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/* igamil()
*
* Inverse of complemented imcomplete gamma integral
*
*
*
* SYNOPSIS:
*
* long double a, x, y, igamil();
*
* x = igamil( a, y );
*
*
*
* DESCRIPTION:
*
* Given y, the function finds x such that
*
* igamc( a, x ) = y.
*
* Starting with the approximate value
*
* 3
* x = a t
*
* where
*
* t = 1 - d - ndtri(y) sqrt(d)
*
* and
*
* d = 1/9a,
*
* the routine performs up to 10 Newton iterations to find the
* root of igamc(a,x) - y = 0.
*
*
* ACCURACY:
*
* Tested for a ranging from 0.5 to 30 and x from 0 to 0.5.
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0,0.5 3400 8.8e-16 1.3e-16
* IEEE 0,0.5 10000 1.1e-14 1.0e-15
*
*/
/*
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 ndtril ( 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 igamcl ( long double, long double );
#else
long double ndtril(), expl(), fabsl(), logl(), sqrtl(), lgaml();
long double igamcl();
#endif
long double igamil( a, y0 )
long double a, y0;
{
long double x0, x1, x, yl, yh, y, d, lgm, dithresh;
int i, dir;
/* bound the solution */
x0 = MAXNUML;
yl = 0.0L;
x1 = 0.0L;
yh = 1.0L;
dithresh = 4.0 * MACHEPL;
/* approximation to inverse function */
d = 1.0L/(9.0L*a);
y = ( 1.0L - d - ndtril(y0) * sqrtl(d) );
x = a * y * y * y;
lgm = lgaml(a);
for( i=0; i<10; i++ )
{
if( x > x0 || x < x1 )
goto ihalve;
y = igamcl(a,x);
if( y < yl || y > yh )
goto ihalve;
if( y < y0 )
{
x0 = x;
yl = y;
}
else
{
x1 = x;
yh = y;
}
/* compute the derivative of the function at this point */
d = (a - 1.0L) * logl(x0) - x0 - lgm;
if( d < -MAXLOGL )
goto ihalve;
d = -expl(d);
/* compute the step to the next approximation of x */
d = (y - y0)/d;
x = x - d;
if( i < 3 )
continue;
if( fabsl(d/x) < dithresh )
goto done;
}
/* Resort to interval halving if Newton iteration did not converge. */
ihalve:
d = 0.0625L;
if( x0 == MAXNUML )
{
if( x <= 0.0L )
x = 1.0L;
while( x0 == MAXNUML )
{
x = (1.0L + d) * x;
y = igamcl( a, x );
if( y < y0 )
{
x0 = x;
yl = y;
break;
}
d = d + d;
}
}
d = 0.5L;
dir = 0;
for( i=0; i<400; i++ )
{
x = x1 + d * (x0 - x1);
y = igamcl( a, x );
lgm = (x0 - x1)/(x1 + x0);
if( fabsl(lgm) < dithresh )
break;
lgm = (y - y0)/y0;
if( fabsl(lgm) < dithresh )
break;
if( x <= 0.0L )
break;
if( y > y0 )
{
x1 = x;
yh = y;
if( dir < 0 )
{
dir = 0;
d = 0.5L;
}
else if( dir > 1 )
d = 0.5L * d + 0.5L;
else
d = (y0 - yl)/(yh - yl);
dir += 1;
}
else
{
x0 = x;
yl = y;
if( dir > 0 )
{
dir = 0;
d = 0.5L;
}
else if( dir < -1 )
d = 0.5L * d;
else
d = (y0 - yl)/(yh - yl);
dir -= 1;
}
}
if( x == 0.0L )
mtherr( "igamil", UNDERFLOW );
done:
return( x );
}
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