,b,x0);
/* compute the derivative of the function at this point */
d = (a - 1.0) * logf(x0) + (b - 1.0) * logf(1.0-x0) + lgm;
if( d < MINLOGF )
{
x0 = 0.0;
goto under;
}
d = expf(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( fabsf(d/x0) < 256.0 * MACHEPF )
goto done;
}
done:
if( rflg )
x0 = 1.0 - x0;
return( x0 );
}
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