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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_sort_realvec_rank (m01da)

## Purpose

nag_sort_realvec_rank (m01da) ranks a vector of double numbers in ascending or descending order.

## Syntax

[irank, ifail] = m01da(rv, m1, order, 'm2', m2)
[irank, ifail] = nag_sort_realvec_rank(rv, m1, order, 'm2', m2)

## Description

nag_sort_realvec_rank (m01da) uses a variant of list-merging, as described on pages 165–166 in Knuth (1973). The function takes advantage of natural ordering in the data, and uses a simple list insertion in a preparatory pass to generate ordered lists of length at least $10$. The ranking is stable: equal elements preserve their ordering in the input data.

## References

Knuth D E (1973) The Art of Computer Programming (Volume 3) (2nd Edition) Addison–Wesley

## Parameters

### Compulsory Input Parameters

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

### Optional Input Parameters

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

### Output Parameters

1:     $\mathrm{irank}\left({\mathbf{m2}}\right)$int64int32nag_int array
Elements m1 to m2 of irank contain the ranks of the corresponding elements of rv. Note that the ranks are in the range m1 to m2: thus, if ${\mathbf{rv}}\left(i\right)$ is the first element in the rank order, ${\mathbf{irank}}\left(i\right)$ is set to m1.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{m2}}<1$, or ${\mathbf{m1}}<1$, or ${\mathbf{m1}}>{\mathbf{m2}}$.
${\mathbf{ifail}}=2$
 On entry, order is not 'A' or 'D'.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

Not applicable.

The average time taken by the function is approximately proportional to $n×\mathrm{log}\left(n\right)$, where $n={\mathbf{m2}}-{\mathbf{m1}}+1$.

## Example

This example reads a list of double numbers and ranks them in ascending order.
```function m01da_example

fprintf('m01da example results\n\n');

rv = [5.3     4.6     7.8     1.7     5.3     9.9 ...
3.2     4.3     7.8     4.5     1.2     7.6];

m1 = int64(1);
order = 'Ascending';
[irank, ifail] = m01da(rv, m1, order);

fprintf('   Data   Ranks\n\n');
for i = 1:numel(rv)
fprintf('%7.1f%7d\n',rv(i),irank(i));
end

```
```m01da example results

Data   Ranks

5.3      7
4.6      6
7.8     10
1.7      2
5.3      8
9.9     12
3.2      3
4.3      4
7.8     11
4.5      5
1.2      1
7.6      9
```