NAG Library Routine Document

G05TKF

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

λ \geq 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: $M \geq 1$ .
2: VLAMDA(M) – REAL (KIND=nag_wp) arrayInput: On entry: the means, $λ_{j}$ , for $j = 1, 2, \dots, M$ , of the Poisson distributions.
Constraint: $0.0 \leq VLAMDA (j) \leq X02BBF / 2.0$ , for $j = 1, 2, \dots, M$ .
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 $0$ , $- 1 or 1$ . 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 $- 1 or 1$ 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 $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit: $IFAIL = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, $M < 1$ .

$IFAIL = 2$: On entry, $VLAMDA (j) < 0.0$ for at least one value of $j$ .

On entry, $2 \times VLAMDA (j) > X02BBF ()$ for at least one value of $j$ .

$IFAIL = 3$

On entry,

STATE vector was not initialized or has been corrupted.

7 Accuracy

Not applicable.

8 Further Comments

None.

9 Example

This example prints ten pseudorandom integers from five Poisson distributions with means

λ_{1} = 0.5

λ_{2} = 5

λ_{3} = 10

λ_{4} = 500

and

λ_{5} = 1000

. These are generated by ten calls to G05TKF, after initialization by G05KFF.

NAG Library Routine DocumentG05TKF

+− Contents

1 Purpose

2 Specification

3 Description