# NAG CL Interfaceg01gcc (prob_​chisq_​noncentral)

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## 1Purpose

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

## 2Specification

 #include
 double g01gcc (double x, double df, double lambda, double tol, Integer max_iter, NagError *fail)
The function may be called by the names: g01gcc, nag_stat_prob_chisq_noncentral or nag_prob_non_central_chi_sq.

## 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
 $P(X≤x:ν;λ)=∑j=0∞e-λ/2(λ/2)jj! P(X≤x:ν+2j;0),$ (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:
 $P(X≤x:a+2;0)=P(X≤x:a;0)-(xa/2)e-x/2 Γ(a+1)$ (2)
is used during the summation in (1).

## 4References

NIST Digital Library of Mathematical Functions

## 5Arguments

1: $\mathbf{x}$double 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: $\mathbf{df}$double Input
On entry: $\nu$, the degrees of freedom of the noncentral ${\chi }^{2}$-distribution.
Constraint: ${\mathbf{df}}\ge 0.0$.
3: $\mathbf{lambda}$double Input
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: $\mathbf{tol}$double Input
On entry: the required accuracy of the solution. If g01gcc is entered with tol greater than or equal to $1.0$ or less than (see X02AJC), the value of is used instead.
5: $\mathbf{max_iter}$Integer Input
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 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

If on exit ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_2_REAL_ARG_CONS, NE_CHI_PROB, NE_INT_ARG_LT, NE_POISSON_WEIGHT, NE_REAL_ARG_LT or NE_TERM_LARGE, then g01gcc returns $0.0$.
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 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface 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

g01gcc is not threaded in any implementation.

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 $\lambda$.
2. (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)