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/* igami()
*
* Inverse of complemented imcomplete gamma integral
*
*
*
* SYNOPSIS:
*
* double a, x, p, igami();
*
* x = igami( a, p );
*
* DESCRIPTION:
*
* Given p, the function finds x such that
*
* igamc( a, x ) = p.
*
* Starting with the approximate value
*
* 3
* x = a t
*
* where
*
* t = 1 - d - ndtri(p) sqrt(d)
*
* and
*
* d = 1/9a,
*
* the routine performs up to 10 Newton iterations to find the
* root of igamc(a,x) - p = 0.
*
* ACCURACY:
*
* Tested at random a, p in the intervals indicated.
*
* a p Relative error:
* arithmetic domain domain # trials peak rms
* IEEE 0.5,100 0,0.5 100000 1.0e-14 1.7e-15
* IEEE 0.01,0.5 0,0.5 100000 9.0e-14 3.4e-15
* IEEE 0.5,10000 0,0.5 20000 2.3e-13 3.8e-14
*/
/*
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*/
#include "mconf.h"
double igami( double a, double y0 )
{
double x0, x1, x, yl, yh, y, d, lgm, dithresh;
int i, dir;
/* bound the solution */
x0 = MAXNUM;
yl = 0;
x1 = 0;
yh = 1.0;
dithresh = 5.0 * MACHEP;
/* approximation to inverse function */
d = 1.0/(9.0*a);
y = (1.0 - d - ndtri(y0) * sqrt(d));
x = a * y * y * y;
lgm = lgam(a);
for (i=0; i<10; i++) {
if (x > x0 || x < x1)
goto ihalve;
y = igamc(a,x);
if (y < yl || y > yh)
goto ihalve;
if (y < y0) {
x0 = x;
yl = y;
} else {
x1 = x;
yh = y;
}
/* compute the derivative of the function at this point */
d = (a - 1.0) * log(x) - x - lgm;
if (d < -MAXLOG)
goto ihalve;
d = -exp(d);
/* compute the step to the next approximation of x */
d = (y - y0)/d;
if (fabs(d/x) < MACHEP)
goto done;
x = x - d;
}
/* Resort to interval halving if Newton iteration did not converge. */
ihalve:
d = 0.0625;
if (x0 == MAXNUM) {
if (x <= 0.0)
x = 1.0;
while (x0 == MAXNUM) {
x = (1.0 + d) * x;
y = igamc(a, x);
if (y < y0) {
x0 = x;
yl = y;
break;
}
d = d + d;
}
}
d = 0.5;
dir = 0;
for (i=0; i<400; i++) {
x = x1 + d * (x0 - x1);
y = igamc(a, x);
lgm = (x0 - x1)/(x1 + x0);
if (fabs(lgm) < dithresh)
break;
lgm = (y - y0)/y0;
if (fabs(lgm) < dithresh)
break;
if (x <= 0.0)
break;
if (y >= y0) {
x1 = x;
yh = y;
if (dir < 0) {
dir = 0;
d = 0.5;
}
else if (dir > 1)
d = 0.5 * d + 0.5;
else
d = (y0 - yl)/(yh - yl);
dir += 1;
} else {
x0 = x;
yl = y;
if (dir > 0) {
dir = 0;
d = 0.5;
} else if (dir < -1) {
d = 0.5 * d;
} else {
d = (y0 - yl)/(yh - yl);
}
dir -= 1;
}
}
if (x == 0.0)
mtherr("igami", CEPHES_UNDERFLOW);
done:
return x;
}
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