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.

4

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

5

Arguments

1: $rv (m2)$ – 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: $m1$ – IntegerInput: On entry: the index of the first element of rv to be sorted.

Constraint: $m1 > 0$ .
3: $m2$ – IntegerInput: On entry: the index of the last element of rv to be sorted.

Constraint: $m2 \geq m1$ .
4: $order$ – Character(1)Input: On entry: if $order ='A'$ , the values will be sorted into ascending (i.e., nondecreasing) order.
If $order ='D'$ , into descending order.

Constraint: $order ='A'$ or $'D'$ .
5: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . 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 $- 1 or 1$ 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 $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$

On entry,	$m2 < 1$ ,
or	$m1 < 1$ ,
or	$m1 > m2$ .

$ifail = 2$

On entry,

order is not 'A' or 'D'.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

Not applicable.

8

Parallelism 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.

9

Further Comments

The average time taken by m01caf is approximately proportional to

n \times \log (n)

, where

n = m2 - m1 + 1

. The worst case time is proportional to

n^{2}

but this is extremely unlikely to occur.

10

Example

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

NAG Library Routine Document

m01caf (realvec_sort)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description