/*							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 2.4:  March,1996
Copyright 1984, 1996 by Stephen L. Moshier
*/

#include "mconf.h"

extern double MACHEP, MAXNUM, MAXLOG, MINLOG;

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", MATHERR_PLOSS);
  if (x0 >= 1.0) {
    x = 1.0 - MACHEP;
    goto done;
  }
  if (x <= 0.0) {
  under:
    mtherr ("incbi", MATHERR_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);
}