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NAG Library Routine Document

c06rhf (fft_qtrcosine)

▸▿ 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

c06rhf computes the discrete quarter-wave Fourier cosine transforms of

m

sequences of real data values. The elements of each sequence and its transform are stored contiguously.

2

Specification

Fortran Interface

Subroutine c06rhf (

idir, m, n, x, ifail)

Integer, Intent (In)	::	idir, m, n
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (Inout)	::	x(0:n-1,m)

C Header Interface

#include nagmk26.h

void	c06rhf_ ( const Integer idir, const Integer m, const Integer n, double x[], Integer ifail)

3

Description

Given

m

sequences of

n

real data values

x_{j}^{p}

, for

j = 0, 1, \dots, n - 1

and

p = 1, 2, \dots, m

, c06rhf simultaneously calculates the quarter-wave Fourier cosine transforms of all the sequences defined by

{\hat{x}}_{k}^{p} = \frac{1}{\sqrt{n}} (\frac{1}{2} x_{0}^{p} + \sum_{j = 1}^{n - 1} x_{j}^{p} \times \cos (j (2 k + 1) \frac{π}{2 n})),   if ​ idir = 1,

or its inverse

x_{k}^{p} = \frac{2}{\sqrt{n}} \sum_{j = 0}^{n - 1} {\hat{x}}_{j}^{p} \times \cos ((2 j + 1) k \frac{π}{2 n}),   if ​ idir = - 1,

where

k = 0, 1, \dots, n - 1

and

p = 1, 2, \dots, m

(Note the scale factor

\frac{1}{\sqrt{n}}

in this definition.)

A call of c06rhf with

idir = 1

followed by a call with

idir = - 1

will restore the original data.

The two transforms are also known as type-III DCT and type-II DCT, respectively.

The transform calculated by this routine can be used to solve Poisson's equation when the derivative of the solution is specified at the left boundary, and the solution is specified at the right boundary (see Swarztrauber (1977)).

The routine uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, described in Temperton (1983), together with pre- and post-processing stages described in Swarztrauber (1982). Special coding is provided for the factors

2

3

4

and

5

4

References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall

Swarztrauber P N (1977) The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle SIAM Rev. 19(3) 490–501

Swarztrauber P N (1982) Vectorizing the FFT's Parallel Computation (ed G Rodrique) 51–83 Academic Press

Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys. 52 340–350

5

Arguments

1: $idir$ – IntegerInput

On entry: indicates the transform, as defined in Section 3, to be computed.

$idir = 1$: Forward transform.
$idir = - 1$: Inverse transform.

Constraint:

idir = 1

- 1

2: $m$ – IntegerInput

On entry:

m

, the number of sequences to be transformed.

Constraint:

m \geq 1

3: $n$ – IntegerInput

On entry:

n

, the number of real values in each sequence.

Constraint:

n \geq 1

4: $x (0 : n - 1, m)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: the data values of the

p

th sequence to be transformed, denoted by

x_{j}^{p}

, for

j = 0, 1, \dots, n - 1

and

p = 1, 2, \dots, m

, must be stored in

x (j, p)

On exit: the

n

components of the

p

th quarter-wave cosine transform, denoted by

{\hat{x}}_{k}^{p}

, for

k = 0, 1, \dots, n - 1

and

p = 1, 2, \dots, m

, are stored in

x (k, p)

, overwriting the corresponding original values.

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, $m = 〈value〉$ .
Constraint: $m \geq 1$ .

$ifail = 2$: On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .

$ifail = 3$: On entry, $idir = 〈value〉$ .
Constraint: $idir = - 1$ or $1$ .

$ifail = 4$: An internal error has occurred in this routine. Check the routine call and any array sizes. If the call is correct then please contact NAG for assistance.

$ifail = - 99$: 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

Some indication of accuracy can be obtained by performing a subsequent inverse transform and comparing the results with the original sequence (in exact arithmetic they would be identical).

8

Parallelism and Performance

c06rhf 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 time taken by c06rhf is approximately proportional to

n m \log (n)

, but also depends on the factors of

n

. c06rhf is fastest if the only prime factors of

n

are

2

3

and

5

, and is particularly slow if

n

is a large prime, or has large prime factors. Workspace is internally allocated by this routine. The total amount of memory allocated is

O (n)

10

Example

This example reads in sequences of real data values and prints their quarter-wave cosine transforms as computed by c06rhf with

idir = 1

. It then calls the routine again with

idir = - 1

and prints the results which may be compared with the original data.

NAG Library Routine Document

c06rhf (fft_qtrcosine)

▸▿ 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

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