# NAG Library Routine Document

## 1Purpose

g05ndf selects a pseudorandom sample without replacement from an integer vector.

## 2Specification

Fortran Interface
 Subroutine g05ndf ( ipop, n, m,
 Integer, Intent (In) :: ipop(n), n, m Integer, Intent (Inout) :: state(*), ifail Integer, Intent (Out) :: isampl(m)
#include <nagmk26.h>
 void g05ndf_ (const Integer ipop[], const Integer *n, Integer isampl[], const Integer *m, Integer state[], Integer *ifail)

## 3Description

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.

## 4References

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

## 5Arguments

1:     $\mathbf{ipop}\left({\mathbf{n}}\right)$ – Integer arrayInput
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:     $\mathbf{isampl}\left({\mathbf{m}}\right)$ – Integer arrayOutput
On exit: the selected sample.
4:     $\mathbf{m}$ – IntegerInput
On entry: the sample size.
Constraint: $1\le {\mathbf{m}}\le {\mathbf{n}}$.
5:     $\mathbf{state}\left(*\right)$ – 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:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . 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  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  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).

## 6Error 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, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: $1\le {\mathbf{m}}\le {\mathbf{n}}$.
${\mathbf{ifail}}=5$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=-99$
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.

Not applicable.

## 8Parallelism 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 implementation-specific 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.

## 10Example

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.

### 10.1Program Text

Program Text (g05ndfe.f90)

### 10.2Program Data

Program Data (g05ndfe.d)

### 10.3Program Results

Program Results (g05ndfe.r)