NAG Library Function Document

nag_rand_f (g05shc)

+− 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

nag_rand_f (g05shc) generates a vector of pseudorandom numbers taken from an

F

(or Fisher's variance ratio) distribution with

μ

and

ν

degrees of freedom.

2 Specification

#include <nag.h>

#include <nagg05.h>

void	nag_rand_f (Integer n, Integer df1, Integer df2, Integer state[], double x[], NagError *fail)

3 Description

The distribution has PDF (probability density function)

\begin{array}{l} f (x) = \frac{(\frac{μ + ν - 2}{2})! x^{\frac{1}{2} μ - 1}}{(\frac{1}{2} μ - 1)! (\frac{1}{2} ν - 1)! {(1 + \frac{μ}{ν} x)}^{\frac{1}{2} (μ + ν)}} \times {(\frac{μ}{ν})}^{\frac{1}{2} μ} & if ​ x > 0, \\ f (x) = 0 & otherwise. \end{array}

nag_rand_f (g05shc) calculates the values

\frac{ν y_{i}}{μ z_{i}}, i = 1, 2, \dots, n,

where

y_{i}

and

z_{i}

are generated by nag_rand_gamma (g05sjc) from gamma distributions with parameters

(\frac{1}{2} μ, 2)

and

(\frac{1}{2} ν, 2)

respectively (i.e., from

χ^{2}

-distributions with

μ

and

ν

degrees of freedom).

One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_f (g05shc).

4 References

Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

5 Arguments

1: n – IntegerInput: On entry: $n$ , the number of pseudorandom numbers to be generated.
Constraint: $n \geq 0$ .
2: df1 – IntegerInput: On entry: $μ$ , the number of degrees of freedom of the distribution.
Constraint: $df1 \geq 1$ .
3: df2 – IntegerInput: On entry: $ν$ , the number of degrees of freedom of the distribution.
Constraint: $df2 \geq 1$ .
4: state[ $\dim$ ] – IntegerCommunication Array: Note: the dimension, $\dim$ , of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).
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[n] – doubleOutput: On exit: the $n$ pseudorandom numbers from the specified $F$ -distribution.
6: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $df1 = ⟨value⟩$ .
Constraint: $df1 \geq 1$ .
On entry, $df2 = ⟨value⟩$ .
Constraint: $df2 \geq 1$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_INVALID_STATE: On entry, state vector has been corrupted or not initialized.

7 Accuracy

Not applicable.

8 Parallelism and Performance

nag_rand_f (g05shc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken by nag_rand_f (g05shc) increases with

μ

and

ν

10 Example

This example prints five pseudorandom numbers from an

F

-distribution with two and three degrees of freedom, generated by a single call to nag_rand_f (g05shc), after initialization by nag_rand_init_repeatable (g05kfc).