# NAG CL Interfaceg05ndc (sample)

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## 1Purpose

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

## 2Specification

 #include
 void g05ndc (const Integer ipop[], Integer n, Integer isampl[], Integer m, Integer state[], NagError *fail)
The function may be called by the names: g05ndc or nag_rand_sample.

## 3Description

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 g05kfc (for a repeatable sequence if computed sequentially) or g05kgc (for a non-repeatable sequence) must be called prior to the first call to g05ndc.

## 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: $\mathbf{ipop}\left[{\mathbf{n}}\right]$const Integer 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 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: $\mathbf{state}\left[\mathit{dim}\right]$Integer Communication 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 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface 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 8 in the Introduction to the NAG Library CL Interface for further information.

Not applicable.

## 8Parallelism and Performance

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 g05ndc 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 ${\mathbf{ipop}}\left[\mathit{i}-1\right]=\mathit{i}$, for $\mathit{i}=1,2,\dots ,n$;
2. (b)use g05ndc 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, g05ncc 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 g05ndc after initialization by g05kfc.

### 10.1Program Text

Program Text (g05ndce.c)

None.

### 10.3Program Results

Program Results (g05ndce.r)