# NAG Library Routine Document

## 1Purpose

g05thf generates a vector of pseudorandom integers from the discrete negative binomial distribution with parameter $m$ and probability $p$ of success at a trial.

## 2Specification

Fortran Interface
 Subroutine g05thf ( mode, n, m, p, r, lr, x,
 Integer, Intent (In) :: mode, n, m, lr Integer, Intent (Inout) :: state(*), ifail Integer, Intent (Out) :: x(n) Real (Kind=nag_wp), Intent (In) :: p Real (Kind=nag_wp), Intent (Inout) :: r(lr)
#include nagmk26.h
 void g05thf_ (const Integer *mode, const Integer *n, const Integer *m, const double *p, double r[], const Integer *lr, Integer state[], Integer x[], Integer *ifail)

## 3Description

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.

## 4References

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

## 5Arguments

1:     $\mathbf{mode}$ – 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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of failures of the distribution.
Constraint: ${\mathbf{m}}\ge 0$.
4:     $\mathbf{p}$ – 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:     $\mathbf{r}\left({\mathbf{lr}}\right)$ – 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.
On exit: if ${\mathbf{mode}}\ne 3$, the reference vector.
6:     $\mathbf{lr}$ – 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:     $\mathbf{state}\left(*\right)$ – 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:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Integer arrayOutput
On exit: the $n$ pseudorandom numbers from the specified negative binomial distribution.
9:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, 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, ${\mathbf{mode}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mode}}=0$, $1$, $2$ or $3$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 0$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: $0.0\le {\mathbf{p}}<1.0$.
${\mathbf{ifail}}=5$
On entry, some of the elements of the array r have been corrupted or have not been initialized.
p or m is not the same as when r was set up in a previous call.
Previous value of ${\mathbf{p}}=〈\mathit{\text{value}}〉$ and ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Previous value of ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=6$
On entry, lr is too small when ${\mathbf{mode}}=0$ or $2$: ${\mathbf{lr}}=〈\mathit{\text{value}}〉$, minimum length required $\text{}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=7$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

Not applicable.

## 8Parallelism and Performance

g05thf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10Example

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.

### 10.1Program Text

Program Text (g05thfe.f90)

### 10.2Program Data

Program Data (g05thfe.d)

### 10.3Program Results

Program Results (g05thfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017