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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_rand_int_hypergeom (g05te)

## Purpose

nag_rand_int_hypergeom (g05te) 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.

## Syntax

[r, state, x, ifail] = g05te(mode, n, ns, np, m, r, state)
[r, state, x, ifail] = nag_rand_int_hypergeom(mode, n, ns, np, m, r, state)

## Description

nag_rand_int_hypergeom (g05te) 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_int_hypergeom (g05te) 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_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_int_hypergeom (g05te).

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{mode}$int64int32nag_int scalar
A code for selecting the operation to be performed by the function.
${\mathbf{mode}}=0$
Set up reference vector only.
${\mathbf{mode}}=1$
Generate variates using reference vector set up in a prior call to nag_rand_int_hypergeom (g05te).
${\mathbf{mode}}=2$
Set up reference vector and generate variates.
${\mathbf{mode}}=3$
Generate variates without using the reference vector.
Constraint: ${\mathbf{mode}}=0$, $1$, $2$ or $3$.
2:     $\mathrm{n}$int64int32nag_int scalar
$n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathrm{ns}$int64int32nag_int scalar
$l$, the sample size of the hypergeometric distribution.
Constraint: $0\le {\mathbf{ns}}\le {\mathbf{np}}$.
4:     $\mathrm{np}$int64int32nag_int scalar
$k$, the population size of the hypergeometric distribution.
Constraint: ${\mathbf{np}}\ge 0$.
5:     $\mathrm{m}$int64int32nag_int scalar
$m$, the number of specified items of the hypergeometric distribution.
Constraint: $0\le {\mathbf{m}}\le {\mathbf{np}}$.
6:     $\mathrm{r}\left(\mathit{lr}\right)$ – double array
lr, the dimension of the array, must satisfy the constraint
• if ${\mathbf{mode}}=0$ or $2$, lr must not be too small, but the limit is too complicated to specify;
• if ${\mathbf{mode}}=1$, lr must remain unchanged from the previous call to nag_rand_int_hypergeom (g05te).
If ${\mathbf{mode}}=1$, the reference vector from the previous call to nag_rand_int_hypergeom (g05te).
If ${\mathbf{mode}}=3$, r is not referenced.
7:     $\mathrm{state}\left(:\right)$int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains information on the selected base generator and its current state.

None.

### Output Parameters

1:     $\mathrm{r}\left(\mathit{lr}\right)$ – double array
If ${\mathbf{mode}}\ne 3$, the reference vector.
2:     $\mathrm{state}\left(:\right)$int64int32nag_int array
Contains updated information on the state of the generator.
3:     $\mathrm{x}\left({\mathbf{n}}\right)$int64int32nag_int array
The pseudorandom numbers from the specified hypergeometric distribution.
4:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
Constraint: ${\mathbf{mode}}=0$, $1$, $2$ or $3$.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=3$
Constraint: $0\le {\mathbf{ns}}\le {\mathbf{np}}$.
${\mathbf{ifail}}=4$
Constraint: ${\mathbf{np}}\ge 0$.
${\mathbf{ifail}}=5$
Constraint: $0\le {\mathbf{m}}\le {\mathbf{np}}$.
${\mathbf{ifail}}=6$
On entry, some of the elements of the array r have been corrupted or have not been initialized.
The value of ns, np or m is not the same as when r was set up in a previous call with ${\mathbf{mode}}=0$ or $2$.
${\mathbf{ifail}}=7$
On entry, lr is too small when ${\mathbf{mode}}=0$ or $2$.
${\mathbf{ifail}}=8$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

Not applicable.

None.

## 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_int_hypergeom (g05te), after initialization by nag_rand_init_repeat (g05kf).
function g05te_example

fprintf('g05te example results\n\n');

% Initialize the base generator to a repeatable sequence
seed  = [int64(1762543)];
genid = int64(1);
subid = int64(1);
[state, ifail] = g05kf( ...
genid, subid, seed);

% Number of variates
n = int64(20);

% Parameters
ns = int64(500);
np = int64(1000);
m = int64(900);

% Generate variates from hypergeomtric distribution
mode = int64(2);
r = zeros(200, 1);
[r, state, x, ifail] = g05te( ...
mode, n, ns, np, m, r, state);

disp('Variates');
disp(double(x));

g05te example results

Variates
452
444
453
454
444
450
449
454
450
452
442
447
451
442
451
447
447
462
456
450