nag_rand_geom (g05tcc) generates a vector of pseudorandom integers from the discrete geometric distribution with probability $p$ of success at a trial.
nag_rand_geom (g05tcc) generates
$n$ integers
${x}_{i}$ from a discrete geometric distribution, where the probability of
${x}_{i}=I$ (a first success after
$I+1$ trials) is
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_geom (g05tcc) with the same parameter value can then use this reference vector to generate further variates. If the search table is not used (as recommended for small values of $p$) then a direct transformation of uniform variates is used.
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 nonrepeatable sequence) must be called prior to the first call to nag_rand_geom (g05tcc).
 1:
$\mathbf{mode}$ – Nag_ModeRNGInput

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 nag_rand_geom (g05tcc).
 ${\mathbf{mode}}=\mathrm{Nag\_InitializeAndGenerate}$
 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:
$\mathbf{n}$ – IntegerInput

On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint:
${\mathbf{n}}\ge 0$.
 3:
$\mathbf{p}$ – doubleInput

On entry: the parameter $p$ of the geometric distribution representing the probability of success at a single trial.
Constraint:
$\mathit{machineprecision}\le {\mathbf{p}}\le 1.0$ (see
nag_machine_precision (X02AJC)).
 4:
$\mathbf{r}\left[{\mathbf{lr}}\right]$ – doubleCommunication Array

On entry: if
${\mathbf{mode}}=\mathrm{Nag\_GenerateFromReference}$, the reference vector from the previous call to nag_rand_geom (g05tcc).
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:
$\mathbf{lr}$ – IntegerInput

On entry: the dimension of the array
r.
Suggested values:
 if ${\mathbf{mode}}\ne \mathrm{Nag\_GenerateWithoutReference}$, ${\mathbf{lr}}=8+42/{\mathbf{p}}$ approximately (see Section 9);
 otherwise ${\mathbf{lr}}=1$.
Constraints:
 if ${\mathbf{mode}}=\mathrm{Nag\_InitializeReference}$ or $\mathrm{Nag\_InitializeAndGenerate}$, ${\mathbf{lr}}\ge 30/{\mathbf{p}}+8$;
 if ${\mathbf{mode}}=\mathrm{Nag\_GenerateFromReference}$, lr should remain unchanged from the previous call to nag_rand_geom (g05tcc).
 6:
$\mathbf{state}\left[\mathit{dim}\right]$ – IntegerCommunication 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:
$\mathbf{x}\left[{\mathbf{n}}\right]$ – IntegerOutput

On exit: the $n$ pseudorandom numbers from the specified geometric distribution.
 8:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 2.7 in How to Use the NAG Library and its Documentation).
Not applicable.
nag_rand_geom (g05tcc) 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 implementationspecific information.
The time taken to set up the reference vector, if used, increases with the length of array
r. However, if the reference vector is used, the time taken to generate numbers decreases as the space allotted to the index part of
r increases. Nevertheless, there is a point, depending on the distribution, where this improvement becomes very small and the suggested value for the length of array
r is designed to approximate this point.
If
p is very small then the storage requirements for the reference vector and the time taken to set up the reference vector becomes prohibitive. In this case it is recommended that the reference vector is not used. This is achieved by selecting
${\mathbf{mode}}=\mathrm{Nag\_GenerateWithoutReference}$.
This example prints
$10$ pseudorandom integers from a geometric distribution with parameter
$p=0.001$, generated by a single call to nag_rand_geom (g05tcc), after initialization by
nag_rand_init_repeatable (g05kfc).
None.