# NAG FL Interfaceg05sef (dist_​dirichlet)

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## 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,*)
#include <nag.h>
 void g05sef_ (const Integer *n, const Integer *m, const double a[], Integer state[], double x[], const Integer *ldx, Integer *ifail)
The routine may be called by the names g05sef or nagf_rand_dist_dirichlet.

## 3Description

The distribution has PDF (probability density function)
 $f(x) = 1 B(α) ∏ i=1 m x i αi - 1 and B(α) = ∏ i=1 m Γ (αi) Γ (∑ i=1 m αi)$
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: $\mathbf{n}$Integer Input
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{m}$Integer Input
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) array Input
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 array Communication 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: $\mathbf{x}\left({\mathbf{ldx}},*\right)$Real (Kind=nag_wp) array Output
Note: the second dimension of the array x must be at least ${\mathbf{m}}$.
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: $\mathbf{ldx}$Integer Input
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: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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)