G05 Chapter Contents
G05 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG05THF

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

G05THF generates a vector of pseudorandom integers from the discrete negative binomial distribution with parameter $m$ and probability $p$ of success at a trial.

## 2  Specification

 SUBROUTINE G05THF ( MODE, N, M, P, R, LR, STATE, X, IFAIL)
 INTEGER MODE, N, M, LR, STATE(*), X(N), IFAIL REAL (KIND=nag_wp) P, R(LR)

## 3  Description

G05THF generates $n$ integers ${x}_{i}$ from a discrete negative binomial distribution, where the probability of ${x}_{i}=I$ ($I$ successes before $m$ failures) is
 $Pxi=I= m+I-1! I!m-1! ×pI×1-pm, I=0,1,….$
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 G05THF with the same parameter value can then use this reference vector to generate further variates.
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 G05THF.

## 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 G05THF.
${\mathbf{MODE}}=2$
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:     M – INTEGERInput
On entry: $m$, the number of failures of the distribution.
Constraint: ${\mathbf{M}}\ge 0$.
4:     P – REAL (KIND=nag_wp)Input
On entry: $p$, the parameter of the negative binomial distribution representing the probability of success at a single trial.
Constraint: $0.0\le {\mathbf{P}}<1.0$.
5:     R(LR) – REAL (KIND=nag_wp) arrayCommunication Array
On entry: if ${\mathbf{MODE}}=1$, the reference vector from the previous call to G05THF.
If ${\mathbf{MODE}}=3$, R is not referenced by G05THF.
On exit: the reference vector.
6:     LR – INTEGERInput
On entry: the dimension of the array R as declared in the (sub)program from which G05THF is called.
Suggested values:
• if ${\mathbf{MODE}}\ne 3$,
${\mathbf{LR}}=28+\left(20×\sqrt{{\mathbf{M}}×{\mathbf{P}}}+30×{\mathbf{P}}\right)/\left(1-{\mathbf{P}}\right)\text{​ approximately}$;
• otherwise ${\mathbf{LR}}=1$.
Constraints:
• if ${\mathbf{MODE}}=0$ or $2$,
$\begin{array}{cc}{\mathbf{LR}}>& \mathrm{int}\left(\begin{array}{c}\frac{{\mathbf{M}}×{\mathbf{P}}+7.15×\sqrt{{\mathbf{M}}×{\mathbf{P}}}+20.15×{\mathbf{P}}}{1-{\mathbf{P}}}+8.5\end{array}\right)\\ \\ & -\mathrm{max}\phantom{\rule{0.25em}{0ex}}\left(\begin{array}{c}0,\mathrm{int}\left(\begin{array}{c}\frac{{\mathbf{M}}×{\mathbf{P}}-7.15×\sqrt{{\mathbf{M}}×{\mathbf{P}}}}{1-{\mathbf{P}}}\end{array}\right)\end{array}\right)+9\\ \end{array}$;
• if ${\mathbf{MODE}}=1$, LR must remain unchanged from the previous call to G05THF.
7:     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.
8:     X(N) – INTEGER arrayOutput
On exit: the $n$ pseudorandom numbers from the specified negative binomial distribution.
9:     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, ${\mathbf{M}}<0$.
${\mathbf{IFAIL}}=4$
 On entry, ${\mathbf{P}}<0.0$, or ${\mathbf{P}}\ge 1.0$.
${\mathbf{IFAIL}}=5$
On entry, P or M is not the same as when R was set up in a previous call to G05THF with ${\mathbf{MODE}}=0$ or $2$.
On entry, the R vector was not initialized correctly, or has been corrupted.
${\mathbf{IFAIL}}=6$
On entry, LR is too small when ${\mathbf{MODE}}=0$ or $2$.
${\mathbf{IFAIL}}=7$
 On entry, STATE vector was not initialized or has been corrupted.

Not applicable.

None.

## 9  Example

This example prints $20$ pseudorandom integers from a negative binomial distribution with parameters $m=60$ and $p=0.999$, generated by a single call to G05THF, after initialization by G05KFF.

### 9.1  Program Text

Program Text (g05thfe.f90)

### 9.2  Program Data

Program Data (g05thfe.d)

### 9.3  Program Results

Program Results (g05thfe.r)