/*							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 2.3:  March, 1995
Copyright 1984, 1987, 1995 by Stephen L. Moshier
*/

#include "mconf.h"

extern double MACHEP, MAXNUM, MAXLOG, MINLOG;

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", MATHERR_UNDERFLOW);

done:
  return (x);
}