incbet.c

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/*                                                     incbet.c
 *
 *     Incomplete beta integral
 *
 *
 * SYNOPSIS:
 *
 * double a, b, x, y, incbet();
 *
 * y = incbet( a, b, x );
 *
 *
 * DESCRIPTION:
 *
 * Returns incomplete beta integral of the arguments, evaluated
 * from zero to x.  The function is defined as
 *
 *                  x
 *     -            -
 *    | (a+b)      | |  a-1     b-1
 *  -----------    |   t   (1-t)   dt.
 *   -     -     | |
 *  | (a) | (b)   -
 *                 0
 *
 * The domain of definition is 0 <= x <= 1.  In this
 * implementation a and b are restricted to positive values.
 * The integral from x to 1 may be obtained by the symmetry
 * relation
 *
 *    1 - incbet( a, b, x )  =  incbet( b, a, 1-x ).
 *
 * The integral is evaluated by a continued fraction expansion
 * or, when b*x is small, by a power series.
 *
 * ACCURACY:
 *
 * Tested at uniformly distributed random points (a,b,x) with a and b
 * in "domain" and x between 0 and 1.
 *                                        Relative error
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0,5         10000       6.9e-15     4.5e-16
 *    IEEE      0,85       250000       2.2e-13     1.7e-14
 *    IEEE      0,1000      30000       5.3e-12     6.3e-13
 *    IEEE      0,10000    250000       9.3e-11     7.1e-12
 *    IEEE      0,100000    10000       8.7e-10     4.8e-11
 * Outputs smaller than the IEEE gradual underflow threshold
 * were excluded from these statistics.
 *
 * ERROR MESSAGES:
 *   message         condition      value returned
 * incbet domain      x<0, x>1          0.0
 * incbet underflow                     0.0
 */

 
/*
 * Cephes Math Library, Release 2.3:  March, 1995
 * Copyright 1984, 1995 by Stephen L. Moshier
 */
 
#include "mconf.h"
 
#define MAXGAM 171.624376956302725
 
extern double MACHEP, MINLOG, MAXLOG;
 
static double big = 4.503599627370496e15;
static double biginv = 2.22044604925031308085e-16;
 
 
/* Power series for incomplete beta integral.
 * Use when b*x is small and x not too close to 1.  */
 
static double pseries(double a, double b, double x)
{
    double s, t, u, v, n, t1, z, ai;
 
    ai = 1.0 / a;
    u = (1.0 - b) * x;
    v = u / (a + 1.0);
    t1 = v;
    t = u;
    n = 2.0;
    s = 0.0;
    z = MACHEP * ai;
    while (fabs(v) > z) {
        u = (n - b) * x / n;
        t *= u;
        v = t / (a + n);
        s += v;
        n += 1.0;
    }
    s += t1;
    s += ai;
 
    u = a * log(x);
    if ((a + b) < MAXGAM && fabs(u) < MAXLOG) {
        t = 1.0 / beta(a, b);
        s = s * t * pow(x, a);
    }
    else {
        t = -lbeta(a,b) + u + log(s);
        if (t < MINLOG)
            s = 0.0;
        else
            s = exp(t);
    }
    return (s);
}
 
 
/* Continued fraction expansion #1 for incomplete beta integral */
 
static double incbcf(double a, double b, double x)
{
    double xk, pk, pkm1, pkm2, qk, qkm1, qkm2;
    double k1, k2, k3, k4, k5, k6, k7, k8;
    double r, t, ans, thresh;
    int n;
 
    k1 = a;
    k2 = a + b;
    k3 = a;
    k4 = a + 1.0;
    k5 = 1.0;
    k6 = b - 1.0;
    k7 = k4;
    k8 = a + 2.0;
 
    pkm2 = 0.0;
    qkm2 = 1.0;
    pkm1 = 1.0;
    qkm1 = 1.0;
    ans = 1.0;
    r = 1.0;
    n = 0;
    thresh = 3.0 * MACHEP;
    do {
 
        xk = -(x * k1 * k2) / (k3 * k4);
        pk = pkm1 + pkm2 * xk;
        qk = qkm1 + qkm2 * xk;
        pkm2 = pkm1;
        pkm1 = pk;
        qkm2 = qkm1;
        qkm1 = qk;
 
        xk = (x * k5 * k6) / (k7 * k8);
        pk = pkm1 + pkm2 * xk;
        qk = qkm1 + qkm2 * xk;
        pkm2 = pkm1;
        pkm1 = pk;
        qkm2 = qkm1;
        qkm1 = qk;
 
        if (qk != 0)
            r = pk / qk;
        if (r != 0) {
            t = fabs((ans - r) / r);
            ans = r;
        }
        else
            t = 1.0;
 
        if (t < thresh)
            goto cdone;
 
        k1 += 1.0;
        k2 += 1.0;
        k3 += 2.0;
        k4 += 2.0;
        k5 += 1.0;
        k6 -= 1.0;
        k7 += 2.0;
        k8 += 2.0;
 
