G05 Chapter Contents
G05 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG05SCF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G05SCF generates a vector of pseudorandom numbers from a Cauchy distribution with median $a$ and semi-interquartile range $b$.

## 2  Specification

 SUBROUTINE G05SCF ( N, XMED, SEMIQR, STATE, X, IFAIL)
 INTEGER N, STATE(*), IFAIL REAL (KIND=nag_wp) XMED, SEMIQR, X(N)

## 3  Description

The distribution has PDF (probability density function)
 $fx=1πb 1+ x-ab 2 .$
G05SCF 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 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 G05SCF.

## 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  Parameters

1:     N – INTEGERInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{N}}\ge 0$.
2:     XMED – REAL (KIND=nag_wp)Input
On entry: $a$, the median of the distribution.
3:     SEMIQR – REAL (KIND=nag_wp)Input
On entry: $b$, the semi-interquartile range of the distribution.
Constraint: ${\mathbf{SEMIQR}}\ge 0.0$.
4:     STATE($*$) – INTEGER arrayCommunication 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:     X(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the $n$ pseudorandom numbers from the specified Cauchy distribution.
6:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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, ${\mathbf{N}}<0$.
${\mathbf{IFAIL}}=3$
On entry, ${\mathbf{SEMIQR}}<0.0$.
${\mathbf{IFAIL}}=4$
 On entry, STATE vector was not initialized or has been corrupted.

Not applicable.

None.

## 9  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 G05SCF, after initialization by G05KFF.

### 9.1  Program Text

Program Text (g05scfe.f90)

### 9.2  Program Data

Program Data (g05scfe.d)

### 9.3  Program Results

Program Results (g05scfe.r)