g05 Chapter Contents
g05 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_rand_sample (g05ndc)

## 1  Purpose

nag_rand_sample (g05ndc) selects a pseudorandom sample without replacement from an integer vector.

## 2  Specification

 #include #include
 void nag_rand_sample (const Integer ipop[], Integer n, Integer isampl[], Integer m, Integer state[], NagError *fail)

## 3  Description

nag_rand_sample (g05ndc) selects $m$ elements from a population vector ipop of length $n$ and places them in a sample vector isampl. Their order in ipop will be preserved in isampl. Each of the $\left(\begin{array}{c}n\\ m\end{array}\right)$ possible combinations of elements of isampl may be regarded as being equally probable.
For moderate or large values of $n$ it is theoretically impossible that all combinations of size $m$ may occur, unless $m$ is near 1 or near $n$. This is because $\left(\begin{array}{c}n\\ m\end{array}\right)$ exceeds the cycle length of any of the base generators. For practical purposes this is irrelevant, as the time taken to generate all possible combinations is many millenia.
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_sample (g05ndc).

## 4  References

Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin
Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

## 5  Arguments

1:    $\mathbf{ipop}\left[{\mathbf{n}}\right]$const IntegerInput
On entry: the population to be sampled.
2:    $\mathbf{n}$IntegerInput
On entry: the number of elements in the population vector to be sampled.
Constraint: ${\mathbf{n}}\ge 1$.
3:    $\mathbf{isampl}\left[{\mathbf{m}}\right]$IntegerOutput
On exit: the selected sample.
4:    $\mathbf{m}$IntegerInput
On entry: the sample size.
Constraint: $1\le {\mathbf{m}}\le {\mathbf{n}}$.
5:    $\mathbf{state}\left[\mathit{dim}\right]$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.
6:    $\mathbf{fail}$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, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: $1\le {\mathbf{m}}\le {\mathbf{n}}$.
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.

Not applicable.

## 8  Parallelism and Performance

nag_rand_sample (g05ndc) 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.

The time taken by nag_rand_sample (g05ndc) is of order $n$.
In order to sample other kinds of vectors, or matrices of higher dimension, the following technique may be used:
 (a) set ${\mathbf{ipop}}\left[\mathit{i}-1\right]=\mathit{i}$, for $\mathit{i}=1,2,\dots ,n$; (b) use nag_rand_sample (g05ndc) to take a sample from ipop and put it into isampl; (c) use the contents of isampl as a set of indices to access the relevant vector or matrix.
In order to divide a population into several groups, nag_rand_permute (g05ncc) is more efficient.

## 10  Example

In the example program random samples of size $1,2,\dots ,8$ are selected from a vector containing the first eight positive integers in ascending order. The samples are generated and printed for each sample size by a call to nag_rand_sample (g05ndc) after initialization by nag_rand_init_repeatable (g05kfc).

### 10.1  Program Text

Program Text (g05ndce.c)

None.

### 10.3  Program Results

Program Results (g05ndce.r)