# NAG FL Interfaceg05shf (dist_​f)

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## 1Purpose

g05shf generates a vector of pseudorandom numbers taken from an $F$ (or Fisher's variance ratio) distribution with $\mu$ and $\nu$ degrees of freedom.

## 2Specification

Fortran Interface
 Subroutine g05shf ( n, df1, df2, x,
 Integer, Intent (In) :: n, df1, df2 Integer, Intent (Inout) :: state(*), ifail Real (Kind=nag_wp), Intent (Out) :: x(n)
#include <nag.h>
 void g05shf_ (const Integer *n, const Integer *df1, const Integer *df2, Integer state[], double x[], Integer *ifail)
The routine may be called by the names g05shf or nagf_rand_dist_f.

## 3Description

The distribution has PDF (probability density function)
 $f (x) = ( μ+ν-2 2 ) ! x 12 μ-1 (12μ-1)! (12ν-1) ! (1+μνx) 12 (μ+ν) × (μν) 12μ if ​ x>0 , f(x)=0 otherwise.$
g05shf calculates the values
 $ν yi μ zi , i=1,2,…,n ,$
where ${y}_{i}$ and ${z}_{i}$ are generated by g05sjf 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 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 g05shf.

## 4References

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

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{df1}$Integer Input
On entry: $\mu$, the number of degrees of freedom of the distribution.
Constraint: ${\mathbf{df1}}\ge 1$.
3: $\mathbf{df2}$Integer Input
On entry: $\nu$, the number of degrees of freedom of the distribution.
Constraint: ${\mathbf{df2}}\ge 1$.
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: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: the $n$ pseudorandom numbers from the specified $F$-distribution.
6: $\mathbf{ifail}$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, ${\mathbf{df1}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{df1}}\ge 1$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{df2}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{df2}}\ge 1$.
${\mathbf{ifail}}=4$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=-99$
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

g05shf 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.

The time taken by g05shf increases with $\mu$ and $\nu$.

## 10Example

This example prints five pseudorandom numbers from an $F$-distribution with two and three degrees of freedom, generated by a single call to g05shf, after initialization by g05kff.

### 10.1Program Text

Program Text (g05shfe.f90)

### 10.2Program Data

Program Data (g05shfe.d)

### 10.3Program Results

Program Results (g05shfe.r)