incbi.c

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/*                                                     incbi()
 *
 *      Inverse of incomplete 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, 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;
        }
    }
    sf_error("incbi", SF_ERROR_LOSS, NULL);
    if (x0 >= 1.0) {
        x = 1.0 - MACHEP;
        goto done;
    }
    if (x <= 0.0) {
      under:
        sf_error("incbi", SF_ERROR_UNDERFLOW, NULL);
        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);
}