NAG Library Routine Document

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,*)

C Header Interface

#include <nagmk26.h>

void	g05sef_ (const Integer n, const Integer m, const double a[], Integer state[], double x[], const Integer ldx, Integer ifail)

3

Description

The distribution has PDF (probability density function)

\begin{array}{l} f (x) & = & \frac{1}{B (α)} \prod_{i = 1}^{m} x_{i}^{α_{i} - 1} and \\ B (α) & = & \frac{\prod_{i = 1}^{m} Γ (α_{i})}{Γ (\sum_{i = 1}^{m} α_{i})} \end{array}

where

x = \{x_{1}, x_{2}, \dots, x_{m}\}

is a vector of dimension

m

, such that

x_{i} > 0

for all

i

and

\sum_{i = 1}^{m} x_{i} = 1

g05sef generates a draw from a Dirichlet distribution by first drawing

m

independent samples,

y_{i} \sim gamma (α_{i}, 1)

, i.e., independent draws from a gamma distribution with parameters

α_{i} > 0

and one, and then setting

x_{i} = y_{i} / \sum_{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

Arguments

1: $n$ – IntegerInput: On entry: $n$ , the number of pseudorandom numbers to be generated.

Constraint: $n \geq 0$ .
2: $m$ – IntegerInput: On entry: $m$ , the number of dimensions of the distribution.

Constraint: $m > 0$ .
3: $a (m)$ – Real (Kind=nag_wp) arrayInput: On entry: the parameter vector for the distribution.

Constraint: $a (i) > 0.0$ , for $i = 1, 2, \dots, m$ .
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: $x (ldx *)$ – Real (Kind=nag_wp) arrayOutput: Note: the second dimension of the array x must be at least $m$ .

On exit: the $n$ pseudorandom numbers from the specified Dirichlet distribution, with $x (i, j)$ 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: $ldx \geq n$ .
7: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . 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 $- 1 or 1$ 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 $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

$ifail = 2$: On entry, $m = 〈value〉$ .
Constraint: $m > 0$ .

$ifail = 3$: On entry, at least one $a (i) \leq 0$ .

$ifail = 4$: On entry, state vector has been corrupted or not initialized.

$ifail = 6$: On entry, $ldx = 〈value〉$ and $n = 〈value〉$ .
Constraint: $ldx \geq n$ .

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

Not applicable.

8

Parallelism 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.

9

Further Comments

None.

10

Example

This example prints a set of five pseudorandom numbers from a Dirichlet distribution with parameters

m = 4

and

α = \{2.0, 2.0, 2.0, 2.0\}

, generated by a single call to g05sef, after initialization by g05kff.

NAG Library Routine Document

g05sef (dist_dirichlet)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification