NAG Library Routine Document
g05tkf (int_poisson_varmean)
1
Purpose
g05tkf generates a vector of pseudorandom integers, each from a discrete Poisson distribution with differing parameter.
2
Specification
Fortran Interface
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
${\lambda}_{j}$, where the probability of
${x}_{j}=I$ is
where
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:
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 nonrepeatable 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: $\mathbf{m}$ – IntegerInput

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) arrayInput

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 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: $\mathbf{x}\left({\mathbf{m}}\right)$ – Integer arrayOutput

On exit: the $m$ pseudorandom numbers from the specified Poisson distributions.
 5: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{ or}1$. 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
$1\text{ or}1$ 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 $\mathbf{1}\text{ or}\mathbf{1}$ 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).
6
Error 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, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
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.
 ${\mathbf{ifail}}=99$
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.
 ${\mathbf{ifail}}=399$
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.
 ${\mathbf{ifail}}=999$
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 implementationspecific information.
None.
10
Example
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.
10.1
Program Text
Program Text (g05tkfe.f90)
10.2
Program Data
Program Data (g05tkfe.d)
10.3
Program Results
Program Results (g05tkfe.r)