/* 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 interval halving or 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 .5,10000 50000 5.8e-12 1.3e-13
* IEEE 0,1 .25,100 100000 1.8e-13 3.9e-15
* IEEE 0,1 0,5 50000 1.1e-12 5.5e-15
* VAX 0,1 .5,100 25000 3.5e-14 1.1e-15
* With a and b constrained to half-integer or integer values:
* IEEE 0,1 .5,10000 50000 5.8e-12 1.1e-13
* IEEE 0,1 .5,100 100000 1.7e-14 7.9e-16
* With a = .5, b constrained to half-integer or integer values:
* IEEE 0,1 .5,10000 10000 8.3e-11 1.0e-11
*/
/*
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1996, 2000 by Stephen L. Moshier
*/
#include "mconf.h"
extern double MACHEP, MAXNUM, MAXLOG, MINLOG;
#ifdef ANSIPROT
extern double ndtri ( double );
extern double md_exp ( double );
extern double md_fabs ( double );
extern double md_log ( double );
extern double sqrt ( double );
extern double lgam ( double );
extern double incbet ( double, double, double );
#else
double ndtri(), md_exp(), md_fabs(), md_log(), sqrt(), lgam(), incbet();
#endif
double incbi( aa, bb, yy0 )
double aa, bb, yy0;
{
double a, b, md_y0, d, y, x, x0, x1, lgm, yp, di, dithresh, yl, yh, xt;
int i, rflg, dir, nflg;
i = 0;
if( yy0 <= 0 )
return(0.0);
if( yy0 >= 1.0 )
return(1.0);
x0 = 0.0;
yl = 0.0;
x1 = 1.0;
yh = 1.0;
nflg = 0;
if( aa <= 1.0 || bb <= 1.0 )
{
dithresh = 1.0e-6;
rflg = 0;
a = aa;
b = bb;
md_y0 = yy0;
x = a/(a+b);
y = incbet( a, b, x );
goto ihalve;
}
else
{
dithresh = 1.0e-4;
}
/* approximation to inverse function */
yp = -ndtri(yy0);
if( yy0 > 0.5 )
{
rflg = 1;
a = bb;
b = aa;
md_y0 = 1.0 - yy0;
yp = -yp;
}
else
{
rflg = 0;
a = aa;
b = bb;
md_y0 = yy0;
}
lgm = (yp * yp - 3.0)/6.0;
x = 2.0/( 1.0/(2.0*a-1.0) + 1.0/(2.0*b-1.0) );
d = yp * sqrt( x + lgm ) / x
- ( 1.0/(2.0*b-1.0) - 1.0/(2.0*a-1.0) )
* (lgm + 5.0/6.0 - 2.0/(3.0*x));
d = 2.0 * d;
if( d < MINLOG )
{
x = 1.0;
goto under;
}
x = a/( a + b * md_exp(d) );
y = incbet( a, b, x );
yp = (y - md_y0)/md_y0;
if( md_fabs(yp) < 0.2 )
goto newt;
/* Resort to interval halving if not close enough. */
ihalve:
dir = 0;
di = 0.5;
for( i=0; i<100; i++ )
{
if( i != 0 )
{
x = x0 + di * (x1 - x0);
if( x == 1.0 )
x = 1.0 - MACHEP;
if( x == 0.0 )
{
di = 0.5;
x = x0 + di * (x1 - x0);
if( x == 0.0 )
goto under;
}
y = incbet( a, b, x );
yp = (x1 - x0)/(x1 + x0);
if( md_fabs(yp) < dithresh )
goto newt;
yp = (y-md_y0)/md_y0;
if( md_fabs(yp) < dithresh )
goto newt;
}
if( y < md_y0 )
{
x0 = x;
yl = y;
if( dir < 0 )
{
dir = 0;
di = 0.5;
}
else if( dir > 3 )
di = 1.0 - (1.0 - di) * (1.0 - di);
else if( dir > 1 )
di = 0.5 * di + 0.5;
else
di = (md_y0 - y)/(yh - yl);
dir += 1;
if( x0 > 0.75 )
{
if( rflg == 1 )
{
rflg = 0;
a = aa;
b = bb;
md_y0 = yy0;
}
else
{
rflg = 1;
a = bb;
b = aa;
md_y0 = 1.0 - yy0;
}
x = 1.0 - x;
y = incbet( a, b, x );
x0 = 0.0;
yl = 0.0;
x1 = 1.0;
yh = 1.0;
goto ihalve;
}
}
else
{
x1 = x;
if( rflg == 1 && x1 < MACHEP )
{
x = 0.0;
goto done;
}
yh = y;
if( dir > 0 )
{
dir = 0;
di = 0.5;
}
else if( dir < -3 )
di = di * di;
else if( dir < -1 )
di = 0.5 * di;
else
di = (y - md_y0)/(yh - yl);
dir -= 1;
}
}
mtherr( "incbi", PLOSS );
if( x0 >= 1.0 )
{
x = 1.0 - MACHEP;
goto done;
}
if( x <= 0.0 )
{
under:
mtherr( "incbi", UNDERFLOW );
x = 0.0;
goto done;
}
newt:
if( nflg )
goto done;
nflg = 1;
lgm = lgam(a+b) - lgam(a) - lgam(b);
for( i=0; i<8; i++ )
{
/* Compute the function at this point. */
if( i != 0 )
y = incbet(a,b,x);
if( y < yl )
{
x = x0;
y = yl;
}
else if( y > yh )
{
x = x1;
y = yh;
}
else if( y < md_y0 )
{
x0 = x;
yl = y;
}
else
{
x1 = x;
yh = y;
}
if( x == 1.0 || x == 0.0 )
break;
/* Compute the derivative of the function at this point. */
d = (a - 1.0) * md_log(x) + (b - 1.0) * md_log(1.0-x) + lgm;
if( d < MINLOG )
goto done;
if( d > MAXLOG )
break;
d = md_exp(d);
/* Compute the step to the next approximation of x. */
d = (y - md_y0)/d;
xt = x - d;
if( xt <= x0 )
{
y = (x - x0) / (x1 - x0);
xt = x0 + 0.5 * y * (x - x0);
if( xt <= 0.0 )
break;
}
if( xt >= x1 )
{
y = (x1 - x) / (x1 - x0);
xt = x1 - 0.5 * y * (x1 - x);
if( xt >= 1.0 )
break;
}
x = xt;
if( md_fabs(d/x) < 128.0 * MACHEP )
goto done;
}
/* Did not converge. */
dithresh = 256.0 * MACHEP;
goto ihalve;
done:
if( rflg )
{
if( x <= MACHEP )
x = 1.0 - MACHEP;
else
x = 1.0 - x;
}
return( x );
}