# NAG FL Interfaceg05ndf (sample)

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## 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 <nag.h>
 void g05ndf_ (const Integer ipop[], const Integer *n, Integer isampl[], const Integer *m, Integer state[], Integer *ifail)
The routine may be called by the names g05ndf or nagf_rand_sample.

## 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.
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 array Input
On entry: the population to be sampled.
2: $\mathbf{n}$Integer Input
On entry: the number of elements in the population to be sampled.
Constraint: ${\mathbf{n}}\ge 1$.
3: $\mathbf{isampl}\left({\mathbf{m}}\right)$Integer array Output
On exit: the selected sample.
4: $\mathbf{m}$Integer Input
On entry: the sample size.
Constraint: $1\le {\mathbf{m}}\le {\mathbf{n}}$.
5: $\mathbf{state}\left(*\right)$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.
6: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ 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 $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).

## 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 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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:
1. (a)set ${\mathbf{ipop}}\left(\mathit{i}\right)=\mathit{i}$, for $\mathit{i}=1,2,\dots ,n$;
2. (b)use g05ndf to take a sample from ipop and put it into isampl;
3. (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)