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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_sort_realvec_sort (m01ca)

## Purpose

nag_sort_realvec_sort (m01ca) rearranges a vector of double numbers into ascending or descending order.

## Syntax

[rv, ifail] = m01ca(rv, m1, order, 'm2', m2)
[rv, ifail] = nag_sort_realvec_sort(rv, m1, order, 'm2', m2)

## Description

nag_sort_realvec_sort (m01ca) is based on Singleton's implementation of the ‘median-of-three’ Quicksort algorithm (see Singleton (1969)), but with two additional modifications. First, small subfiles are sorted by an insertion sort on a separate final pass (see Sedgewick (1978)). Second, if a subfile is partitioned into two very unbalanced subfiles, the larger of them is flagged for special treatment: before it is partitioned, its end points are swapped with two random points within it; this makes the worst case behaviour extremely unlikely.

## References

Sedgewick R (1978) Implementing Quicksort programs Comm. ACM 21 847–857
Singleton R C (1969) An efficient algorithm for sorting with minimal storage: Algorithm 347 Comm. ACM 12 185–187

## Parameters

### Compulsory Input Parameters

1:     rv(m2) – double array
m2, the dimension of the array, must satisfy the constraint m2m1${\mathbf{m2}}\ge {\mathbf{m1}}$.
Elements m1 to m2 of rv must contain double values to be sorted.
2:     m1 – int64int32nag_int scalar
The index of the first element of rv to be sorted.
Constraint: m1 > 0${\mathbf{m1}}>0$.
3:     order – string (length ≥ 1)
If order = 'A'${\mathbf{order}}=\text{'A'}$, the values will be sorted into ascending (i.e., nondecreasing) order.
If order = 'D'${\mathbf{order}}=\text{'D'}$, into descending order.
Constraint: order = 'A'${\mathbf{order}}=\text{'A'}$ or 'D'$\text{'D'}$.

### Optional Input Parameters

1:     m2 – int64int32nag_int scalar
Default: The dimension of the array rv.
The index of the last element of rv to be sorted.
Constraint: m2m1${\mathbf{m2}}\ge {\mathbf{m1}}$.

None.

### Output Parameters

1:     rv(m2) – double array
These values are rearranged into sorted order.
2:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, m2 < 1${\mathbf{m2}}<1$, or m1 < 1${\mathbf{m1}}<1$, or m1 > m2${\mathbf{m1}}>{\mathbf{m2}}$.
ifail = 2${\mathbf{ifail}}=2$
 On entry, order is not 'A' or 'D'.

## Accuracy

Not applicable.

The average time taken by nag_sort_realvec_sort (m01ca) is approximately proportional to n × log(n)$n×\mathrm{log}\left(n\right)$, where n = m2m1 + 1$n={\mathbf{m2}}-{\mathbf{m1}}+1$. The worst case time is proportional to n2${n}^{2}$ but this is extremely unlikely to occur.

## Example

```function nag_sort_realvec_sort_example
rv = [1.3;
5.9;
4.1;
2.3;
0.5;
5.8;
1.3;
6.5;
2.3;
0.5;
6.5;
9.9;
2.1;
1.1;
1.2;
8.6];
m1 = int64(1);
order = 'Ascending';
[rvOut, ifail] = nag_sort_realvec_sort(rv, m1, order)
```
```

rvOut =

0.5000
0.5000
1.1000
1.2000
1.3000
1.3000
2.1000
2.3000
2.3000
4.1000
5.8000
5.9000
6.5000
6.5000
8.6000
9.9000

ifail =

0

```
```function m01ca_example
rv = [1.3;
5.9;
4.1;
2.3;
0.5;
5.8;
1.3;
6.5;
2.3;
0.5;
6.5;
9.9;
2.1;
1.1;
1.2;
8.6];
m1 = int64(1);
order = 'Ascending';
[rvOut, ifail] = m01ca(rv, m1, order)
```
```

rvOut =

0.5000
0.5000
1.1000
1.2000
1.3000
1.3000
2.1000
2.3000
2.3000
4.1000
5.8000
5.9000
6.5000
6.5000
8.6000
9.9000

ifail =

0

```