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/* fdtr.c
*
* F distribution
*
*
*
* SYNOPSIS:
*
* double df1, df2;
* double x, y, fdtr();
*
* y = fdtr( df1, df2, x );
*
* DESCRIPTION:
*
* Returns the area from zero to x under the F density
* function (also known as Snedcor's density or the
* variance ratio density). This is the density
* of x = (u1/df1)/(u2/df2), where u1 and u2 are random
* variables having Chi square distributions with df1
* and df2 degrees of freedom, respectively.
*
* The incomplete beta integral is used, according to the
* formula
*
* P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ).
*
*
* The arguments a and b are greater than zero, and x is
* nonnegative.
*
* ACCURACY:
*
* Tested at random points (a,b,x).
*
* x a,b Relative error:
* arithmetic domain domain # trials peak rms
* IEEE 0,1 0,100 100000 9.8e-15 1.7e-15
* IEEE 1,5 0,100 100000 6.5e-15 3.5e-16
* IEEE 0,1 1,10000 100000 2.2e-11 3.3e-12
* IEEE 1,5 1,10000 100000 1.1e-11 1.7e-13
* See also incbet.c.
*
*
* ERROR MESSAGES:
*
* message condition value returned
* fdtr domain a<0, b<0, x<0 0.0
*
*/
/* fdtrc()
*
* Complemented F distribution
*
*
*
* SYNOPSIS:
*
* double df1, df2;
* double x, y, fdtrc();
*
* y = fdtrc( df1, df2, x );
*
* DESCRIPTION:
*
* Returns the area from x to infinity under the F density
* function (also known as Snedcor's density or the
* variance ratio density).
*
*
* inf.
* -
* 1 | | a-1 b-1
* 1-P(x) = ------ | t (1-t) dt
* B(a,b) | |
* -
* x
*
*
* The incomplete beta integral is used, according to the
* formula
*
* P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ).
*
*
* ACCURACY:
*
* Tested at random points (a,b,x) in the indicated intervals.
* x a,b Relative error:
* arithmetic domain domain # trials peak rms
* IEEE 0,1 1,100 100000 3.7e-14 5.9e-16
* IEEE 1,5 1,100 100000 8.0e-15 1.6e-15
* IEEE 0,1 1,10000 100000 1.8e-11 3.5e-13
* IEEE 1,5 1,10000 100000 2.0e-11 3.0e-12
* See also incbet.c.
*
* ERROR MESSAGES:
*
* message condition value returned
* fdtrc domain a<0, b<0, x<0 0.0
*
*/
/* fdtri()
*
* Inverse of F distribution
*
*
*
* SYNOPSIS:
*
* double df1, df2;
* double x, p, fdtri();
*
* x = fdtri( df1, df2, p );
*
* DESCRIPTION:
*
* Finds the F density argument x such that the integral
* from -infinity to x of the F density is equal to the
* given probability p.
*
* This is accomplished using the inverse beta integral
* function and the relations
*
* z = incbi( df2/2, df1/2, p )
* x = df2 (1-z) / (df1 z).
*
* Note: the following relations hold for the inverse of
* the uncomplemented F distribution:
*
* z = incbi( df1/2, df2/2, p )
* x = df2 z / (df1 (1-z)).
*
* ACCURACY:
*
* Tested at random points (a,b,p).
*
* a,b Relative error:
* arithmetic domain # trials peak rms
* For p between .001 and 1:
* IEEE 1,100 100000 8.3e-15 4.7e-16
* IEEE 1,10000 100000 2.1e-11 1.4e-13
* For p between 10^-6 and 10^-3:
* IEEE 1,100 50000 1.3e-12 8.4e-15
* IEEE 1,10000 50000 3.0e-12 4.8e-14
* See also fdtrc.c.
*
* ERROR MESSAGES:
*
* message condition value returned
* fdtri domain p <= 0 or p > 1 NaN
* v < 1
*
*/
/*
* Cephes Math Library Release 2.3: March, 1995
* Copyright 1984, 1987, 1995 by Stephen L. Moshier
*/
#include "mconf.h"
double fdtrc(double a, double b, double x)
{
double w;
if ((a <= 0.0) || (b <= 0.0) || (x < 0.0)) {
mtherr("fdtrc", DOMAIN);
return NPY_NAN;
}
w = b / (b + a * x);
return incbet(0.5 * b, 0.5 * a, w);
}
double fdtr(double a, double b, double x)
{
double w;
if ((a <= 0.0) || (b <= 0.0) || (x < 0.0)) {
mtherr("fdtr", DOMAIN);
return NPY_NAN;
}
w = a * x;
w = w / (b + w);
return incbet(0.5 * a, 0.5 * b, w);
}
double fdtri(double a, double b, double y)
{
double w, x;
if ((a <= 0.0) || (b <= 0.0) || (y <= 0.0) || (y > 1.0)) {
mtherr("fdtri", DOMAIN);
return NPY_NAN;
}
y = 1.0 - y;
/* Compute probability for x = 0.5. */
w = incbet(0.5 * b, 0.5 * a, 0.5);
/* If that is greater than y, then the solution w < .5.
* Otherwise, solve at 1-y to remove cancellation in (b - b*w). */
if (w > y || y < 0.001) {
w = incbi(0.5 * b, 0.5 * a, y);
x = (b - b * w) / (a * w);
}
else {
w = incbi(0.5 * a, 0.5 * b, 1.0 - y);
x = b * w / (a * (1.0 - w));
}
return x;
}
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