g05 Chapter Contents
g05 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_rand_f (g05shc)

## 1  Purpose

nag_rand_f (g05shc) generates a vector of pseudorandom numbers taken from an $F$ (or Fisher's variance ratio) distribution with $\mu$ and $\nu$ degrees of freedom.

## 2  Specification

 #include #include
 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)
 $f x = μ+ν-2 2 ! x 12 μ-1 12 μ-1! 12 ν-1 ! 1+ μν x 12 μ+ν × μν 12μ if ​ x>0 , fx=0 otherwise.$
nag_rand_f (g05shc) calculates the values
 $ν yi μ zi , i=1,2,…,n ,$
where ${y}_{i}$ and ${z}_{i}$ are generated by nag_rand_gamma (g05sjc) from gamma distributions with parameters $\left(\frac{1}{2}\mu ,2\right)$ and $\left(\frac{1}{2}\nu ,2\right)$ respectively (i.e., from ${\chi }^{2}$-distributions with $\mu$ and $\nu$ 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).
Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

## 5  Arguments

1:     nIntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
2:     df1IntegerInput
On entry: $\mu$, the number of degrees of freedom of the distribution.
Constraint: ${\mathbf{df1}}\ge 1$.
3:     df2IntegerInput
On entry: $\nu$, the number of degrees of freedom of the distribution.
Constraint: ${\mathbf{df2}}\ge 1$.
4:     state[$\mathit{dim}$]IntegerCommunication Array
Note: the dimension, $\mathit{dim}$, 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:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{df1}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{df1}}\ge 1$.
On entry, ${\mathbf{df2}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{df2}}\ge 1$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
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.

Not applicable.

## 8  Parallelism and Performance

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

The time taken by nag_rand_f (g05shc) increases with $\mu$ and $\nu$.

## 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).

### 10.1  Program Text

Program Text (g05shce.c)

None.

### 10.3  Program Results

Program Results (g05shce.r)