# NAG Library Function Document

## 1Purpose

nag_prob_non_central_chi_sq (g01gcc) returns the probability associated with the lower tail of the noncentral ${\chi }^{2}$-distribution .

## 2Specification

 #include #include
 double nag_prob_non_central_chi_sq (double x, double df, double lambda, double tol, Integer max_iter, NagError *fail)

## 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).

## 4References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

## 5Arguments

1:    $\mathbf{x}$doubleInput
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:    $\mathbf{df}$doubleInput
On entry: $\nu$, the degrees of freedom of the noncentral ${\chi }^{2}$-distribution.
Constraint: ${\mathbf{df}}\ge 0.0$.
3:    $\mathbf{lambda}$doubleInput
On entry: $\lambda$, the noncentrality parameter of the noncentral ${\chi }^{2}$-distribution.
Constraint: ${\mathbf{lambda}}\ge 0.0$ if ${\mathbf{df}}>0.0$ or ${\mathbf{lambda}}>0.0$ if ${\mathbf{df}}=0.0$.
4:    $\mathbf{tol}$doubleInput
On entry: the required accuracy of the solution. If nag_prob_non_central_chi_sq (g01gcc) is entered with tol greater than or equal to $1.0$ or less than  (see nag_machine_precision (X02AJC)), the value of  is used instead.
5:    $\mathbf{max_iter}$IntegerInput
On entry: the maximum number of iterations to be performed.
Suggested value: $100$. See Section 9 for further discussion.
Constraint: ${\mathbf{max_iter}}\ge 1$.
6:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_2_REAL_ARG_CONS
On entry, ${\mathbf{df}}=0.0$ and ${\mathbf{lambda}}=0.0$.
Constraint: ${\mathbf{lambda}}>0.0$ if ${\mathbf{df}}=0.0$.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_CHI_PROB
The calculations for the central chi-square probability has failed to converge. A larger value of tol should be used.
NE_CONV
The solution has failed to converge in $〈\mathit{\text{value}}〉$ iterations. Consider increasing max_iter or tol.
NE_INT_ARG_LT
On entry, ${\mathbf{max_iter}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{max_iter}}\ge 1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_POISSON_WEIGHT
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.
NE_REAL_ARG_LT
On entry, ${\mathbf{df}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df}}\ge 0.0$.
On entry, ${\mathbf{lambda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lambda}}\ge 0.0$.
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\ge 0.0$.
NE_TERM_LARGE
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 lambda are too large.

## 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

nag_prob_non_central_chi_sq (g01gcc) 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 (g01gcce.c)

### 10.2Program Data

Program Data (g01gcce.d)

### 10.3Program Results

Program Results (g01gcce.r)

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