g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_rngs_geom (g05mbc)

## 1  Purpose

nag_rngs_geom (g05mbc) 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_rngs_geom (Integer mode, double p, Integer n, Integer x[], Integer igen, Integer iseed[], double r[], NagError *fail)

## 3  Description

nag_rngs_geom (g05mbc) generates $n$ integers ${x}_{I}$ from a discrete geometric distribution, where the probability of ${x}_{i}=I$ (a first success after $I$ trials) is
 $Pxi=I=p×1-pI-1, I=1,2,….$
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_rngs_geom (g05mbc) 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_rngs_init_repeatable (g05kbc) (for a repeatable sequence if computed sequentially) or nag_rngs_init_nonrepeatable (g05kcc) (for a non-repeatable sequence) must be called prior to the first call to nag_rngs_geom (g05mbc).

## 4  References

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

## 5  Arguments

1:     modeIntegerInput
On entry: a code for selecting the operation to be performed by the function.
${\mathbf{mode}}=0$
Set up reference vector only.
${\mathbf{mode}}=1$
Generate variates using reference vector set up in a prior call to nag_rngs_geom (g05mbc).
${\mathbf{mode}}=2$
Set up reference vector and generate variates.
${\mathbf{mode}}=3$
Generate variates without using the reference vector.
Constraint: ${\mathbf{mode}}=0$, $1$, $2$ or $3$.
2:     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)).
3:     nIntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 1$.
4:     x[n]IntegerOutput
On exit: the $n$ pseudorandom numbers from the specified geometric distribution.
5:     igenIntegerInput
On entry: must contain the identification number for the generator to be used to return a pseudorandom number and should remain unchanged following initialization by a prior call to nag_rngs_init_repeatable (g05kbc) or nag_rngs_init_nonrepeatable (g05kcc).
6:     iseed[$4$]IntegerCommunication Array
On entry: contains values which define the current state of the selected generator.
On exit: contains updated values defining the new state of the selected generator.
7:     r[$6+42/{\mathbf{p}}$]doubleCommunication Array
On entry: if ${\mathbf{mode}}=1$, the reference vector from the previous call to nag_rngs_geom (g05mbc).
On exit: the reference vector.
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_DIM_INFEASIBLE
p is so small that the reference vector length would exceed integer range We recommend setting ${\mathbf{mode}}=3$. ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
NE_INT
On entry, ${\mathbf{mode}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mode}}=0$, $1$, $2$ or $3$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_INT_2
On entry, $6+42/{\mathbf{p}}<\mathit{int}\left[30/{\mathbf{p}}-1\right]+6$ $6+42/{\mathbf{p}}=〈\mathit{\text{value}}〉$ and $\mathit{int}\left[30/{\mathbf{p}}-1\right]+6=〈\mathit{\text{value}}〉$.
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_PREV_CALL
p is not the same as when r was set up in a previous call. Previous value $\text{}=〈\mathit{\text{value}}〉$ and ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
NE_REAL
On entry,  or ${\mathbf{p}}>1.0$: ${\mathbf{p}}=〈\mathit{\text{value}}〉$.

## 7  Accuracy

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. There is a point, depending on the distribution, where this improvement becomes very small and the recommended values for the length of array r in other functions are 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 ${\mathbf{mode}}=3$.

## 9  Example

This example prints five pseudorandom integers from a geometric distribution with parameter $p=0.001$, generated by a single call to nag_rngs_geom (g05mbc), after initialization by nag_rngs_init_repeatable (g05kbc).

### 9.1  Program Text

Program Text (g05mbce.c)

None.

### 9.3  Program Results

Program Results (g05mbce.r)