g05 Chapter Contents
g05 Chapter Introduction
NAG 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:    $\mathbf{mode}$Nag_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:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
3:    $\mathbf{ns}$IntegerInput
On entry: $l$, the sample size of the hypergeometric distribution.
Constraint: $0\le {\mathbf{ns}}\le {\mathbf{np}}$.
4:    $\mathbf{np}$IntegerInput
On entry: $k$, the population size of the hypergeometric distribution.
Constraint: ${\mathbf{np}}\ge 0$.
5:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of specified items of the hypergeometric distribution.
Constraint: $0\le {\mathbf{m}}\le {\mathbf{np}}$.
6:    $\mathbf{r}\left[{\mathbf{lr}}\right]$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 and may be NULL.
On exit: if ${\mathbf{mode}}\ne \mathrm{Nag_GenerateWithoutReference}$, the reference vector.
7:    $\mathbf{lr}$IntegerInput
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:    $\mathbf{state}\left[\mathit{dim}\right]$IntegerCommunication Array
Note: the dimension, $\mathit{dim}$, 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:    $\mathbf{x}\left[{\mathbf{n}}\right]$IntegerOutput
On exit: the pseudorandom numbers from the specified hypergeometric distribution.
10:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation 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 2.7.5 in How to Use the NAG Library and its Documentation 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 ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ 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.

## 8  Parallelism and Performance

nag_rand_hypergeometric (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.

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)

None.

### 10.3  Program Results

Program Results (g05tece.r)