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