G05 Chapter Contents
G05 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG05TJF

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

G05TJF generates a vector of pseudorandom integers from the discrete Poisson distribution with mean $\lambda$.

## 2  Specification

 SUBROUTINE G05TJF ( MODE, N, LAMBDA, R, LR, STATE, X, IFAIL)
 INTEGER MODE, N, LR, STATE(*), X(N), IFAIL REAL (KIND=nag_wp) LAMBDA, R(LR)

## 3  Description

G05TJF generates $n$ integers ${x}_{i}$ from a discrete Poisson distribution with mean $\lambda$, where the probability of ${x}_{i}=I$ is
 $Pxi=I= λI×e-λ I! , I=0,1,…,$
where $\lambda \ge 0$.
The variates can be generated with or without using a search table and index. If a search table is used then it is stored with the index in a reference vector and subsequent calls to G05TJF with the same parameter values can then use this reference vector to generate further variates. The reference array is found using a recurrence relation if $\lambda$ is less than $50$ and by Stirling's formula otherwise.
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 G05TJF.

## 4  References

Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin
Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

## 5  Parameters

1:     MODE – INTEGERInput
On entry: a code for selecting the operation to be performed by the routine.
${\mathbf{MODE}}=0$
Set up reference vector only.
${\mathbf{MODE}}=1$
Generate variates using reference vector set up in a prior call to G05TJF.
${\mathbf{MODE}}=2$
Set up reference vector and generate variates.
${\mathbf{MODE}}=3$
Generate variates without using the reference vector.
Constraint: ${\mathbf{MODE}}=0$, $1$, $2$ or $3$.
2:     N – INTEGERInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{N}}\ge 0$.
3:     LAMBDA – REAL (KIND=nag_wp)Input
On entry: $\lambda$, the mean of the Poisson distribution.
Constraint: ${\mathbf{LAMBDA}}\ge 0.0$.
4:     R(LR) – REAL (KIND=nag_wp) arrayCommunication Array
On entry: if ${\mathbf{MODE}}=1$, the reference vector from the previous call to G05TJF.
If ${\mathbf{MODE}}=3$, R is not referenced by G05TJF.
On exit: the reference vector.
5:     LR – INTEGERInput
On entry: the dimension of the array R as declared in the (sub)program from which G05TJF is called.
Suggested values:
• if ${\mathbf{MODE}}\ne 3$, ${\mathbf{LR}}=30+20×\sqrt{{\mathbf{LAMBDA}}}+{\mathbf{LAMBDA}}$;
• otherwise ${\mathbf{LR}}=1$.
Constraints:
• if ${\mathbf{MODE}}=0$ or $2$,
• if $\sqrt{{\mathbf{LAMBDA}}}>7.15$, ${\mathbf{LR}}>9+\mathrm{int}\left(8.5+14.3×\sqrt{{\mathbf{LAMBDA}}}\right)$;
• otherwise ${\mathbf{LR}}>9+\mathrm{int}\left({\mathbf{LAMBDA}}+7.15×\sqrt{{\mathbf{LAMBDA}}}+8.5\right)$;
• if ${\mathbf{MODE}}=1$, LR must remain unchanged from the previous call to G05TJF.
6:     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.
7:     X(N) – INTEGER arrayOutput
On exit: the $n$ pseudorandom numbers from the specified Poisson distribution.
8:     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, ${\mathbf{MODE}}\ne 0$, $1$, $2$ or $3$.
${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{N}}<0$.
${\mathbf{IFAIL}}=3$
On entry, ${\mathbf{LAMBDA}}<0.0$.
${\mathbf{MODE}}=0$ or $2$ and LAMBDA is such that LR would have to be larger than the largest representable integer. Use ${\mathbf{MODE}}=3$ in this case.
${\mathbf{IFAIL}}=4$
On entry, LAMBDA is not the same as when R was set up in a previous call to G05TJF with ${\mathbf{MODE}}=0$ or $2$.
On entry, the R vector was not initialized correctly, or has been corrupted.
${\mathbf{IFAIL}}=5$
On entry, LR is too small when ${\mathbf{MODE}}=0$ or $2$.
${\mathbf{IFAIL}}=6$
 On entry, STATE vector was not initialized or has been corrupted.

Not applicable.

None.

## 9  Example

This example prints $10$ pseudorandom integers from a Poisson distribution with mean $\lambda =20$, generated by a single call to G05TJF, after initialization by G05KFF.

### 9.1  Program Text

Program Text (g05tjfe.f90)

### 9.2  Program Data

Program Data (g05tjfe.d)

### 9.3  Program Results

Program Results (g05tjfe.r)