# NAG Library Routine Document

## 1Purpose

m01dkf ranks the columns of a matrix of integer numbers in ascending or descending order.

## 2Specification

Fortran Interface
 Subroutine m01dkf ( im, ldm, m1, m2, n1, n2,
 Integer, Intent (In) :: im(ldm,n2), ldm, m1, m2, n1, n2 Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: irank(n2) Character (1), Intent (In) :: order
#include nagmk26.h
 void m01dkf_ (const Integer im[], const Integer *ldm, const Integer *m1, const Integer *m2, const Integer *n1, const Integer *n2, const char *order, Integer irank[], Integer *ifail, const Charlen length_order)

## 3Description

m01dkf ranks columns n1 to n2 of a matrix, using the data in rows m1 to m2 of those columns. The ordering is determined by first ranking the data in row m1, then ranking any tied columns according to the data in row ${\mathbf{m1}}+1$, and so on up to row m2.
m01dkf uses a variant of list-merging, as described on pages 165–166 in Knuth (1973). The routine 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 columns preserve their ordering in the input data.

## 4References

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

## 5Arguments

1:     $\mathbf{im}\left({\mathbf{ldm}},{\mathbf{n2}}\right)$ – Integer arrayInput
On entry: rows m1 to m2 of columns n1 to n2 of im must contain integer data to be ranked.
2:     $\mathbf{ldm}$ – IntegerInput
On entry: the first dimension of the array im as declared in the (sub)program from which m01dkf is called.
Constraint: ${\mathbf{ldm}}\ge {\mathbf{m2}}$.
3:     $\mathbf{m1}$ – IntegerInput
On entry: the index of the first row of im to be used.
Constraint: ${\mathbf{m1}}>0$.
4:     $\mathbf{m2}$ – IntegerInput
On entry: the index of the last row of im to be used.
Constraint: ${\mathbf{m2}}\ge {\mathbf{m1}}$.
5:     $\mathbf{n1}$ – IntegerInput
On entry: the index of the first column of im to be ranked.
Constraint: ${\mathbf{n1}}>0$.
6:     $\mathbf{n2}$ – IntegerInput
On entry: the index of the last column of im to be ranked.
Constraint: ${\mathbf{n2}}\ge {\mathbf{n1}}$.
7:     $\mathbf{order}$ – Character(1)Input
On entry: if ${\mathbf{order}}=\text{'A'}$, the columns will be ranked in ascending (i.e., nondecreasing) order.
If ${\mathbf{order}}=\text{'D'}$, into descending order.
Constraint: ${\mathbf{order}}=\text{'A'}$ or $\text{'D'}$.
8:     $\mathbf{irank}\left({\mathbf{n2}}\right)$ – Integer arrayOutput
On exit: elements n1 to n2 of irank contain the ranks of the corresponding columns of im. Note that the ranks are in the range n1 to n2: thus, if the $i$th column of im is the first in the rank order, ${\mathbf{irank}}\left(i\right)$ is set to n1.
9:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ 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 $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{m2}}<1$, or ${\mathbf{n2}}<1$, or ${\mathbf{m1}}<1$, or ${\mathbf{m1}}>{\mathbf{m2}}$, or ${\mathbf{n1}}<1$, or ${\mathbf{n1}}>{\mathbf{n2}}$, or ${\mathbf{ldm}}<{\mathbf{m2}}$.
${\mathbf{ifail}}=2$
 On entry, order is not 'A' or 'D'.
${\mathbf{ifail}}=-99$
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

m01dkf is not threaded in any implementation.

The average time taken by the routine is approximately proportional to $n×\mathrm{log}\left(n\right)$, where $n={\mathbf{n2}}-{\mathbf{n1}}+1$.

## 10Example

This example reads a matrix of integers and ranks the columns in descending order.

### 10.1Program Text

Program Text (m01dkfe.f90)

### 10.2Program Data

Program Data (m01dkfe.d)

### 10.3Program Results

Program Results (m01dkfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017