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/* 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, MAXLOG, MINLOG;
double incbi(aa, bb, yy0)
double aa, bb, 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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