# NAG Library Routine Document

## 1Purpose

g05sef generates a vector of pseudorandom numbers taken from a Dirichlet distribution.

## 2Specification

Fortran Interface
 Subroutine g05sef ( n, m, a, x, ldx,
 Integer, Intent (In) :: n, m, ldx Integer, Intent (Inout) :: state(*), ifail Real (Kind=nag_wp), Intent (In) :: a(m) Real (Kind=nag_wp), Intent (Inout) :: x(ldx,m)
#include nagmk26.h
 void g05sef_ (const Integer *n, const Integer *m, const double a[], Integer state[], double x[], const Integer *ldx, Integer *ifail)

## 3Description

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.

## 4References

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:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of dimensions of the distribution.
Constraint: ${\mathbf{m}}>0$.
3:     $\mathbf{a}\left({\mathbf{m}}\right)$ – 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:     $\mathbf{state}\left(*\right)$ – 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:     $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{m}}\right)$ – 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:     $\mathbf{ldx}$ – 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:     $\mathbf{ifail}$ – 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, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}>0$.
${\mathbf{ifail}}=3$
On entry, at least one ${\mathbf{a}}\left(i\right)\le 0$.
${\mathbf{ifail}}=4$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=6$
On entry, ${\mathbf{ldx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
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

g05sef 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 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.

### 10.1Program Text

Program Text (g05sefe.f90)

### 10.2Program Data

Program Data (g05sefe.d)

### 10.3Program Results

Program Results (g05sefe.r)

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