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 nag_rand_poisson (g05tjc) 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

λ

is less than

50

and by Stirling's formula otherwise.

One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_poisson (g05tjc).

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 Arguments

1: $mode$ – Nag_ModeRNGInput

On entry: a code for selecting the operation to be performed by the function.

$mode = Nag_InitializeReference$: Set up reference vector only.
$mode = Nag_GenerateFromReference$: Generate variates using reference vector set up in a prior call to nag_rand_poisson (g05tjc).
$mode = Nag_InitializeAndGenerate$: Set up reference vector and generate variates.
$mode = Nag_GenerateWithoutReference$: Generate variates without using the reference vector.

Constraint:

mode = Nag_InitializeReference

Nag_GenerateFromReference

Nag_InitializeAndGenerate

Nag_GenerateWithoutReference

2: $n$ – IntegerInput

On entry:

n

, the number of pseudorandom numbers to be generated.

Constraint:

n \geq 0

3: $lambda$ – doubleInput

On entry:

λ

, the mean of the Poisson distribution.

Constraint:

lambda \geq 0.0

4: $r [lr]$ – doubleCommunication Array

On entry: if

mode = Nag_GenerateFromReference

, the reference vector from the previous call to nag_rand_poisson (g05tjc).

mode = Nag_GenerateWithoutReference

, r is not referenced and may be NULL.

On exit: if

mode \neq Nag_GenerateWithoutReference

, the reference vector.

5: $lr$ – IntegerInput

On entry: the dimension of the array r.

Suggested values:

if $mode \neq Nag_GenerateWithoutReference$ , $lr = 30 + 20 \times \sqrt{lambda} + lambda$ ;
otherwise $lr = 1$ .

Constraints:

if mode=Nag_InitializeReference or Nag_InitializeAndGenerate,
- if $\sqrt{lambda} > 7.15$ , $lr > 9 + int (8.5 + 14.3 \times \sqrt{lambda})$ ;
- otherwise $lr > 9 + int (lambda + 7.15 \times \sqrt{lambda} + 8.5)$ ;
if $mode = Nag_GenerateFromReference$ , lr must remain unchanged from the previous call to nag_rand_poisson (g05tjc).

6: $state [\dim]$ – IntegerCommunication Array

Note: the dimension,

\dim

, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).

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: $x [n]$ – IntegerOutput

On exit: the

n

pseudorandom numbers from the specified Poisson distribution.

8: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, lr is too small when $mode = Nag_InitializeReference$ or $Nag_InitializeAndGenerate$ : $lr = 〈value〉$ , minimum length required $= 〈value〉$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_INVALID_STATE: On entry, state vector has been corrupted or not initialized.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_PREV_CALL: lambda is not the same as when r was set up in a previous call.
Previous value of $lambda = 〈value〉$ and $lambda = 〈value〉$ .
NE_REAL: lambda is such that lr would have to be larger than the largest representable integer. Use $mode = Nag_GenerateWithoutReference$ instead. $lambda = 〈value〉$ .

On entry, $lambda = 〈value〉$ .
Constraint: $lambda \geq 0.0$ .
NE_REF_VEC: On entry, some of the elements of the array r have been corrupted or have not been initialized.

7 Accuracy

Not applicable.

8 Parallelism and Performance

nag_rand_poisson (g05tjc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.