NAG Library Routine Document
G05NDF
1 Purpose
G05NDF selects a pseudorandom sample without replacement from an integer vector.
2 Specification
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 nonrepeatable 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 Arguments
 1: $\mathrm{IPOP}\left({\mathbf{N}}\right)$ – INTEGER arrayInput

On entry: the population to be sampled.
 2: $\mathrm{N}$ – INTEGERInput

On entry: the number of elements in the population vector to be sampled.
Constraint:
${\mathbf{N}}\ge 1$.
 3: $\mathrm{ISAMPL}\left({\mathbf{M}}\right)$ – INTEGER arrayOutput

On exit: the selected sample.
 4: $\mathrm{M}$ – INTEGERInput

On entry: the sample size.
Constraint:
$1\le {\mathbf{M}}\le {\mathbf{N}}$.
 5: $\mathrm{STATE}\left(*\right)$ – 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: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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
${\mathbf{IFAIL}}=0$ or
$1$, 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, ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{N}}\ge 1$.
 ${\mathbf{IFAIL}}=4$

On entry, ${\mathbf{M}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: $1\le {\mathbf{M}}\le {\mathbf{N}}$.
 ${\mathbf{IFAIL}}=5$

On entry,
STATE vector has been corrupted or not initialized.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7 Accuracy
Not applicable.
8 Parallelism and Performance
G05NDF 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 implementationspecific information.
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.
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 G05NDF after initialization by
G05KFF.
10.1 Program Text
Program Text (g05ndfe.f90)
10.2 Program Data
Program Data (g05ndfe.d)
10.3 Program Results
Program Results (g05ndfe.r)