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incbi.c
/*                                        incbi()
 *
 *      Inverse of imcomplete beta integral
 *
 *
 *
 * SYNOPSIS:
 *
 * double a, b, x, y, incbi();
 *
 * x = incbi( a, b, y );
 *
 *
 *
 * DESCRIPTION:
 *
 * Given y, the function finds x such that
 *
 *  incbet( a, b, x ) = y .
 *
 * The routine performs interval halving or Newton iterations to find the
 * root of incbet(a,b,x) - y = 0.
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 *                x     a,b
 * arithmetic   domain  domain  # trials    peak       rms
 *    IEEE      0,1    .5,10000   50000    5.8e-12   1.3e-13
 *    IEEE      0,1   .25,100    100000    1.8e-13   3.9e-15
 *    IEEE      0,1     0,5       50000    1.1e-12   5.5e-15
 *    VAX       0,1    .5,100     25000    3.5e-14   1.1e-15
 * With a and b constrained to half-integer or integer values:
 *    IEEE      0,1    .5,10000   50000    5.8e-12   1.1e-13
 *    IEEE      0,1    .5,100    100000    1.7e-14   7.9e-16
 * With a = .5, b constrained to half-integer or integer values:
 *    IEEE      0,1    .5,10000   10000    8.3e-11   1.0e-11
 */


/*
Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1996, 2000 by Stephen L. Moshier
*/

#include "mconf.h"

double incbi( double aa, double bb, double yy0)
{
    double a, b, y0, d, y, x, x0, x1, lgm, yp, di, dithresh, yl, yh, xt;
    int i, rflg, dir, nflg;

    i = 0;

    if (yy0 <= 0)
      return 0.0;
    if (yy0 >= 1.0)
      return 1.0;

    x0 = 0.0;
    yl = 0.0;
    x1 = 1.0;
    yh = 1.0;
    nflg = 0;

    if (aa <= 1.0 || bb <= 1.0) {
      dithresh = 1.0e-6;
      rflg = 0;
      a = aa;
      b = bb;
      y0 = yy0;
      x = a/(a+b);
      y = incbet(a, b, x);
      goto ihalve;
    } else {
      dithresh = 1.0e-4;
    }

    /* approximation to inverse function */

    yp = -ndtri(yy0);

    if (yy0 > 0.5) {
      rflg = 1;
      a = bb;
      b = aa;
      y0 = 1.0 - yy0;
      yp = -yp;
    } else {
      rflg = 0;
      a = aa;
      b = bb;
      y0 = yy0;
    }

    lgm = (yp * yp - 3.0)/6.0;
    x = 2.0/(1.0/(2.0*a-1.0)  +  1.0/(2.0*b-1.0));
    d = yp * sqrt(x + lgm) / x
      - (1.0/(2.0*b-1.0) - 1.0/(2.0*a-1.0))
      * (lgm + 5.0/6.0 - 2.0/(3.0*x));
    d = 2.0 * d;

    if (d < MINLOG) {
      x = 1.0;
      goto under;
    }

    x = a/(a + b * exp(d));
    y = incbet(a, b, x);
    yp = (y - y0)/y0;

    if (fabs(yp) < 0.2)
      goto newt;

    /* Resort to interval halving if not close enough. */
 ihalve:

    dir = 0;
    di = 0.5;
    for(i=0; i<100; i++) {
      if (i != 0) {
          x = x0  +  di * (x1 - x0);
          if (x == 1.0)
            x = 1.0 - MACHEP;
          if (x == 0.0) {
            di = 0.5;
            x = x0  +  di * (x1 - x0);
            if (x == 0.0)
                goto under;
          }
          y = incbet(a, b, x);
          yp = (x1 - x0)/(x1 + x0);
          if (fabs(yp) < dithresh)
            goto newt;
          yp = (y-y0)/y0;
          if (fabs(yp) < dithresh)
            goto newt;
      }
      if (y < y0) {
          x0 = x;
          yl = y;
          if (dir < 0) {
            dir = 0;
            di = 0.5;
          }
          else if (dir > 3)
            di = 1.0 - (1.0 - di) * (1.0 - di);
          else if (dir > 1)
            di = 0.5 * di + 0.5; 
          else
            di = (y0 - y)/(yh - yl);
          dir += 1;
          if (x0 > 0.75) {
            if (rflg == 1) {
                rflg = 0;
                a = aa;
                b = bb;
                y0 = yy0;
            } else {
                rflg = 1;
                a = bb;
                b = aa;
                y0 = 1.0 - yy0;
            }
            x = 1.0 - x;
            y = incbet(a, b, x);
            x0 = 0.0;
            yl = 0.0;
            x1 = 1.0;
            yh = 1.0;
            goto ihalve;
          }
      } else {
          x1 = x;
          if (rflg == 1 && x1 < MACHEP) {
            x = 0.0;
            goto done;
          }
          yh = y;
          if (dir > 0) {
            dir = 0;
            di = 0.5;
          }
          else if (dir < -3)
            di = di * di;
          else if (dir < -1)
            di = 0.5 * di;
          else
            di = (y - y0)/(yh - yl);
          dir -= 1;
      }
    }

    mtherr("incbi", CEPHES_PLOSS);

    if (x0 >= 1.0) {
      x = 1.0 - MACHEP;
      goto done;
    }

    if (x <= 0.0) {
    under:
      mtherr("incbi", CEPHES_UNDERFLOW);
      x = 0.0;
      goto done;
    }

 newt:

    if (nflg)
      goto done;

    nflg = 1;
    lgm = lgam(a+b) - lgam(a) - lgam(b);

    for (i=0; i<8; i++) {
      /* Compute the function at this point. */
      if (i != 0)
          y = incbet(a,b,x);
      if (y < yl) {
          x = x0;
          y = yl;
      } else if (y > yh) {
          x = x1;
          y = yh;
      } else if (y < y0) {
          x0 = x;
          yl = y;
      } else {
          x1 = x;
          yh = y;
      }
      if (x == 1.0 || x == 0.0)
          break;
      /* Compute the derivative of the function at this point. */
      d = (a - 1.0) * log(x) + (b - 1.0) * log(1.0-x) + lgm;
      if (d < MINLOG)
          goto done;
      if (d > MAXLOG)
          break;
      d = exp(d);
      /* Compute the step to the next approximation of x. */
      d = (y - y0)/d;
      xt = x - d;
      if (xt <= x0) {
          y = (x - x0) / (x1 - x0);
          xt = x0 + 0.5 * y * (x - x0);
          if (xt <= 0.0)
            break;
      }
      if (xt >= x1) {
          y = (x1 - x) / (x1 - x0);
          xt = x1 - 0.5 * y * (x1 - x);
          if (xt >= 1.0)
            break;
      }
      x = xt;
      if (fabs(d/x) < 128.0 * MACHEP)
          goto done;
    }

    /* Did not converge */
    dithresh = 256.0 * MACHEP;
    goto ihalve;

 done:

    if (rflg) {
      if (x <= MACHEP)
          x = 1.0 - MACHEP;
      else
          x = 1.0 - x;
    }

    return x;
}

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