g05 Chapter Contents
g05 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_rand_cauchy (g05scc)

## 1  Purpose

nag_rand_cauchy (g05scc) generates a vector of pseudorandom numbers from a Cauchy distribution with median $a$ and semi-interquartile range $b$.

## 2  Specification

 #include #include
 void nag_rand_cauchy (Integer n, double xmed, double semiqr, Integer state[], double x[], NagError *fail)

## 3  Description

The distribution has PDF (probability density function)
 $fx=1πb 1+ x-ab 2 .$
nag_rand_cauchy (g05scc) 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_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_cauchy (g05scc).

## 4  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

## 5  Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
2:    $\mathbf{xmed}$doubleInput
On entry: $a$, the median of the distribution.
3:    $\mathbf{semiqr}$doubleInput
On entry: $b$, the semi-interquartile range of the distribution.
Constraint: ${\mathbf{semiqr}}\ge 0.0$.
4:    $\mathbf{state}\left[\mathit{dim}\right]$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:    $\mathbf{x}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the $n$ pseudorandom numbers from the specified Cauchy distribution.
6:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
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.
See Section 3.6.6 in the Essential Introduction for further information.
NE_INVALID_STATE
On entry, state vector has been corrupted or not initialized.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_REAL
On entry, ${\mathbf{semiqr}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{semiqr}}\ge 0.0$.

Not applicable.

## 8  Parallelism and Performance

nag_rand_cauchy (g05scc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10  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_cauchy (g05scc), after initialization by nag_rand_init_repeatable (g05kfc).

### 10.1  Program Text

Program Text (g05scce.c)

None.

### 10.3  Program Results

Program Results (g05scce.r)