# NAG Library Function Document

## 1Purpose

nag_rand_gamma (g05sjc) generates a vector of pseudorandom numbers taken from a gamma distribution with parameters $a$ and $b$.

## 2Specification

 #include #include
 void nag_rand_gamma (Integer n, double a, double b, Integer state[], double x[], NagError *fail)

## 3Description

The gamma distribution has PDF (probability density function)
 $fx= 1baΓa xa-1e-x/b if ​x≥0; a,b>0 fx=0 otherwise.$
One of three algorithms is used to generate the variates depending upon the value of $a$:
 (i) if $a<1$, a switching algorithm described by Dagpunar (1988) (called G6) is used. The target distributions are ${f}_{1}\left(x\right)=ca{x}^{a-1}/{t}^{a}$ and ${f}_{2}\left(x\right)=\left(1-c\right){e}^{-\left(x-t\right)}$, where $c=t/\left(t+a{e}^{-t}\right)$, and the switching argument, $t$, is taken as $1-a$. This is similar to Ahrens and Dieter's GS algorithm (see Ahrens and Dieter (1974)) in which $t=1$; (ii) if $a=1$, the gamma distribution reduces to the exponential distribution and the method based on the logarithmic transformation of a uniform random variate is used; (iii) if $a>1$, the algorithm given by Best (1978) is used. This is based on using a Student's $t$-distribution with two degrees of freedom as the target distribution in an envelope rejection method.
One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_gamma (g05sjc).

## 4References

Ahrens J H and Dieter U (1974) Computer methods for sampling from gamma, beta, Poisson and binomial distributions Computing 12 223–46
Best D J (1978) Letter to the Editor Appl. Statist. 27 181
Dagpunar J (1988) Principles of Random Variate Generation Oxford University Press
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

## 5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
2:    $\mathbf{a}$doubleInput
On entry: $a$, the parameter of the gamma distribution.
Constraint: ${\mathbf{a}}>0.0$.
3:    $\mathbf{b}$doubleInput
On entry: $b$, the parameter of the gamma distribution.
Constraint: ${\mathbf{b}}>0.0$.
4:    $\mathbf{state}\left[\mathit{dim}\right]$IntegerCommunication Array
Note: the dimension, $\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
5:    $\mathbf{x}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the $n$ pseudorandom numbers from the specified gamma distribution.
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_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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
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_INVALID_STATE
On entry, state vector has been corrupted or not initialized.
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_REAL
On entry, ${\mathbf{a}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}>0.0$.
On entry, ${\mathbf{b}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{b}}>0.0$.

Not applicable.

## 8Parallelism and Performance

nag_rand_gamma (g05sjc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10Example

This example prints a set of five pseudorandom numbers from a gamma distribution with parameters $a=5.0$ and $b=1.0$, generated by a single call to nag_rand_gamma (g05sjc), after initialization by nag_rand_init_repeatable (g05kfc).

### 10.1Program Text

Program Text (g05sjce.c)

None.

### 10.3Program Results

Program Results (g05sjce.r)

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