G05 Chapter Contents
G05 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG05TKF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G05TKF generates a vector of pseudorandom integers, each from a discrete Poisson distribution with differing parameter.

## 2  Specification

 SUBROUTINE G05TKF ( M, VLAMDA, STATE, X, IFAIL)
 INTEGER M, STATE(*), X(M), IFAIL REAL (KIND=nag_wp) VLAMDA(M)

## 3  Description

G05TKF generates $m$ integers ${x}_{j}$, each from a discrete Poisson distribution with mean ${\lambda }_{j}$, where the probability of ${x}_{j}=I$ is
 $P xj=I = λjI × e -λj I! , I=0,1,… ,$
where
 $λj ≥ 0 , j=1,2,…,m .$
The methods used by this routine have low set up times and are designed for efficient use when the value of the parameter $\lambda$ changes during the simulation. For large samples from a distribution with fixed $\lambda$ using G05TJF to set up and use a reference vector may be more efficient.
When $\lambda <7.5$ the product of uniforms method is used, see for example Dagpunar (1988). For larger values of $\lambda$ an envelope rejection method is used with a target distribution:
 $fx=13 if ​x≤1, fx=13x-3 otherwise.$
This distribution is generated using a ratio of uniforms method. A similar approach has also been suggested by Ahrens and Dieter (1989). The basic method is combined with quick acceptance and rejection tests given by Maclaren (1990). For values of $\lambda \ge 87$ Stirling's approximation is used in the computation of the Poisson distribution function, otherwise tables of factorials are used as suggested by Maclaren (1990).
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 G05TKF.

## 4  References

Ahrens J H and Dieter U (1989) A convenient sampling method with bounded computation times for Poisson distributions Amer. J. Math. Management Sci. 1–13
Dagpunar J (1988) Principles of Random Variate Generation Oxford University Press
Maclaren N M (1990) A Poisson random number generator Personal Communication

## 5  Parameters

1:     M – INTEGERInput
On entry: $m$, the number of Poisson distributions for which pseudorandom variates are required.
Constraint: ${\mathbf{M}}\ge 1$.
2:     VLAMDA(M) – REAL (KIND=nag_wp) arrayInput
On entry: the means, ${\lambda }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,{\mathbf{M}}$, of the Poisson distributions.
Constraint: $0.0\le {\mathbf{VLAMDA}}\left(\mathit{j}\right)\le {\mathbf{X02BBF}}/2.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{M}}$.
3:     STATE($*$) – 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.
4:     X(M) – INTEGER arrayOutput
On exit: the $m$ pseudorandom numbers from the specified Poisson distributions.
5:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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, ${\mathbf{M}}<1$.
${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{VLAMDA}}\left(j\right)<0.0$ for at least one value of $j$.
On entry, $2×{\mathbf{VLAMDA}}\left(j\right)>{\mathbf{X02BBF}}\left(\right)$ for at least one value of $j$.
${\mathbf{IFAIL}}=3$
 On entry, STATE vector was not initialized or has been corrupted.

Not applicable.

None.

## 9  Example

This example prints ten pseudorandom integers from five Poisson distributions with means ${\lambda }_{1}=0.5$, ${\lambda }_{2}=5$, ${\lambda }_{3}=10$, ${\lambda }_{4}=500$ and ${\lambda }_{5}=1000$. These are generated by ten calls to G05TKF, after initialization by G05KFF.

### 9.1  Program Text

Program Text (g05tkfe.f90)

### 9.2  Program Data

Program Data (g05tkfe.d)

### 9.3  Program Results

Program Results (g05tkfe.r)