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g05 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_rand_geom (g05tcc)

## 1  Purpose

nag_rand_geom (g05tcc) generates a vector of pseudorandom integers from the discrete geometric distribution with probability $p$ of success at a trial.

## 2  Specification

 #include #include
 void nag_rand_geom (Nag_ModeRNG mode, Integer n, double p, double r[], Integer lr, Integer state[], Integer x[], NagError *fail)

## 3  Description

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
 $P xi=I = p × 1-p I , I=0,1,… .$
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 non-repeatable sequence) must be called prior to the first call to nag_rand_geom (g05tcc).

## 4  References

Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

## 5  Arguments

1:     modeNag_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:     nIntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
3:     pdoubleInput
On entry: the parameter $p$ of the geometric distribution representing the probability of success at a single trial.
Constraint:  (see nag_machine_precision (X02AJC)).
4:     r[lr]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:     lrIntegerInput
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:     state[$\mathit{dim}$]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:     x[n]IntegerOutput
On exit: the $n$ pseudorandom numbers from the specified geometric distribution.
8:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

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.
NE_INVALID_STATE
On entry, state vector has been corrupted or not initialized.
NE_PREV_CALL
p is not the same as when r was set up in a previous call.
Previous value of ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
NE_REAL
On entry, ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: .
p is so small that lr would have to be larger than the largest representable integer. Use ${\mathbf{mode}}=\mathrm{Nag_GenerateWithoutReference}$ instead. ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$
NE_REF_VEC
On entry, some of the elements of the array r have been corrupted or have not been initialized.

Not applicable.

## 8  Parallelism and Performance

Not applicable.

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 ${\mathbf{mode}}=\mathrm{Nag_GenerateWithoutReference}$.

## 10  Example

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).

### 10.1  Program Text

Program Text (g05tcce.c)

None.

### 10.3  Program Results

Program Results (g05tcce.r)