g05tec generates a vector of pseudorandom integers from the discrete hypergeometric distribution of the number of specified items in a sample of size

l

, taken from a population of size

k

with

m

specified items in it.

2 Specification

#include <nag.h>

void	g05tec (Nag_ModeRNG mode, Integer n, Integer ns, Integer np, Integer m, double r[], Integer lr, Integer state[], Integer x[], NagError *fail)

The function may be called by the names: g05tec, nag_rand_int_hypergeom or nag_rand_hypergeometric.

3 Description

g05tec generates

n

integers

x_{i}

from a discrete hypergeometric distribution, where the probability of

x_{i} = I

\begin{matrix} P (i = I) = \frac{l! m! (k - l)! (k - m)!}{I! (l - I)! (m - I)! (k - m - l + I)! k!} & if I = \max (0, m + l - k), \dots, \min (l, m), \\ P (i = I) = 0 & otherwise. \end{matrix}

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 g05tec with the same parameter values can then use this reference vector to generate further variates. The reference array is generated by a recurrence relation if

l m (k - l) (k - m) < 50 k^{3}

, otherwise Stirling's approximation 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 g05tec.

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 g05tec.
$mode = Nag_InitializeAndGenerate$: 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: $ns$ – Integer Input

On entry:

l

, the sample size of the hypergeometric distribution.

Constraint:

0 \leq ns \leq np

4: $np$ – Integer Input

On entry:

k

, the population size of the hypergeometric distribution.

Constraint:

np \geq 0

5: $m$ – Integer Input

On entry:

m

, the number of specified items of the hypergeometric distribution.

Constraint:

0 \leq m \leq np

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

On entry: if

mode = Nag_GenerateFromReference

, the reference vector from the previous call to g05tec.

mode = Nag_GenerateWithoutReference

, r is not referenced and may be NULL.

On exit: if

mode \neq Nag_GenerateWithoutReference

, the reference vector.

7: $lr$ – Integer Input

On entry: the dimension of the array r.

Suggested values:

if $mode \neq Nag_GenerateWithoutReference$ , $lr = 28 + 20 \times \sqrt{(ns \times m \times (np - m) \times (np - ns)) / {np}^{3}}$ approximately;
otherwise $lr = 1$ .

Constraints:

if $mode = Nag_InitializeReference$ or $Nag_InitializeAndGenerate$ , lr must not be too small, but the limit is too complicated to specify;
if $mode = Nag_GenerateFromReference$ , lr must remain unchanged from the previous call to g05tec.

8: $state [\dim]$ – 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.

9: $x [n]$ – Integer Output

On exit: the pseudorandom numbers from the specified hypergeometric distribution.

10: $fail$ – 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: $n \geq 0$ .

On entry, $np = ⟨ value ⟩$ .
Constraint: $np \geq 0$ .
NE_INT_2: On entry, $m = ⟨ value ⟩$ and $np = ⟨ value ⟩$ .
Constraint: $0 \leq m \leq np$ .

On entry, $ns = ⟨ value ⟩$ and $np = ⟨ value ⟩$ .
Constraint: $0 \leq ns \leq np$ .
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: The value of ns, np or m is not the same as when r was set up in a previous call with $mode = Nag_InitializeReference$ or $Nag_InitializeAndGenerate$ .
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.

g05tec 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.