# NAG CL Interfaceg05tkc (int_​poisson_​varmean)

## 1Purpose

g05tkc generates a vector of pseudorandom integers, each from a discrete Poisson distribution with differing parameter.

## 2Specification

 #include
 void g05tkc (Integer m, const double vlamda[], Integer state[], Integer x[], NagError *fail)
The function may be called by the names: g05tkc, nag_rand_int_poisson_varmean or nag_rand_compd_poisson.

## 3Description

g05tkc generates $m$ integers ${x}_{j}$, each from a discrete Poisson distribution with mean ${\lambda }_{j}$, where the probability of ${x}_{j}=I$ is
 $P xj=I = λjI × e -λj I! , I=0,1,… ,$
where
 $λj ≥ 0 , j=1,2,…,m .$
The methods used by this function have low set up times and are designed for efficient use when the value of the parameter $\lambda$ changes during the simulation. For large samples from a distribution with fixed $\lambda$ using g05tjc to set up and use a reference vector may be more efficient.
When $\lambda <7.5$ the product of uniforms method is used, see for example Dagpunar (1988). For larger values of $\lambda$ an envelope rejection method is used with a target distribution:
 $fx=13 if ​x≤1, fx=13x-3 otherwise.$
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 $\lambda \ge 87$ 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 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 g05tkc.

## 4References

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

## 5Arguments

1: $\mathbf{m}$Integer Input
On entry: $m$, the number of Poisson distributions for which pseudorandom variates are required.
Constraint: ${\mathbf{m}}\ge 1$.
2: $\mathbf{vlamda}\left[{\mathbf{m}}\right]$const double Input
On entry: the means, ${\lambda }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,{\mathbf{m}}$, of the Poisson distributions.
Constraint: $0.0\le {\mathbf{vlamda}}\left[\mathit{j}-1\right]\le {\mathbf{nag_max_integer}}/2.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{m}}$.
3: $\mathbf{state}\left[\mathit{dim}\right]$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.
4: $\mathbf{x}\left[{\mathbf{m}}\right]$Integer Output
On exit: the $m$ pseudorandom numbers from the specified Poisson distributions.
5: $\mathbf{fail}$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.
NE_BAD_PARAM
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
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_REAL_ARRAY
On entry, at least one element of vlamda is less than zero.
On entry, at least one element of vlamda is too large.

Not applicable.

## 8Parallelism and Performance

g05tkc 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 ten pseudorandom integers from five Poisson distributions with means ${\lambda }_{1}=0.5$, ${\lambda }_{2}=5$, ${\lambda }_{3}=10$, ${\lambda }_{4}=500$ and ${\lambda }_{5}=1000$. These are generated by ten calls to g05tkc, after initialization by g05kfc.

### 10.1Program Text

Program Text (g05tkce.c)

None.

### 10.3Program Results

Program Results (g05tkce.r)