# NAG FL Interfaceg05rff (copula_​frank_​bivar)

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## 1Purpose

g05rff generates pseudorandom uniform bivariates with joint distribution of a Frank Archimedean copula.

## 2Specification

Fortran Interface
 Subroutine g05rff ( n, x, ldx, sdx,
 Integer, Intent (In) :: n, sorder, ldx, sdx Integer, Intent (Inout) :: state(*), ifail Real (Kind=nag_wp), Intent (In) :: theta Real (Kind=nag_wp), Intent (Inout) :: x(ldx,sdx)
#include <nag.h>
 void g05rff_ (const Integer *n, const double *theta, const Integer *sorder, Integer state[], double x[], const Integer *ldx, const Integer *sdx, Integer *ifail)
The routine may be called by the names g05rff or nagf_rand_copula_frank_bivar.

## 3Description

Generates pseudorandom uniform bivariates $\left\{{u}_{1},{u}_{2}\right\}\in {\left[0,1\right]}^{2}$ whose joint distribution is the Frank Archimedean copula ${C}_{\theta }$ with parameter $\theta$, given by
 $Cθ = - 1θ ln[1+ ( e -θu1 -1) ( e -θu2 -1) e-θ-1 ] , θ ∈ (-∞,∞) ∖ {0}$
with the special cases:
• ${C}_{-\infty }=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({u}_{1}+{u}_{2}-1,0\right)$, the Fréchet–Hoeffding lower bound;
• ${C}_{0}={u}_{1}{u}_{2}$, the product copula;
• ${C}_{\infty }=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({u}_{1},{u}_{2}\right)$, the Fréchet–Hoeffding upper bound.
The generation method uses conditional sampling.
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 g05rff.

## 4References

Nelsen R B (2006) An Introduction to Copulas (2nd Edition) Springer Series in Statistics

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of bivariates to generate.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{theta}$Real (Kind=nag_wp) Input
On entry: $\theta$, the copula parameter.
3: $\mathbf{sorder}$Integer Input
On entry: determines the storage order of variates; the $\left(\mathit{i},\mathit{j}\right)$th variate is stored in ${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$ if ${\mathbf{sorder}}=1$, and ${\mathbf{x}}\left(\mathit{j},\mathit{i}\right)$ if ${\mathbf{sorder}}=2$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2$.
Constraint: ${\mathbf{sorder}}=1$ or $2$.
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: $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{sdx}}\right)$Real (Kind=nag_wp) array Output
On exit: the $n$ bivariate uniforms with joint distribution described by ${C}_{\theta }$, with ${\mathbf{x}}\left(i,j\right)$ holding the $i$th value for the $j$th dimension if ${\mathbf{sorder}}=1$ and the $j$th value for the $i$th dimension if ${\mathbf{sorder}}=2$.
6: $\mathbf{ldx}$Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which g05rff is called.
Constraints:
• if ${\mathbf{sorder}}=1$, ${\mathbf{ldx}}\ge {\mathbf{n}}$;
• if ${\mathbf{sorder}}=2$, ${\mathbf{ldx}}\ge 2$.
7: $\mathbf{sdx}$Integer Input
On entry: the second dimension of the array x as declared in the (sub)program from which g05rff is called.
Constraints:
• if ${\mathbf{sorder}}=1$, ${\mathbf{sdx}}\ge 2$;
• if ${\mathbf{sorder}}=2$, ${\mathbf{sdx}}\ge {\mathbf{n}}$.
8: $\mathbf{ifail}$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, corrupt state argument.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=4$
On entry, invalid sorder.
Constraint: ${\mathbf{sorder}}=1$ or $2$.
${\mathbf{ifail}}=6$
On entry, ldx must be at least $⟨\mathit{\text{value}}⟩$: ${\mathbf{ldx}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=7$
On entry, sdx must be at least $⟨\mathit{\text{value}}⟩$: ${\mathbf{sdx}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=-99$
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

g05rff 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.

In practice, the need for numerical stability restricts the range of $\theta$ such that:
• if $\theta <\mathrm{ln}{\epsilon }_{s}$, the routine returns pseudorandom uniform variates with ${C}_{-\infty }$ joint distribution;
• if $|\theta |<1.0×{10}^{-6}$, the routine returns pseudorandom uniform variates with ${C}_{0}$ joint distribution;
• if $\theta >\mathrm{ln}\epsilon$, the routine returns pseudorandom uniform variates with ${C}_{\infty }$ joint distribution;
where ${\epsilon }_{s}$ is the safe-range parameter, the value of which is returned by x02amf; and $\epsilon$ is the machine precision returned by x02ajf.

## 10Example

This example generates thirteen variates for copula ${C}_{-12.0}$.

### 10.1Program Text

Program Text (g05rffe.f90)

### 10.2Program Data

Program Data (g05rffe.d)

### 10.3Program Results

Program Results (g05rffe.r)