NAG CL Interfaceg05scc (dist_​cauchy)

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1Purpose

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

2Specification

 #include
 void g05scc (Integer n, double xmed, double semiqr, Integer state[], double x[], NagError *fail)
The function may be called by the names: g05scc, nag_rand_dist_cauchy or nag_rand_cauchy.

3Description

The distribution has PDF (probability density function)
 $f(x)=1πb (1+ (x-ab) 2) .$
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 g05kfc (for a repeatable sequence if computed sequentially) or g05kgc (for a non-repeatable sequence) must be called prior to the first call to g05scc.

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: $\mathbf{xmed}$double Input
On entry: $a$, the median of the distribution.
3: $\mathbf{semiqr}$double Input
On entry: $b$, the semi-interquartile range of the distribution.
Constraint: ${\mathbf{semiqr}}\ge 0.0$.
4: $\mathbf{state}\left[\mathit{dim}\right]$Integer Communication 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: $\mathbf{x}\left[{\mathbf{n}}\right]$double Output
On exit: the $n$ pseudorandom numbers from the specified Cauchy distribution.
6: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface 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 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, ${\mathbf{semiqr}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{semiqr}}\ge 0.0$.

Not applicable.

8Parallelism and Performance

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.

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 g05scc, after initialization by g05kfc.

10.1Program Text

Program Text (g05scce.c)

None.

10.3Program Results

Program Results (g05scce.r)