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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$a$ and semi-interquartile range b$b$.

## 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)
 f(x) = 1/(πb (1 + ((x − a)/b)2) ). $f(x)=1πb (1+ (x-ab) 2) .$
nag_rand_dist_cauchy (g05sc) returns the value
 a + b(2y1 − 1)/(y2), $a+b2y1- 1y2,$
where y1${y}_{1}$ and y2${y}_{2}$ are a pair of consecutive pseudorandom numbers from a uniform distribution over (0,1)$\left(0,1\right)$, such that
 (2y1 − 1)2 + y22 ≤ 1. $(2y1-1) 2+y22≤1.$
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:     n – int64int32nag_int scalar
n$n$, the number of pseudorandom numbers to be generated.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     xmed – double scalar
a$a$, the median of the distribution.
3:     semiqr – double scalar
b$b$, the semi-interquartile range of the distribution.
Constraint: semiqr0.0${\mathbf{semiqr}}\ge 0.0$.
4:     state( : $:$) – 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.

None.

### Output Parameters

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

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
On entry, n < 0${\mathbf{n}}<0$.
ifail = 3${\mathbf{ifail}}=3$
On entry, semiqr < 0.0${\mathbf{semiqr}}<0.0$.
ifail = 4${\mathbf{ifail}}=4$
 On entry, state vector was not initialized or has been corrupted.

Not applicable.

None.

## Example

```function nag_rand_dist_cauchy_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
xmed = 1;
semiqr = 2;
% Initialize the generator to a repeatable sequence
[state, ifail] = nag_rand_init_repeat(genid, subid, seed);
[state, x, ifail] = nag_rand_dist_cauchy(n, xmed, semiqr, state)
```
```

state =

17
1234
1
0
9910
16740
20386
10757
17917
13895
19930
8
0
1234
1
1
1234

x =

6.1229
2.2328
-2.2118
0.4118
0.9892

ifail =

0

```
```function g05sc_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
xmed = 1;
semiqr = 2;
% Initialize the generator to a repeatable sequence
[state, ifail] = g05kf(genid, subid, seed);
[state, x, ifail] = g05sc(n, xmed, semiqr, state)
```
```

state =

17
1234
1
0
9910
16740
20386
10757
17917
13895
19930
8
0
1234
1
1
1234

x =

6.1229
2.2328
-2.2118
0.4118
0.9892

ifail =

0

```