g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_rand_hypergeometric (g05tec)

## 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 #include
 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 ${x}_{i}=I$ is
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 $lm\left(k-l\right)\left(k-m\right)<50{k}^{3}$, 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:     modeNag_ModeRNGInput
On entry: a code for selecting the operation to be performed by the function.
${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$
Set up reference vector only.
${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$
Generate variates using reference vector set up in a prior call to nag_rand_hypergeometric (g05tec).
${\mathbf{mode}}=\mathrm{Nag_InitializeAndGenerate}$
Set up reference vector and generate variates.
${\mathbf{mode}}=\mathrm{Nag_GenerateWithoutReference}$
Generate variates without using the reference vector.
Constraint: ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$, $\mathrm{Nag_GenerateFromReference}$, $\mathrm{Nag_InitializeAndGenerate}$ or $\mathrm{Nag_GenerateWithoutReference}$.
2:     nIntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
3:     nsIntegerInput
On entry: $l$, the sample size of the hypergeometric distribution.
Constraint: $0\le {\mathbf{ns}}\le {\mathbf{np}}$.
4:     npIntegerInput
On entry: $k$, the population size of the hypergeometric distribution.
Constraint: ${\mathbf{np}}\ge 0$.
5:     mIntegerInput
On entry: $m$, the number of specified items of the hypergeometric distribution.
Constraint: $0\le {\mathbf{m}}\le {\mathbf{np}}$.
6:     r[lr]doubleCommunication Array
On entry: if ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$, the reference vector from the previous call to nag_rand_hypergeometric (g05tec).
If ${\mathbf{mode}}=\mathrm{Nag_GenerateWithoutReference}$, r is not referenced by nag_rand_hypergeometric (g05tec).
On exit: the reference vector.
7:     lrIntegerInput
On entry: the dimension of the array r.
Suggested values:
• if ${\mathbf{mode}}\ne \mathrm{Nag_GenerateWithoutReference}$, ${\mathbf{lr}}=28+20×\sqrt{\left({\mathbf{ns}}×{\mathbf{m}}×\left({\mathbf{np}}-{\mathbf{m}}\right)×\left({\mathbf{np}}-{\mathbf{ns}}\right)\right)/{{\mathbf{np}}}^{3}}$ approximately;
• otherwise ${\mathbf{lr}}=1$.
Constraints:
• if ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ or $\mathrm{Nag_InitializeAndGenerate}$, lr must not be too small, but the limit is too complicated to specify;
• if ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$, lr must remain unchanged from the previous call to nag_rand_hypergeometric (g05tec).
8:     state[$\mathit{dim}$]IntegerCommunication Array
Note: the actual argument supplied must be the array state supplied to the initialization functions 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:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, lr is too small when ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ or $\mathrm{Nag_InitializeAndGenerate}$: ${\mathbf{lr}}=〈\mathit{\text{value}}〉$, minimum length required $\text{}=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{np}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{np}}\ge 0$.
NE_INT_2
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{np}}=〈\mathit{\text{value}}〉$.
Constraint: $0\le {\mathbf{m}}\le {\mathbf{np}}$.
On entry, ${\mathbf{ns}}=〈\mathit{\text{value}}〉$ and ${\mathbf{np}}=〈\mathit{\text{value}}〉$.
Constraint: $0\le {\mathbf{ns}}\le {\mathbf{np}}$.
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 ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ or $\mathrm{Nag_InitializeAndGenerate}$.
NE_REF_VEC
On entry, some of the elements of the array r have been corrupted or have not been initialized.

Not applicable.

None.

## 9  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).

### 9.1  Program Text

Program Text (g05tece.c)

None.

### 9.3  Program Results

Program Results (g05tece.r)