# NAG FL Interfaceg05scf (dist_​cauchy)

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## 1Purpose

g05scf generates a vector of pseudorandom numbers from a Cauchy distribution with median $a$ and semi-interquartile range $b$.

## 2Specification

Fortran Interface
 Subroutine g05scf ( n, xmed, x,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: state(*), ifail Real (Kind=nag_wp), Intent (In) :: xmed, semiqr Real (Kind=nag_wp), Intent (Out) :: x(n)
#include <nag.h>
 void g05scf_ (const Integer *n, const double *xmed, const double *semiqr, Integer state[], double x[], Integer *ifail)
The routine may be called by the names g05scf or nagf_rand_dist_cauchy.

## 3Description

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

## 4References

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

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{xmed}$Real (Kind=nag_wp) Input
On entry: $a$, the median of the distribution.
3: $\mathbf{semiqr}$Real (Kind=nag_wp) Input
On entry: $b$, the semi-interquartile range of the distribution.
Constraint: ${\mathbf{semiqr}}\ge 0.0$.
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 Cauchy 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: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{semiqr}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{semiqr}}\ge 0.0$.
${\mathbf{ifail}}=4$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
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

Background information to multithreading can be found in the Multithreading documentation.
g05scf 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.

None.

## 10Example

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.

### 10.1Program Text

Program Text (g05scfe.f90)

### 10.2Program Data

Program Data (g05scfe.d)

### 10.3Program Results

Program Results (g05scfe.r)