# NAG Library Routine Document

## 1Purpose

g05sbf generates a vector of pseudorandom numbers taken from a beta distribution with parameters $a$ and $b$.

## 2Specification

Fortran Interface
 Subroutine g05sbf ( n, a, b, x,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: state(*), ifail Real (Kind=nag_wp), Intent (In) :: a, b Real (Kind=nag_wp), Intent (Out) :: x(n)
#include nagmk26.h
 void g05sbf_ (const Integer *n, const double *a, const double *b, Integer state[], double x[], Integer *ifail)

## 3Description

The beta distribution has PDF (probability density function)
One of four algorithms is used to generate the variates depending on the values of $a$ and $b$. Let $\alpha$ be the maximum and $\beta$ be the minimum of $a$ and $b$. Then the algorithms are as follows:
 (i) if $\alpha <0.5$, Johnk's algorithm is used, see for example Dagpunar (1988). This generates the beta variate as ${u}_{1}^{1/a}/\left(\begin{array}{c}{u}_{1}^{1/a}+{u}_{2}^{1/b}\end{array}\right)$, where ${u}_{1}$ and ${u}_{2}$ are uniformly distributed random variates; (ii) if $\beta >1$, the algorithm BB given by Cheng (1978) is used. This involves the generation of an observation from a beta distribution of the second kind by the envelope rejection method using a log-logistic target distribution and then transforming it to a beta variate; (iii) if $\alpha >1$ and $\beta <1$, the switching algorithm given by Atkinson (1979) is used. The two target distributions used are ${f}_{1}\left(x\right)=\beta {x}^{\beta }$ and ${f}_{2}\left(x\right)=\alpha {\left(1-x\right)}^{\beta -1}$, along with the approximation to the switching argument of $t=\left(1-\beta \right)/\left(\alpha +1-\beta \right)$; (iv) in all other cases, Cheng's BC algorithm (see Cheng (1978)) is used with modifications suggested by Dagpunar (1988). This algorithm is similar to BB, used when $\beta >1$, but is tuned for small values of $a$ and $b$.
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 g05sbf.

## 4References

Atkinson A C (1979) A family of switching algorithms for the computer generation of beta random variates Biometrika 66 141–5
Cheng R C H (1978) Generating beta variates with nonintegral shape parameters Comm. ACM 21 317–322
Dagpunar J (1988) Principles of Random Variate Generation Oxford University Press
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{a}$ – Real (Kind=nag_wp)Input
On entry: $a$, the parameter of the beta distribution.
Constraint: ${\mathbf{a}}>0.0$.
3:     $\mathbf{b}$ – Real (Kind=nag_wp)Input
On entry: $b$, the parameter of the beta distribution.
Constraint: ${\mathbf{b}}>0.0$.
4:     $\mathbf{state}\left(*\right)$ – 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.
5:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the $n$ pseudorandom numbers from the specified beta distribution.
6:     $\mathbf{ifail}$ – 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, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{a}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}>0.0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{b}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{b}}>0.0$.
${\mathbf{ifail}}=4$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
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

g05sbf 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.

To generate an observation, $y$, from the beta distribution of the second kind from an observation, $x$, generated by g05sbf the transformation, $y=x/\left(1-x\right)$, may be used.

## 10Example

This example prints a set of five pseudorandom numbers from a beta distribution with parameters $a=2.0$ and $b=2.0$, generated by a single call to g05sbf, after initialization by g05kff.

### 10.1Program Text

Program Text (g05sbfe.f90)

### 10.2Program Data

Program Data (g05sbfe.d)

### 10.3Program Results

Program Results (g05sbfe.r)

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