G05 Chapter Contents
G05 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG05NDF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G05NDF selects a pseudorandom sample without replacement from an integer vector.

## 2  Specification

 SUBROUTINE G05NDF ( IPOP, N, ISAMPL, M, STATE, IFAIL)
 INTEGER IPOP(N), N, ISAMPL(M), M, STATE(*), IFAIL

## 3  Description

G05NDF 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 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 G05NDF.

## 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  Parameters

1:     IPOP(N) – INTEGER arrayInput
On entry: the population to be sampled.
2:     N – INTEGERInput
On entry: the number of elements in the population vector to be sampled.
Constraint: ${\mathbf{N}}\ge 1$.
3:     ISAMPL(M) – INTEGER arrayOutput
On exit: the selected sample.
4:     M – INTEGERInput
On entry: the sample size.
Constraint: $1\le {\mathbf{M}}\le {\mathbf{N}}$.
5:     STATE($*$) – INTEGER arrayCommunication 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.
6:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{N}}<1$.
${\mathbf{IFAIL}}=4$
 On entry, ${\mathbf{M}}<1$, or ${\mathbf{M}}>{\mathbf{N}}$.
${\mathbf{IFAIL}}=5$
 On entry, STATE vector was not initialized or has been corrupted.

## 7  Accuracy

Not applicable.

The time taken by G05NDF 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 ${\mathbf{IPOP}}\left(\mathit{i}\right)=\mathit{i}$, for $\mathit{i}=1,2,\dots ,n$; (b) use G05NDF 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, G05NCF is more efficient.

## 9  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 G05NDF after initialization by G05KFF.

### 9.1  Program Text

Program Text (g05ndfe.f90)

### 9.2  Program Data

Program Data (g05ndfe.d)

### 9.3  Program Results

Program Results (g05ndfe.r)