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 g05tcc 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 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 g05tcc.

4 References

Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

5 Arguments

1: $mode$ – Nag_ModeRNG Input

On entry: a code for selecting the operation to be performed by the function.

$mode = Nag_InitializeReference$: Set up reference vector only.
$mode = Nag_GenerateFromReference$: Generate variates using reference vector set up in a prior call to g05tcc.
$mode = Nag_InitializeAndGenerate$: Set up reference vector and generate variates.
$mode = Nag_GenerateWithoutReference$: Generate variates without using the reference vector.

Constraint:

mode = Nag_InitializeReference

Nag_GenerateFromReference

Nag_InitializeAndGenerate

Nag_GenerateWithoutReference

2: $n$ – Integer Input

On entry:

n

, the number of pseudorandom numbers to be generated.

Constraint:

n \geq 0

3: $p$ – double Input

On entry: the parameter

p

of the geometric distribution representing the probability of success at a single trial.

Constraint:

machine precision \leq p \leq 1.0

(see X02AJC).

4: $r [lr]$ – double Communication Array

On entry: if

mode = Nag_GenerateFromReference

, the reference vector from the previous call to g05tcc.

mode = Nag_GenerateWithoutReference

, r is not referenced and may be NULL.

On exit: if

mode \neq Nag_GenerateWithoutReference

, the reference vector.

5: $lr$ – Integer Input

On entry: the dimension of the array r.

Suggested values:

if $mode \neq Nag_GenerateWithoutReference$ , $lr = 8 + 42 / p$ approximately (see Section 9);
otherwise $lr = 1$ .

Constraints:

if $mode = Nag_InitializeReference$ or $Nag_InitializeAndGenerate$ , $lr \geq 30 / p + 8$ ;
if $mode = Nag_GenerateFromReference$ , lr should remain unchanged from the previous call to g05tcc.

6: $state [\dim]$ – Integer Communication Array

Note: the dimension,

\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.

7: $x [n]$ – Integer Output

On exit: the

n

pseudorandom numbers from the specified geometric distribution.

8: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, lr is too small when $mode = Nag_InitializeReference$ or $Nag_InitializeAndGenerate$ : $lr = ⟨ value ⟩$ , minimum length required $= ⟨ value ⟩$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .
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_PREV_CALL: p is not the same as when r was set up in a previous call.
Previous value of $p = ⟨ value ⟩$ and $p = ⟨ value ⟩$ .
NE_REAL: On entry, $p = ⟨ value ⟩$ .
Constraint: $machine precision \leq p \leq 1.0$ .

p is so small that lr would have to be larger than the largest representable integer. Use $mode = Nag_GenerateWithoutReference$ instead. $p = ⟨ value ⟩$
NE_REF_VEC: On entry, some of the elements of the array r have been corrupted or have not been initialized.

7 Accuracy

Not applicable.

8 Parallelism and Performance

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

g05tcc 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.

9 Further Comments

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