NAG Library Routine Document

C06HDF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

C06HDF computes the discrete quarter-wave Fourier cosine transforms of

m

sequences of real data values. This routine is designed to be particularly efficient on vector processors.

2 Specification

SUBROUTINE C06HDF (

DIRECT, M, N, X, INIT, TRIG, WORK, IFAIL)

INTEGER	M, N, IFAIL
REAL (KIND=nag_wp)	X(MN), TRIG(2N), WORK(M*N)
CHARACTER(1)	DIRECT, INIT

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

, C06HDF 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 ​ DIRECT ='F',

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 ​ DIRECT ='B',

for

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 C06HDF with

DIRECT ='F'

followed by a call with

DIRECT ='B'

will restore the original data.

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)). (See the C06 Chapter Introduction.)

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

5

and

6

. This routine is designed to be particularly efficient on vector processors, and it becomes especially fast as

m

, the number of transforms to be computed in parallel, increases.

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 Parameters

1: DIRECT – CHARACTER(1)Input

On entry: if the forward transform as defined in Section 3 is to be computed, then DIRECT must be set equal to 'F'.

If the backward transform is to be computed then DIRECT must be set equal to 'B'.

Constraint:

DIRECT ='F'

'B'

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( $M \times N$ ) – REAL (KIND=nag_wp) arrayInput/Output

On entry: the data must be stored in X as if in a two-dimensional array of dimension

(1 : M, 0 : N - 1)

; each of the

m

sequences is stored in a row of the array. In other words, if the data values of the

p

th sequence to be transformed are denoted by

x_{j}^{p}

, for

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

and

p = 1, 2, \dots, m

, then the

m n

elements of the array X must contain the values

x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{m}, x_{1}^{1}, x_{1}^{2}, \dots, x_{1}^{m}, \dots, x_{n - 1}^{1}, x_{n - 1}^{2}, \dots, x_{n - 1}^{m} .

On exit: the

m

quarter-wave cosine transforms stored as if in a two-dimensional array of dimension

(1 : M, 0 : N - 1)

. Each of the

m

transforms is stored in a row of the array, overwriting the corresponding original sequence. If the

n

components of the

p

th quarter-wave cosine transform are denoted by

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

, for

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

and

p = 1, 2, \dots, m

, then the

m n

elements of the array X contain the values

{\hat{x}}_{0}^{1}, {\hat{x}}_{0}^{2}, \dots, {\hat{x}}_{0}^{m}, {\hat{x}}_{1}^{1}, {\hat{x}}_{1}^{2}, \dots, {\hat{x}}_{1}^{m}, \dots, {\hat{x}}_{n - 1}^{1}, {\hat{x}}_{n - 1}^{2}, \dots, {\hat{x}}_{n - 1}^{m} .

5: INIT – CHARACTER(1)Input

On entry: indicates whether trigonometric coefficients are to be calculated.

$INIT ='I'$: Calculate the required trigonometric coefficients for the given value of $n$ , and store in the array TRIG.
$INIT ='S'$ or $'R'$: The required trigonometric coefficients are assumed to have been calculated and stored in the array TRIG in a prior call to one of C06HAF, C06HBF, C06HCF or C06HDF. The routine performs a simple check that the current value of $n$ is consistent with the values stored in TRIG.

Constraint:

INIT ='I'

'S'

'R'

6: TRIG( $2 \times N$ ) – REAL (KIND=nag_wp) arrayInput/Output

On entry: if

INIT ='S'

'R'

, TRIG must contain the required trigonometric coefficients calculated in a previous call of the routine. Otherwise TRIG need not be set.

On exit: contains the required coefficients (computed by the routine if

INIT ='I'

7: WORK( $M \times N$ ) – REAL (KIND=nag_wp) arrayWorkspace

8: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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 < 1

$IFAIL = 2$

On entry,

N < 1

$IFAIL = 3$

On entry,

INIT \neq'I'

'S'

'R'

$IFAIL = 4$: Not used at this Mark.

$IFAIL = 5$

On entry,

INIT ='S'

'R'

, but the array TRIG and the current value of N are inconsistent.

$IFAIL = 6$

On entry,

DIRECT \neq'F'

'B'

$IFAIL = 7$: An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.

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 Further Comments

The time taken by C06HDF is approximately proportional to

n m \log n

, but also depends on the factors of

n

. C06HDF 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.

9 Example

This example reads in sequences of real data values and prints their quarter-wave cosine transforms as computed by C06HDF with

DIRECT ='F'

. It then calls the routine again with

DIRECT ='B'

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

NAG Library Routine DocumentC06HDF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

C06HDF