# NAG Library Routine Document

## 1Purpose

g01gcf returns the probability associated with the lower tail of the noncentral ${\chi }^{2}$-distribution via the routine name.

## 2Specification

Fortran Interface
 Function g01gcf ( x, df, tol,
 Real (Kind=nag_wp) :: g01gcf Integer, Intent (In) :: maxit Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x, df, rlamda, tol
#include nagmk26.h
 double g01gcf_ (const double *x, const double *df, const double *rlamda, const double *tol, const Integer *maxit, Integer *ifail)

## 3Description

The lower tail probability of the noncentral ${\chi }^{2}$-distribution with $\nu$ degrees of freedom and noncentrality parameter $\lambda$, $P\left(X\le x:\nu \text{;}\lambda \right)$, is defined by
 $PX≤x:ν;λ=∑j=0∞e-λ/2λ/2jj! PX≤x:ν+2j;0,$ (1)
where $P\left(X\le x:\nu +2j\text{;}0\right)$ is a central ${\chi }^{2}$-distribution with $\nu +2j$ degrees of freedom.
The value of $j$ at which the Poisson weight, ${e}^{-\lambda /2}\frac{{\left(\lambda /2\right)}^{j}}{j!}$, is greatest is determined and the summation (1) is made forward and backward from that value of $j$.
The recursive relationship:
 $PX≤x:a+2;0=PX≤x:a;0-xa/2e-x/2 Γa+1$ (2)
is used during the summation in (1).
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

## 5Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the deviate from the noncentral ${\chi }^{2}$-distribution with $\nu$ degrees of freedom and noncentrality parameter $\lambda$.
Constraint: ${\mathbf{x}}\ge 0.0$.
2:     $\mathbf{df}$ – Real (Kind=nag_wp)Input
On entry: $\nu$, the degrees of freedom of the noncentral ${\chi }^{2}$-distribution.
Constraint: ${\mathbf{df}}\ge 0.0$.
3:     $\mathbf{rlamda}$ – Real (Kind=nag_wp)Input
On entry: $\lambda$, the noncentrality parameter of the noncentral ${\chi }^{2}$-distribution.
Constraint: ${\mathbf{rlamda}}\ge 0.0$ if ${\mathbf{df}}>0.0$ or ${\mathbf{rlamda}}>0.0$ if ${\mathbf{df}}=0.0$.
4:     $\mathbf{tol}$ – 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  (see x02ajf), the value of  is used instead.
5:     $\mathbf{maxit}$ – IntegerInput
On entry: the maximum number of iterations to be performed.
Suggested value: $100$. See Section 9 for further discussion.
Constraint: ${\mathbf{maxit}}\ge 1$.
6:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{​ or ​}1$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Note: g01gcf may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
If on exit ${\mathbf{ifail}}={\mathbf{1}}$, ${\mathbf{2}}$, ${\mathbf{4}}$ or ${\mathbf{5}}$, then g01gcf returns $0.0$.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{df}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df}}\ge 0.0$.
On entry, ${\mathbf{df}}=0.0$ and ${\mathbf{rlamda}}=0.0$.
Constraint: ${\mathbf{rlamda}}>0.0$ if ${\mathbf{df}}=0.0$.
On entry, ${\mathbf{maxit}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{maxit}}\ge 1$.
On entry, ${\mathbf{rlamda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{rlamda}}\ge 0.0$.
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\ge 0.0$.
${\mathbf{ifail}}=2$
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.
${\mathbf{ifail}}=3$
The solution has failed to converge in $〈\mathit{\text{value}}〉$ iterations. Consider increasing maxit or tol.
${\mathbf{ifail}}=4$
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.
${\mathbf{ifail}}=5$
The calculations for the central chi-square probability has failed to converge. A larger value of tol should be used.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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.

## 8Parallelism and Performance

g01gcf is not threaded in any implementation.

The number of terms in (1) required for a given accuracy will depend on the following factors:
 (i) The rate at which the Poisson weights tend to zero. This will be slower for larger values of $\lambda$. (ii) The rate at which the central ${\chi }^{2}$ probabilities tend to zero. This will be slower for larger values of $\nu$ and $x$.

## 10Example

This example reads values from various noncentral ${\chi }^{2}$-distributions, calculates the lower tail probabilities and prints all these values until the end of data is reached.

### 10.1Program Text

Program Text (g01gcfe.f90)

### 10.2Program Data

Program Data (g01gcfe.d)

### 10.3Program Results

Program Results (g01gcfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017