/* incbil()
*
* Inverse of imcomplete beta integral
*
*
*
* SYNOPSIS:
*
* long double a, b, x, y, incbil();
*
* x = incbil( 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 .5,10000 10000 1.1e-14 1.4e-16
*/
/*
Cephes Math Library Release 2.3: March, 1995
Copyright 1984, 1995 by Stephen L. Moshier
*/
#include <math.h>
extern long double MACHEPL, MAXNUML, MAXLOGL, MINLOGL;
#ifdef ANSIPROT
extern long double incbetl ( long double, long double, long double );
extern long double expl ( long double );
extern long double fabsl ( long double );
extern long double logl ( long double );
extern long double sqrtl ( long double );
extern long double lgaml ( long double );
extern long double ndtril ( long double );
#else
long double incbetl(), expl(), fabsl(), logl(), sqrtl(), lgaml();
long double ndtril();
#endif
long double incbil( aa, bb, yy0 )
long double aa, bb, yy0;
{
long double a, b, y0, d, y, x, x0, x1, lgm, yp, di, dithresh, yl, yh, xt;
int i, rflg, dir, nflg;
if( yy0 <= 0.0L )
return(0.0L);
if( yy0 >= 1.0L )
return(1.0L);
x0 = 0.0L;
yl = 0.0L;
x1 = 1.0L;
yh = 1.0L;
if( aa <= 1.0L || bb <= 1.0L )
{
dithresh = 1.0e-7L;
rflg = 0;
a = aa;
b = bb;
y0 = yy0;
x = a/(a+b);
y = incbetl( a, b, x );
nflg = 0;
goto ihalve;
}
else
{
nflg = 0;
dithresh = 1.0e-4L;
}
/* approximation to inverse function */
yp = -ndtril( yy0 );
if( yy0 > 0.5L )
{
rflg = 1;
a = bb;
b = aa;
y0 = 1.0L - yy0;
yp = -yp;
}
else
{
rflg = 0;
a = aa;
b = bb;
y0 = yy0;
}
lgm = (yp * yp - 3.0L)/6.0L;
x = 2.0L/( 1.0L/(2.0L * a-1.0L) + 1.0L/(2.0L * b - 1.0L) );
d = yp * sqrtl( x + lgm ) / x
- ( 1.0L/(2.0L * b - 1.0L) - 1.0L/(2.0L * a - 1.0L) )
* (lgm + (5.0L/6.0L) - 2.0L/(3.0L * x));
d = 2.0L * d;
if( d < MINLOGL )
{
x = 1.0L;
goto under;
}
x = a/( a + b * expl(d) );
y = incbetl( a, b, x );
yp = (y - y0)/y0;
if( fabsl(yp) < 0.2 )
goto newt;
/* Resort to interval halving if not close enough. */
ihalve:
dir = 0;
di = 0.5L;
for( i=0; i<400; i++ )
{
if( i != 0 )
{
x = x0 + di * (x1 - x0);
if( x == 1.0L )
x = 1.0L - MACHEPL;
if( x == 0.0L )
{
di = 0.5;
x = x0 + di * (x1 - x0);
if( x == 0.0 )
goto under;
}
y = incbetl( a, b, x );
yp = (x1 - x0)/(x1 + x0);
if( fabsl(yp) < dithresh )
goto newt;
yp = (y-y0)/y0;
if( fabsl(yp) < dithresh )
goto newt;
}
if( y < y0 )
{
x0 = x;
yl = y;
if( dir < 0 )
{
dir = 0;
di = 0.5L;
}
else if( dir > 3 )
di = 1.0L - (1.0L - di) * (1.0L - di);
else if( dir > 1 )
di = 0.5L * di + 0.5L;
else
di = (y0 - y)/(yh - yl);
dir += 1;
if( x0 > 0.95L )
{
if( rflg == 1 )
{
rflg = 0;
a = aa;
b = bb;
y0 = yy0;
}
else
{
rflg = 1;
a = bb;
b = aa;
y0 = 1.0 - yy0;
}
x = 1.0L - x;
y = incbetl( a, b, x );
x0 = 0.0;
yl = 0.0;
x1 = 1.0;
yh = 1.0;
goto ihalve;
}
}
else
{
x1 = x;
if( rflg == 1 && x1 < MACHEPL )
{
x = 0.0L;
goto done;
}
yh = y;
if( dir > 0 )
{
dir = 0;
di = 0.5L;
}
else if( dir < -3 )
di = di * di;
else if( dir < -1 )
di = 0.5L * di;
else
di = (y - y0)/(yh - yl);
dir -= 1;
}
}
mtherr( "incbil", PLOSS );
if( x0 >= 1.0L )
{
x = 1.0L - MACHEPL;
goto done;
}
if( x <= 0.0L )
{
under:
mtherr( "incbil", UNDERFLOW );
x = 0.0L;
goto done;
}
newt:
if( nflg )
goto done;
nflg = 1;
lgm = lgaml(a+b) - lgaml(a) - lgaml(b);
for( i=0; i<15; i++ )
{
/* Compute the function at this point. */
if( i != 0 )
y = incbetl(a,b,x);
if( y < yl )
{
x = x0;
y = yl;
}
else if( y > yh )
{
x = x1;
y = yh;
}
else if( y < y0 )
{
x0 = x;
yl = y;
}
else
{
x1 = x;
yh = y;
}
if( x == 1.0L || x == 0.0L )
break;
/* Compute the derivative of the function at this point. */
d = (a - 1.0L) * logl(x) + (b - 1.0L) * logl(1.0L - x) + lgm;
if( d < MINLOGL )
goto done;
if( d > MAXLOGL )
break;
d = expl(d);
/* Compute the step to the next approximation of x. */
d = (y - y0)/d;
xt = x - d;
if( xt <= x0 )
{
y = (x - x0) / (x1 - x0);
xt = x0 + 0.5L * y * (x - x0);
if( xt <= 0.0L )
break;
}
if( xt >= x1 )
{
y = (x1 - x) / (x1 - x0);
xt = x1 - 0.5L * y * (x1 - x);
if( xt >= 1.0L )
break;
}
x = xt;
if( fabsl(d/x) < (128.0L * MACHEPL) )
goto done;
}
/* Did not converge. */
dithresh = 256.0L * MACHEPL;
goto ihalve;
done:
if( rflg )
{
if( x <= MACHEPL )
x = 1.0L - MACHEPL;
else
x = 1.0L - x;
}
return( x );
}