G05 Chapter Contents
G05 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG05TCF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G05TCF generates a vector of pseudorandom integers from the discrete geometric distribution with probability $p$ of success at a trial.

## 2  Specification

 SUBROUTINE G05TCF ( MODE, N, P, R, LR, STATE, X, IFAIL)
 INTEGER MODE, N, LR, STATE(*), X(N), IFAIL REAL (KIND=nag_wp) P, R(LR)

## 3  Description

G05TCF 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 G05TCF 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 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 G05TCF.

## 4  References

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

## 5  Parameters

1:     MODE – INTEGERInput
On entry: a code for selecting the operation to be performed by the routine.
${\mathbf{MODE}}=0$
Set up reference vector only.
${\mathbf{MODE}}=1$
Generate variates using reference vector set up in a prior call to G05TCF.
${\mathbf{MODE}}=2$
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:     N – INTEGERInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{N}}\ge 0$.
3:     P – REAL (KIND=nag_wp)Input
On entry: the parameter $p$ of the geometric distribution representing the probability of success at a single trial.
Constraint:  (see X02AJF).
4:     R(LR) – REAL (KIND=nag_wp) arrayCommunication Array
On entry: if ${\mathbf{MODE}}=1$, the reference vector from the previous call to G05TCF.
If ${\mathbf{MODE}}=3$, R is not referenced by G05TCF.
On exit: the reference vector.
5:     LR – INTEGERInput
On entry: the dimension of the array R as declared in the (sub)program from which G05TCF is called.
Suggested values:
• if ${\mathbf{MODE}}\ne 3$, ${\mathbf{LR}}=8+42/{\mathbf{P}}$ approximately (see Section 8);
• otherwise ${\mathbf{LR}}=1$.
Constraints:
• if ${\mathbf{MODE}}=0$ or $2$, ${\mathbf{LR}}\ge 30/{\mathbf{P}}+8$;
• if ${\mathbf{MODE}}=1$, LR should remain unchanged from the previous call to G05TCF.
6:     STATE($*$) – 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.
7:     X(N) – INTEGER arrayOutput
On exit: the $n$ pseudorandom numbers from the specified geometric distribution.
8:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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 ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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, ${\mathbf{MODE}}\ne 0$, $1$, $2$ or $3$.
${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{N}}<0$.
${\mathbf{IFAIL}}=3$
 On entry, ${\mathbf{P}}<0.0$ or ${\mathbf{P}}>1.0$, or ${\mathbf{MODE}}=0$ or $2$ and P is so small that LR would have to be larger than the largest representable integer. Use ${\mathbf{MODE}}=3$ in this case.
${\mathbf{IFAIL}}=4$
On entry, P is not the same as when R was set up in a previous call to G05TCF with ${\mathbf{MODE}}=0$ or $2$.
On entry, the R vector was not initialized correctly or has been corrupted.
${\mathbf{IFAIL}}=5$
On entry, LR is too small when ${\mathbf{MODE}}=0$ or $2$.
${\mathbf{IFAIL}}=6$
 On entry, STATE vector was not initialized or has been corrupted.

## 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. 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 ${\mathbf{MODE}}=3$.

## 9  Example

This example prints $10$ pseudorandom integers from a geometric distribution with parameter $p=0.001$, generated by a single call to G05TCF, after initialization by G05KFF.

### 9.1  Program Text

Program Text (g05tcfe.f90)

### 9.2  Program Data

Program Data (g05tcfe.d)

### 9.3  Program Results

Program Results (g05tcfe.r)