# NAG Library Routine Document

## 1Purpose

m01caf rearranges a vector of real numbers into ascending or descending order.

## 2Specification

Fortran Interface
 Subroutine m01caf ( rv, m1, m2,
 Integer, Intent (In) :: m1, m2 Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Inout) :: rv(m2) Character (1), Intent (In) :: order
#include nagmk26.h
 void m01caf_ (double rv[], const Integer *m1, const Integer *m2, const char *order, Integer *ifail, const Charlen length_order)

## 3Description

m01caf 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.

## 4References

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

## 5Arguments

1:     $\mathbf{rv}\left({\mathbf{m2}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: elements m1 to m2 of rv must contain real values to be sorted.
On exit: these values are rearranged into sorted order.
2:     $\mathbf{m1}$ – IntegerInput
On entry: the index of the first element of rv to be sorted.
Constraint: ${\mathbf{m1}}>0$.
3:     $\mathbf{m2}$ – IntegerInput
On entry: the index of the last element of rv to be sorted.
Constraint: ${\mathbf{m2}}\ge {\mathbf{m1}}$.
4:     $\mathbf{order}$ – Character(1)Input
On entry: if ${\mathbf{order}}=\text{'A'}$, the values will be sorted into ascending (i.e., nondecreasing) order.
If ${\mathbf{order}}=\text{'D'}$, into descending order.
Constraint: ${\mathbf{order}}=\text{'A'}$ or $\text{'D'}$.
5:     $\mathbf{ifail}$ – 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 ${\mathbf{m1}}<1$, or ${\mathbf{m1}}>{\mathbf{m2}}$.
${\mathbf{ifail}}=2$
 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

m01caf 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 average time taken by m01caf is approximately proportional to $n×\mathrm{log}\left(n\right)$, where $n={\mathbf{m2}}-{\mathbf{m1}}+1$. The worst case time is proportional to ${n}^{2}$ but this is extremely unlikely to occur.

## 10Example

This example reads a list of real numbers and sorts them into ascending order.

### 10.1Program Text

Program Text (m01cafe.f90)

### 10.2Program Data

Program Data (m01cafe.d)

### 10.3Program Results

Program Results (m01cafe.r)

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