/*							igami()
 *
 *      Inverse of complemented imcomplete gamma integral
 *
 *
 *
 * SYNOPSIS:
 *
 * double a, x, y, igami();
 *
 * x = igami( 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 2.0:  April, 1987
Copyright 1984, 1987 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/

#include "mconf.h"

extern double MACHEP, MAXNUM, MAXLOG, MINLOG;

double igami( a, y0 )
double a, y0;
{
double d, y, x0, lgm;
int i;
double igamc();
double ndtri(), exp(), fabs(), log(), sqrt(), lgam();


/* approximation to inverse function */
d = 1.0/(9.0*a);
y = ( 1.0 - d - ndtri(y0) * sqrt(d) );
x0 = a * y * y * y;

lgm = lgam(a);

for( i=0; i<10; i++ )
	{
	if( x0 <= 0.0 )
		{
		mtherr( "igami", UNDERFLOW );
		return(0.0);
		}
	y = igamc(a,x0);
/* compute the derivative of the function at this point */
	d = (a - 1.0) * log(x0) - x0 - lgm;
	if( d < -MAXLOG )
		{
		mtherr( "igami", UNDERFLOW );
		goto done;
		}
	d = -exp(d);
/* compute the step to the next approximation of x */
	if( d == 0.0 )
		goto done;
	d = (y - y0)/d;
	x0 = x0 - d;
	if( i < 3 )
		continue;
	if( fabs(d/x0) < 2.0 * MACHEP )
		goto done;
	}

done:
return( x0 );
}