G05 Chapter Contents
G05 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG05SEF

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

G05SEF generates a vector of pseudorandom numbers taken from a Dirichlet distribution.

## 2  Specification

 SUBROUTINE G05SEF ( N, M, A, STATE, X, LDX, IFAIL)
 INTEGER N, M, STATE(*), LDX, IFAIL REAL (KIND=nag_wp) A(M), X(LDX,M)

## 3  Description

The distribution has PDF (probability density function)
 $fx = 1 Bα ∏ i=1 m x i αi - 1 and Bα = ∏ i=1 m Γ αi Γ ∑ i=1 m αi$
where $x=\left\{{x}_{1},{x}_{2},\dots ,{x}_{m}\right\}$ is a vector of dimension $m$, such that ${x}_{i}>0$ for all $i$ and $\sum _{\mathit{i}=1}^{m}{x}_{i}=1$.
G05SEF generates a draw from a Dirichlet distribution by first drawing $m$ independent samples, ${y}_{i}\sim \mathrm{gamma}\left({\alpha }_{i},1\right)$, i.e., independent draws from a gamma distribution with parameters ${\alpha }_{i}>0$ and one, and then setting ${x}_{i}={y}_{i}/\sum _{\mathit{j}=1}^{m}{y}_{j}$.
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 G05SEF.

## 4  References

Dagpunar J (1988) Principles of Random Variate Generation Oxford University Press
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

## 5  Parameters

1:     N – INTEGERInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{N}}\ge 0$.
2:     M – INTEGERInput
On entry: $m$, the number of dimensions of the distribution.
Constraint: ${\mathbf{M}}>0$.
3:     A(M) – REAL (KIND=nag_wp) arrayInput
On entry: the parameter vector for the distribution.
Constraint: ${\mathbf{A}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{M}}$.
4:     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.
5:     X(LDX,M) – REAL (KIND=nag_wp) arrayOutput
On exit: the $n$ pseudorandom numbers from the specified Dirichlet distribution, with ${\mathbf{X}}\left(i,j\right)$ holding the $j$th dimension for the $i$th variate.
6:     LDX – INTEGERInput
On entry: the first dimension of the array X as declared in the (sub)program from which G05SEF is called.
Constraint: ${\mathbf{LDX}}\ge {\mathbf{N}}$.
7:     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, ${\mathbf{N}}<0$.
${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{M}}<1$.
${\mathbf{IFAIL}}=3$
On entry, at least one ${\mathbf{A}}\left(\mathit{i}\right)\le 0.0$.
${\mathbf{IFAIL}}=4$
 On entry, STATE vector was not initialized or has been corrupted.
${\mathbf{IFAIL}}=6$
On entry, ${\mathbf{LDX}}<{\mathbf{N}}$.

Not applicable.

None.

## 9  Example

This example prints a set of five pseudorandom numbers from a Dirichlet distribution with parameters $m=4$ and $\alpha =\left\{2.0,2.0,2.0,2.0\right\}$, generated by a single call to G05SEF, after initialization by G05KFF.

### 9.1  Program Text

Program Text (g05sefe.f90)

### 9.2  Program Data

Program Data (g05sefe.d)

### 9.3  Program Results

Program Results (g05sefe.r)