# NAG FL Interfaceg05tkf (int_​poisson_​varmean)

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## 1Purpose

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

## 2Specification

Fortran Interface
 Subroutine g05tkf ( m, x,
 Integer, Intent (In) :: m Integer, Intent (Inout) :: state(*), ifail Integer, Intent (Out) :: x(m) Real (Kind=nag_wp), Intent (In) :: vlamda(m)
#include <nag.h>
 void g05tkf_ (const Integer *m, const double vlamda[], Integer state[], Integer x[], Integer *ifail)
The routine may be called by the names g05tkf or nagf_rand_int_poisson_varmean.

## 3Description

g05tkf 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 routine 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 g05tjf 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:
 $f(x)=13 if ​|x|≤1, f(x)=13|x|−3 otherwise.$
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 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.

## 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: $\mathbf{vlamda}\left({\mathbf{m}}\right)$Real (Kind=nag_wp) array Input
On entry: the means, ${\lambda }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,{\mathbf{m}}$, of the Poisson distributions.
Constraint: $0.0\le {\mathbf{vlamda}}\left(\mathit{j}\right)\le {\mathbf{x02bbf}}/2.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{m}}$.
3: $\mathbf{state}\left(*\right)$Integer array Communication 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: $\mathbf{x}\left({\mathbf{m}}\right)$Integer array Output
On exit: the $m$ pseudorandom numbers from the specified Poisson distributions.
5: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, at least one element of vlamda is less than zero.
On entry, at least one element of vlamda is too large.
${\mathbf{ifail}}=3$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

Not applicable.

## 8Parallelism 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.

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 g05tkf, after initialization by g05kff.

### 10.1Program Text

Program Text (g05tkfe.f90)

### 10.2Program Data

Program Data (g05tkfe.d)

### 10.3Program Results

Program Results (g05tkfe.r)