NAG Library Function Document

nag_rand_hypergeometric (g05tec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_rand_hypergeometric (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>

#include <nagg05.h>

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

3 Description

nag_rand_hypergeometric (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 nag_rand_hypergeometric (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 nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_hypergeometric (g05tec).

4 References

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

5 Arguments

1: mode – Nag_ModeRNGInput

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 nag_rand_hypergeometric (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 – IntegerInput

On entry:

n

, the number of pseudorandom numbers to be generated.

Constraint:

n \geq 0

3: ns – IntegerInput

On entry:

l

, the sample size of the hypergeometric distribution.

Constraint:

0 \leq ns \leq np

4: np – IntegerInput

On entry:

k

, the population size of the hypergeometric distribution.

Constraint:

np \geq 0

5: m – IntegerInput

On entry:

m

, the number of specified items of the hypergeometric distribution.

Constraint:

0 \leq m \leq np

6: r[lr] – doubleCommunication Array

On entry: if

mode = Nag_GenerateFromReference

, the reference vector from the previous call to nag_rand_hypergeometric (g05tec).

mode = Nag_GenerateWithoutReference

, r is not referenced and may be NULL.

On exit: if

mode \neq Nag_GenerateWithoutReference

, the reference vector.

7: lr – IntegerInput

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 nag_rand_hypergeometric (g05tec).

8: state[ $\dim$ ] – IntegerCommunication 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] – IntegerOutput

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

10: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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.
NE_INVALID_STATE: On entry, state vector has been corrupted or not initialized.
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

Not applicable.

9 Further Comments

None.

10 Example

The example program prints

20

pseudorandom integers from a hypergeometric distribution with

l = 500

m = 900

and

n = 1000

, generated by a single call to nag_rand_hypergeometric (g05tec), after initialization by nag_rand_init_repeatable (g05kfc).

10.1 Program Text

Program Text (g05tece.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (g05tece.r)

nag_rand_hypergeometric (g05tec) (PDF version)

g05 Chapter Contents

g05 Chapter Introduction

NAG Library Manual

NAG Library Function Documentnag_rand_hypergeometric (g05tec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_rand_hypergeometric (g05tec)