        if ((fabs(qk) + fabs(pk)) > big) {
            pkm2 *= biginv;
            pkm1 *= biginv;
            qkm2 *= biginv;
            qkm1 *= biginv;
        }
        if ((fabs(qk) < biginv) || (fabs(pk) < biginv)) {
            pkm2 *= big;
            pkm1 *= big;
            qkm2 *= big;
            qkm1 *= big;
        }
    }
    while (++n < 300);
 
  cdone:
    return (ans);
}
 
 
/* Continued fraction expansion #2 for incomplete beta integral */
 
static double incbd(double a, double b, double x)
{
    double xk, pk, pkm1, pkm2, qk, qkm1, qkm2;
    double k1, k2, k3, k4, k5, k6, k7, k8;
    double r, t, ans, z, thresh;
    int n;
 
    k1 = a;
    k2 = b - 1.0;
    k3 = a;
    k4 = a + 1.0;
    k5 = 1.0;
    k6 = a + b;
    k7 = a + 1.0;;
    k8 = a + 2.0;
 
    pkm2 = 0.0;
    qkm2 = 1.0;
    pkm1 = 1.0;
    qkm1 = 1.0;
    z = x / (1.0 - x);
    ans = 1.0;
    r = 1.0;
    n = 0;
    thresh = 3.0 * MACHEP;
    do {
 
        xk = -(z * k1 * k2) / (k3 * k4);
        pk = pkm1 + pkm2 * xk;
        qk = qkm1 + qkm2 * xk;
        pkm2 = pkm1;
        pkm1 = pk;
        qkm2 = qkm1;
        qkm1 = qk;
 
        xk = (z * k5 * k6) / (k7 * k8);
        pk = pkm1 + pkm2 * xk;
        qk = qkm1 + qkm2 * xk;
        pkm2 = pkm1;
        pkm1 = pk;
        qkm2 = qkm1;
        qkm1 = qk;
 
        if (qk != 0)
            r = pk / qk;
        if (r != 0) {
            t = fabs((ans - r) / r);
            ans = r;
        }
        else
            t = 1.0;
 
        if (t < thresh)
            goto cdone;
 
        k1 += 1.0;
        k2 -= 1.0;
        k3 += 2.0;
        k4 += 2.0;
        k5 += 1.0;
        k6 += 1.0;
        k7 += 2.0;
        k8 += 2.0;
 
        if ((fabs(qk) + fabs(pk)) > big) {
            pkm2 *= biginv;
            pkm1 *= biginv;
            qkm2 *= biginv;
            qkm1 *= biginv;
        }
        if ((fabs(qk) < biginv) || (fabs(pk) < biginv)) {
            pkm2 *= big;
            pkm1 *= big;
            qkm2 *= big;
            qkm1 *= big;
        }
    }
    while (++n < 300);
  cdone:
    return (ans);
}
 
 
double incbet(double aa, double bb, double xx)
{
    double a, b, t, x, xc, w, y;
    int flag;
 
    if (aa <= 0.0 || bb <= 0.0)
        goto domerr;
 
    if ((xx <= 0.0) || (xx >= 1.0)) {
        if (xx == 0.0)
            return (0.0);
        if (xx == 1.0)
            return (1.0);
      domerr:
        sf_error("incbet", SF_ERROR_DOMAIN, NULL);
        return (NAN);
    }
 
    flag = 0;
    if ((bb * xx) <= 1.0 && xx <= 0.95) {
        t = pseries(aa, bb, xx);
        goto done;
    }
 
    w = 1.0 - xx;
 
    /* Reverse a and b if x is greater than the mean. */
    if (xx > (aa / (aa + bb))) {
        flag = 1;
        a = bb;
        b = aa;
        xc = xx;
        x = w;
    }
    else {
        a = aa;
        b = bb;
        xc = w;
        x = xx;
    }
 
    if (flag == 1 && (b * x) <= 1.0 && x <= 0.95) {
        t = pseries(a, b, x);
        goto done;
    }
 
    /* Choose expansion for better convergence. */
    y = x * (a + b - 2.0) - (a - 1.0);
    if (y < 0.0)
        w = incbcf(a, b, x);
    else
        w = incbd(a, b, x) / xc;
 
    /* Multiply w by the factor
     * a      b   _             _     _
     * x  (1-x)   | (a+b) / ( a | (a) | (b) ) .   */
 
    y = a * log(x);
    t = b * log(xc);
    if ((a + b) < MAXGAM && fabs(y) < MAXLOG && fabs(t) < MAXLOG) {
        t = pow(xc, b);
        t *= pow(x, a);
        t /= a;
        t *= w;
        t *= 1.0 / beta(a, b);
        goto done;
    }
    /* Resort to logarithms.  */
    y += t - lbeta(a,b);
    y += log(w / a);
    if (y < MINLOG)
        t = 0.0;
    else
        t = exp(y);
 
  done:
 
    if (flag == 1) {
        if (t <= MACHEP)
            t = 1.0 - MACHEP;
        else
            t = 1.0 - t;
    }
    return (t);
}