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igami.c

/*                                        igami()
 *
 *      Inverse of complemented imcomplete gamma integral
 *
 *
 *
 * SYNOPSIS:
 *
 * double a, x, p, igami();
 *
 * x = igami( a, p );
 *
 * DESCRIPTION:
 *
 * Given p, the function finds x such that
 *
 *  igamc( a, x ) = p.
 *
 * Starting with the approximate value
 *
 *         3
 *  x = a t
 *
 *  where
 *
 *  t = 1 - d - ndtri(p) sqrt(d)
 * 
 * and
 *
 *  d = 1/9a,
 *
 * the routine performs up to 10 Newton iterations to find the
 * root of igamc(a,x) - p = 0.
 *
 * ACCURACY:
 *
 * Tested at random a, p in the intervals indicated.
 *
 *                a        p                      Relative error:
 * arithmetic   domain   domain     # trials      peak         rms
 *    IEEE     0.5,100   0,0.5       100000       1.0e-14     1.7e-15
 *    IEEE     0.01,0.5  0,0.5       100000       9.0e-14     3.4e-15
 *    IEEE    0.5,10000  0,0.5        20000       2.3e-13     3.8e-14
 */

/*
Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*/

#include "mconf.h"

double igami( double a, double y0 )
{
double x0, x1, x, yl, yh, y, d, lgm, dithresh;
int i, dir;

/* bound the solution */
x0 = MAXNUM;
yl = 0;
x1 = 0;
yh = 1.0;
dithresh = 5.0 * MACHEP;

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

lgm = lgam(a);

for( i=0; i<10; i++ )
      {
      if( x > x0 || x < x1 )
            goto ihalve;
      y = igamc(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 - 1.0) * log(x) - x - lgm;
      if( d < -MAXLOG )
            goto ihalve;
      d = -exp(d);
/* compute the step to the next approximation of x */
      d = (y - y0)/d;
      if( fabs(d/x) < MACHEP )
            goto done;
      x = x - d;
      }

/* Resort to interval halving if Newton iteration did not converge. */
ihalve:

d = 0.0625;
if( x0 == MAXNUM )
      {
      if( x <= 0.0 )
            x = 1.0;
      while( x0 == MAXNUM )
            {
            x = (1.0 + d) * x;
            y = igamc( a, x );
            if( y < y0 )
                  {
                  x0 = x;
                  yl = y;
                  break;
                  }
            d = d + d;
            }
      }
d = 0.5;
dir = 0;

for( i=0; i<400; i++ )
      {
      x = x1  +  d * (x0 - x1);
      y = igamc( a, x );
      lgm = (x0 - x1)/(x1 + x0);
      if( fabs(lgm) < dithresh )
            break;
      lgm = (y - y0)/y0;
      if( fabs(lgm) < dithresh )
            break;
      if( x <= 0.0 )
            break;
      if( y >= y0 )
            {
            x1 = x;
            yh = y;
            if( dir < 0 )
                  {
                  dir = 0;
                  d = 0.5;
                  }
            else if( dir > 1 )
                  d = 0.5 * d + 0.5; 
            else
                  d = (y0 - yl)/(yh - yl);
            dir += 1;
            }
      else
            {
            x0 = x;
            yl = y;
            if( dir > 0 )
                  {
                  dir = 0;
                  d = 0.5;
                  }
            else if( dir < -1 )
                  d = 0.5 * d;
            else
                  d = (y0 - yl)/(yh - yl);
            dir -= 1;
            }
      }
if( x == 0.0 )
      mtherr( "igami", UNDERFLOW );

done:
return( x );
}

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