Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_rand_dist_cauchy (g05sc)

## Purpose

nag_rand_dist_cauchy (g05sc) generates a vector of pseudorandom numbers from a Cauchy distribution with median $a$ and semi-interquartile range $b$.

## Syntax

[state, x, ifail] = g05sc(n, xmed, semiqr, state)
[state, x, ifail] = nag_rand_dist_cauchy(n, xmed, semiqr, state)

## Description

The distribution has PDF (probability density function)
 $fx=1πb 1+ x-ab 2 .$
nag_rand_dist_cauchy (g05sc) returns the value
 $a+b2y1- 1y2,$
where ${y}_{1}$ and ${y}_{2}$ are a pair of consecutive pseudorandom numbers from a uniform distribution over $\left(0,1\right)$, such that
 $2y1-1 2+y22≤1.$
One of the initialization functions nag_rand_init_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_dist_cauchy (g05sc).

## References

Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin
Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
$n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathrm{xmed}$ – double scalar
$a$, the median of the distribution.
3:     $\mathrm{semiqr}$ – double scalar
$b$, the semi-interquartile range of the distribution.
Constraint: ${\mathbf{semiqr}}\ge 0.0$.
4:     $\mathrm{state}\left(:\right)$int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains information on the selected base generator and its current state.

None.

### Output Parameters

1:     $\mathrm{state}\left(:\right)$int64int32nag_int array
Contains updated information on the state of the generator.
2:     $\mathrm{x}\left({\mathbf{n}}\right)$ – double array
The $n$ pseudorandom numbers from the specified Cauchy distribution.
3:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=3$
Constraint: ${\mathbf{semiqr}}\ge 0.0$.
${\mathbf{ifail}}=4$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

Not applicable.

None.

## Example

This example prints the first five pseudorandom real numbers from a Cauchy distribution with median $1.0$ and semi-interquartile range $2.0$, generated by a single call to nag_rand_dist_cauchy (g05sc), after initialization by nag_rand_init_repeat (g05kf).
```function g05sc_example

fprintf('g05sc example results\n\n');

% Initialize the base generator to a repeatable sequence
seed  = [int64(1762543)];
genid = int64(1);
subid = int64(1);
[state, ifail] = g05kf( ...
genid, subid, seed);

% Number of variates
n = int64(5);

% Parameters
xmed = 1;
semiqr = 2;

% Generate variates from Cauchy distribution
[state, x, ifail] = g05sc( ...
n, xmed, semiqr, state);

disp('Variates');
disp(x);

```
```g05sc example results

Variates
6.1229
2.2328
-2.2118
0.4118
0.9892

```