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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", CEPHES_UNDERFLOW);

 done:
    return x;
}

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