/* $Id: incbi.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, 1996, 2000 by Stephen L. Moshier
*/

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

/*							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
 */

extern double MACHEP, MAXNUM, MAXLOG, MINLOG;

double
cephes_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 = cephes_incbet( a, b, x );
    goto ihalve;
  } else {
    dithresh = 1.0e-4;
  }
  /* approximation to inverse function */

  yp = -cephes_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 = cephes_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 = cephes_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 = cephes_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;
    }
  }
  cephes_mtherr( "incbi", PLOSS );
  if( x0 >= 1.0 ) {
    x = 1.0 - MACHEP;
    goto done;
  }
  if( x <= 0.0 ) {
  under:
    cephes_mtherr( "incbi", UNDERFLOW );
    x = 0.0;
    goto done;
  }

 newt:
  if( nflg )
    goto done;
  nflg = 1;
  lgm = cephes_lgam(a+b) - cephes_lgam(a) - cephes_lgam(b);

  for( i=0; i<8; i++ ) {
    /* Compute the function at this point. */
    if( i != 0 )
      y = cephes_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 );
}

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