NAG FL Interface
g01gcf (prob_​chisq_​noncentral)

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1 Purpose

g01gcf returns the probability associated with the lower tail of the noncentral χ2-distribution.

2 Specification

Fortran Interface
Function g01gcf ( x, df, rlamda, tol, maxit, ifail)
Real (Kind=nag_wp) :: g01gcf
Integer, Intent (In) :: maxit
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x, df, rlamda, tol
C Header Interface
#include <nag.h>
double  g01gcf_ (const double *x, const double *df, const double *rlamda, const double *tol, const Integer *maxit, Integer *ifail)
The routine may be called by the names g01gcf or nagf_stat_prob_chisq_noncentral.

3 Description

The lower tail probability of the noncentral χ2-distribution with ν degrees of freedom and noncentrality parameter λ, P(Xx:ν;λ), is defined by
P(Xx:ν;λ)=j=0e-λ/2(λ/2)jj! P(Xx:ν+2j;0), (1)
where P(Xx:ν+2j;0) is a central χ2-distribution with ν+2j degrees of freedom.
The value of j at which the Poisson weight, e-λ/2 (λ/2)jj! , is greatest is determined and the summation (1) is made forward and backward from that value of j.
The recursive relationship:
P(Xx:a+2;0)=P(Xx:a;0)-(xa/2)e-x/2 Γ(a+1) (2)
is used during the summation in (1).

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: x Real (Kind=nag_wp) Input
On entry: the deviate from the noncentral χ2-distribution with ν degrees of freedom and noncentrality parameter λ.
Constraint: x0.0.
2: df Real (Kind=nag_wp) Input
On entry: ν, the degrees of freedom of the noncentral χ2-distribution.
Constraint: df0.0.
3: rlamda Real (Kind=nag_wp) Input
On entry: λ, the noncentrality parameter of the noncentral χ2-distribution.
Constraint: rlamda0.0 if df>0.0 or rlamda>0.0 if df=0.0.
4: tol Real (Kind=nag_wp) Input
On entry: the required accuracy of the solution. If g01gcf is entered with tol greater than or equal to 1.0 or less than 10×machine precision (see x02ajf), the value of 10×machine precision is used instead.
5: maxit Integer Input
On entry: the maximum number of iterations to be performed.
Suggested value: 100. See Section 9 for further discussion.
Constraint: maxit1.
6: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value −1 is recommended since useful values can be provided in some output arguments even when ifail0 on exit. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases g01gcf may return useful information.
If on exit ifail=1, 2, 4 or 5, then g01gcf returns 0.0.
On entry, df=value.
Constraint: df0.0.
On entry, df=0.0 and rlamda=0.0.
Constraint: rlamda>0.0 if df=0.0.
On entry, maxit=value.
Constraint: maxit1.
On entry, rlamda=value.
Constraint: rlamda0.0.
On entry, x=value.
Constraint: x0.0.
The initial value of the Poisson weight used in the summation of (1) (see Section 3) was too small to be calculated. The computed probability is likely to be zero.
The solution has failed to converge in value iterations. Consider increasing maxit or tol.
The value of a term required in (2) (see Section 3) is too large to be evaluated accurately. The most likely cause of this error is both x and rlamda are too large.
The calculations for the central chi-square probability has failed to converge. A larger value of tol should be used.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The summations described in Section 3 are made until an upper bound on the truncation error relative to the current summation value is less than tol.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g01gcf is not threaded in any implementation.

9 Further Comments

The number of terms in (1) required for a given accuracy will depend on the following factors:
  1. (i)The rate at which the Poisson weights tend to zero. This will be slower for larger values of λ.
  2. (ii)The rate at which the central χ2 probabilities tend to zero. This will be slower for larger values of ν and x.

10 Example

This example reads values from various noncentral χ2-distributions, calculates the lower tail probabilities and prints all these values until the end of data is reached.

10.1 Program Text

Program Text (g01gcfe.f90)

10.2 Program Data

Program Data (g01gcfe.d)

10.3 Program Results

Program Results (g01gcfe.r)