/* $Id: igami.c,v 1.1 2003/04/15 03:20:17 cher Exp $ */

/* This file is taken from Cephes math library */

/*
  Cephes Math Library Release 2.8:  June, 2000
  Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*/

#include "cephes.h"
#include "mconf.h"
#include <math.h>

/*							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
 */

extern double MACHEP, MAXNUM, MAXLOG, MINLOG;

double
cephes_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 - cephes_ndtri(y0) * sqrt(d) );
  x = a * y * y * y;

  lgm = cephes_lgam(a);

  for( i=0; i<10; i++ ) {
    if( x > x0 || x < x1 )
      goto ihalve;
    y = cephes_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 = cephes_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 = cephes_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 )
    cephes_mtherr( "igami", UNDERFLOW );

 done:
  return( x );
}

/*
 * Local variables:
 *  compile-command: "make -C .."
 * End:
 */