# NAG CL Interfaceg05tjc (int_​poisson)

## 1Purpose

g05tjc generates a vector of pseudorandom integers from the discrete Poisson distribution with mean $\lambda$.

## 2Specification

 #include
 void g05tjc (Nag_ModeRNG mode, Integer n, double lambda, double r[], Integer lr, Integer state[], Integer x[], NagError *fail)
The function may be called by the names: g05tjc, nag_rand_int_poisson or nag_rand_poisson.

## 3Description

g05tjc 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 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 $\lambda$ is less than $50$ and by Stirling's formula otherwise.
One of the initialization functions g05kfc (for a repeatable sequence if computed sequentially) or g05kgc (for a non-repeatable sequence) must be called prior to the first call to g05tjc.

## 4References

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

## 5Arguments

1: $\mathbf{mode}$Nag_ModeRNG Input
On entry: a code for selecting the operation to be performed by the function.
${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$
Set up reference vector only.
${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$
Generate variates using reference vector set up in a prior call to g05tjc.
${\mathbf{mode}}=\mathrm{Nag_InitializeAndGenerate}$
Set up reference vector and generate variates.
${\mathbf{mode}}=\mathrm{Nag_GenerateWithoutReference}$
Generate variates without using the reference vector.
Constraint: ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$, $\mathrm{Nag_GenerateFromReference}$, $\mathrm{Nag_InitializeAndGenerate}$ or $\mathrm{Nag_GenerateWithoutReference}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{lambda}$double Input
On entry: $\lambda$, the mean of the Poisson distribution.
Constraint: ${\mathbf{lambda}}\ge 0.0$.
4: $\mathbf{r}\left[{\mathbf{lr}}\right]$double Communication Array
On entry: if ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$, the reference vector from the previous call to g05tjc.
If ${\mathbf{mode}}=\mathrm{Nag_GenerateWithoutReference}$, r is not referenced and may be NULL.
On exit: if ${\mathbf{mode}}\ne \mathrm{Nag_GenerateWithoutReference}$, the reference vector.
5: $\mathbf{lr}$Integer Input
On entry: the dimension of the array r.
Suggested values:
• if ${\mathbf{mode}}\ne \mathrm{Nag_GenerateWithoutReference}$, ${\mathbf{lr}}=30+20×\sqrt{{\mathbf{lambda}}}+{\mathbf{lambda}}$;
• otherwise ${\mathbf{lr}}=1$.
Constraints:
• if ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ or $\mathrm{Nag_InitializeAndGenerate}$,
• 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 ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$, lr must remain unchanged from the previous call to g05tjc.
6: $\mathbf{state}\left[\mathit{dim}\right]$Integer Communication Array
Note: the dimension, $\mathit{dim}$, 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: $\mathbf{x}\left[{\mathbf{n}}\right]$Integer Output
On exit: the $n$ pseudorandom numbers from the specified Poisson distribution.
8: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, lr is too small when ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ or $\mathrm{Nag_InitializeAndGenerate}$: ${\mathbf{lr}}=〈\mathit{\text{value}}〉$, minimum length required $\text{}=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface 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 8 in the Introduction to the NAG Library CL Interface for further information.
NE_PREV_CALL
lambda is not the same as when r was set up in a previous call.
Previous value of ${\mathbf{lambda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{lambda}}=〈\mathit{\text{value}}〉$.
NE_REAL
lambda is such that lr would have to be larger than the largest representable integer. Use ${\mathbf{mode}}=\mathrm{Nag_GenerateWithoutReference}$ instead. ${\mathbf{lambda}}=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{lambda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lambda}}\ge 0.0$.
NE_REF_VEC
On entry, some of the elements of the array r have been corrupted or have not been initialized.

Not applicable.

## 8Parallelism and Performance

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.

None.

## 10Example

This example prints $10$ pseudorandom integers from a Poisson distribution with mean $\lambda =20$, generated by a single call to g05tjc, after initialization by g05kfc.

### 10.1Program Text

Program Text (g05tjce.c)

None.

### 10.3Program Results

Program Results (g05tjce.r)