g05tef 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

Fortran Interface

Subroutine g05tef (

mode, n, ns, np, m, r, lr, state, x, ifail)

Integer, Intent (In)	::	mode, n, ns, np, m, lr
Integer, Intent (Inout)	::	state(*), ifail
Integer, Intent (Out)	::	x(n)
Real (Kind=nag_wp), Intent (Inout)	::	r(lr)

C Header Interface

#include <nag.h>

void	g05tef_ (const Integer mode, const Integer n, const Integer ns, const Integer np, const Integer m, double r[], const Integer lr, Integer state[], Integer x[], Integer *ifail)

The routine may be called by the names g05tef or nagf_rand_int_hypergeom.

3 Description

g05tef 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 g05tef 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 routines g05kff (for a repeatable sequence if computed sequentially) or g05kgf (for a non-repeatable sequence) must be called prior to the first call to g05tef.

4 References

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

5 Arguments

1: $mode$ – Integer Input

On entry: a code for selecting the operation to be performed by the routine.

$mode = 0$: Set up reference vector only.
$mode = 1$: Generate variates using reference vector set up in a prior call to g05tef.
$mode = 2$: Set up reference vector and generate variates.
$mode = 3$: Generate variates without using the reference vector.

Constraint:

mode = 0

1

2

3

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: $r (lr)$ – Real (Kind=nag_wp) array Communication Array

On entry: if

mode = 1

, the reference vector from the previous call to g05tef.

mode = 3

, r is not referenced.

On exit: if

mode \neq 3

, the reference vector.

7: $lr$ – Integer Input

On entry: the dimension of the array r as declared in the (sub)program from which g05tef is called.

Suggested values:

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

Constraints:

if $mode = 0$ or $2$ , lr must not be too small, but the limit is too complicated to specify;
if $mode = 1$ , lr must remain unchanged from the previous call to g05tef.

8: $state (*)$ – Integer array Communication Array

Note: the actual argument supplied must be the array state supplied to the initialization routines g05kff or g05kgf.

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 array Output

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

10: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $mode = ⟨ value ⟩$ .
Constraint: $mode = 0$ , $1$ , $2$ or $3$ .

$ifail = 2$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

$ifail = 3$: On entry, $ns = ⟨ value ⟩$ and $np = ⟨ value ⟩$ .
Constraint: $0 \leq ns \leq np$ .

$ifail = 4$: On entry, $np = ⟨ value ⟩$ .
Constraint: $np \geq 0$ .

$ifail = 5$: On entry, $m = ⟨ value ⟩$ and $np = ⟨ value ⟩$ .
Constraint: $0 \leq m \leq np$ .

$ifail = 6$: 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 $mode = 0$ or $2$ .

$ifail = 7$: On entry, lr is too small when $mode = 0$ or $2$ : $lr = ⟨ value ⟩$ , minimum length required $= ⟨ value ⟩$ .

$ifail = 8$: On entry, state vector has been corrupted or not initialized.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g05tef 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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 g05tef, after initialization by g05kff.

g05te: FL CL CPP AD PY MB

NAG FL Interfaceg05tef (int_​hypergeom)

▸▿ 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 FL Interface
g05tef (int_hypergeom)