# NAG FL Interfaceg05tcf (int_​geom)

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## 1Purpose

g05tcf generates a vector of pseudorandom integers from the discrete geometric distribution with probability $p$ of success at a trial.

## 2Specification

Fortran Interface
 Subroutine g05tcf ( mode, n, p, r, lr, x,
 Integer, Intent (In) :: mode, n, lr Integer, Intent (Inout) :: state(*), ifail Integer, Intent (Out) :: x(n) Real (Kind=nag_wp), Intent (In) :: p Real (Kind=nag_wp), Intent (Inout) :: r(lr)
#include <nag.h>
 void g05tcf_ (const Integer *mode, const Integer *n, const double *p, double r[], const Integer *lr, Integer state[], Integer x[], Integer *ifail)
The routine may be called by the names g05tcf or nagf_rand_int_geom.

## 3Description

g05tcf generates $n$ integers ${x}_{i}$ from a discrete geometric distribution, where the probability of ${x}_{i}=I$ (a first success after $I+1$ trials) is
 $P (xi=I) = p × (1-p) I , I=0,1,… .$
The variates can be generated with or without using a search table and index. If a search table is used then it is stored with the index in a reference vector and subsequent calls to g05tcf with the same parameter value can then use this reference vector to generate further variates. If the search table is not used (as recommended for small values of $p$) then a direct transformation of uniform variates is used.
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 g05tcf.
Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

## 5Arguments

1: $\mathbf{mode}$Integer Input
On entry: a code for selecting the operation to be performed by the routine.
${\mathbf{mode}}=0$
Set up reference vector only.
${\mathbf{mode}}=1$
Generate variates using reference vector set up in a prior call to g05tcf.
${\mathbf{mode}}=2$
Set up reference vector and generate variates.
${\mathbf{mode}}=3$
Generate variates without using the reference vector.
Constraint: ${\mathbf{mode}}=0$, $1$, $2$ or $3$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{p}$Real (Kind=nag_wp) Input
On entry: the parameter $p$ of the geometric distribution representing the probability of success at a single trial.
Constraint: (see x02ajf).
4: $\mathbf{r}\left({\mathbf{lr}}\right)$Real (Kind=nag_wp) array Communication Array
On entry: if ${\mathbf{mode}}=1$, the reference vector from the previous call to g05tcf.
If ${\mathbf{mode}}=3$, r is not referenced.
On exit: if ${\mathbf{mode}}\ne 3$, the reference vector.
5: $\mathbf{lr}$Integer Input
On entry: the dimension of the array r as declared in the (sub)program from which g05tcf is called.
Suggested values:
• if ${\mathbf{mode}}\ne 3$, ${\mathbf{lr}}=8+42/{\mathbf{p}}$ approximately (see Section 9);
• otherwise ${\mathbf{lr}}=1$.
Constraints:
• if ${\mathbf{mode}}=0$ or $2$, ${\mathbf{lr}}\ge 30/{\mathbf{p}}+8$;
• if ${\mathbf{mode}}=1$, lr should remain unchanged from the previous call to g05tcf.
6: $\mathbf{state}\left(*\right)$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.
7: $\mathbf{x}\left({\mathbf{n}}\right)$Integer array Output
On exit: the $n$ pseudorandom numbers from the specified geometric distribution.
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, ${\mathbf{mode}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{mode}}=0$, $1$, $2$ or $3$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: .
p is so small that lr would have to be larger than the largest representable integer. Use ${\mathbf{mode}}=3$ instead. ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$
${\mathbf{ifail}}=4$
On entry, some of the elements of the array r have been corrupted or have not been initialized.
p is not the same as when r was set up in a previous call.
Previous value of ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=5$
On entry, lr is too small when ${\mathbf{mode}}=0$ or $2$: ${\mathbf{lr}}=⟨\mathit{\text{value}}⟩$, minimum length required $\text{}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=6$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=-99$
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

g05tcf 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.

The time taken to set up the reference vector, if used, increases with the length of array r. However, if the reference vector is used, the time taken to generate numbers decreases as the space allotted to the index part of r increases. Nevertheless, there is a point, depending on the distribution, where this improvement becomes very small and the suggested value for the length of array r is designed to approximate this point.
If p is very small then the storage requirements for the reference vector and the time taken to set up the reference vector becomes prohibitive. In this case it is recommended that the reference vector is not used. This is achieved by selecting ${\mathbf{mode}}=3$.

## 10Example

This example prints $10$ pseudorandom integers from a geometric distribution with parameter $p=0.001$, generated by a single call to g05tcf, after initialization by g05kff.

### 10.1Program Text

Program Text (g05tcfe.f90)

### 10.2Program Data

Program Data (g05tcfe.d)

### 10.3Program Results

Program Results (g05tcfe.r)