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NAG Library Routine Document

g05tkf (int_poisson_varmean)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

g05tkf generates a vector of pseudorandom integers, each from a discrete Poisson distribution with differing parameter.

2

Specification

Fortran Interface

Subroutine g05tkf (

m, vlamda, state, x, ifail)

Integer, Intent (In)	::	m
Integer, Intent (Inout)	::	state(*), ifail
Integer, Intent (Out)	::	x(m)
Real (Kind=nag_wp), Intent (In)	::	vlamda(m)

C Header Interface

#include nagmk26.h

void	g05tkf_ ( const Integer m, const double vlamda[], Integer state[], Integer x[], Integer ifail)

3

Description

g05tkf generates

m

integers

x_{j}

, each from a discrete Poisson distribution with mean

λ_{j}

, where the probability of

x_{j} = I

P (x_{j} = I) = \frac{λ_{j}^{I} \times e^{- λ_{j}}}{I!}, I = 0, 1, \dots,

where

λ_{j} \geq 0, j = 1, 2, \dots, m .

The methods used by this routine have low set up times and are designed for efficient use when the value of the parameter

λ

changes during the simulation. For large samples from a distribution with fixed

λ

using g05tjf to set up and use a reference vector may be more efficient.

When

λ < 7.5

the product of uniforms method is used, see for example Dagpunar (1988). For larger values of

λ

an envelope rejection method is used with a target distribution:

\begin{matrix} f (x) = \frac{1}{3} & if ​ |x| \leq 1, \\ f (x) = \frac{1}{3} {|x|}^{- 3} & otherwise. \end{matrix}

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

Arguments

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 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 $- 1 or 1$ 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 $- 1 or 1$ 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, $m = 〈value〉$ .
Constraint: $m \geq 1$ .

$ifail = 2$: On entry, at least one element of vlamda is less than zero.

On entry, at least one element of vlamda is too large.

$ifail = 3$: On entry, state vector has been corrupted or not initialized.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

Not applicable.

8

Parallelism and Performance

g05tkf 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.

9

Further Comments

None.

10

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 Document

g05tkf (int_poisson_varmean)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results