# NAG Library Routine Document

## 1Purpose

g05sjf generates a vector of pseudorandom numbers taken from a gamma distribution with parameters $a$ and $b$.

## 2Specification

Fortran Interface
 Subroutine g05sjf ( n, a, b, x,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: state(*), ifail Real (Kind=nag_wp), Intent (In) :: a, b Real (Kind=nag_wp), Intent (Out) :: x(n)
#include nagmk26.h
 void g05sjf_ (const Integer *n, const double *a, const double *b, Integer state[], double x[], Integer *ifail)

## 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 routines g05kff (for a repeatable sequence if computed sequentially) or g05kgf (for a non-repeatable sequence) must be called prior to the first call to g05sjf.

## 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:     $\mathbf{a}$ – Real (Kind=nag_wp)Input
On entry: $a$, the parameter of the gamma distribution.
Constraint: ${\mathbf{a}}>0.0$.
3:     $\mathbf{b}$ – Real (Kind=nag_wp)Input
On entry: $b$, the parameter of the gamma distribution.
Constraint: ${\mathbf{b}}>0.0$.
4:     $\mathbf{state}\left(*\right)$ – Integer arrayCommunication Array
Note: the actual argument supplied must be the array state supplied to the initialization routines g05kff or g05kgf.
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:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the $n$ pseudorandom numbers from the specified gamma distribution.
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, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{a}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}>0.0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{b}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{b}}>0.0$.
${\mathbf{ifail}}=4$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=-99$
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.

Not applicable.

## 8Parallelism and Performance

g05sjf 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 routine. 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 g05sjf, after initialization by g05kff.

### 10.1Program Text

Program Text (g05sjfe.f90)

### 10.2Program Data

Program Data (g05sjfe.d)

### 10.3Program Results

Program Results (g05sjfe.r)

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