NAG Library Routine Document

C06FRF

+− 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

C06FRF computes the discrete Fourier transforms of

m

sequences, each containing

n

complex data values. This routine is designed to be particularly efficient on vector processors.

2 Specification

SUBROUTINE C06FRF (

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

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

3 Description

Given

m

sequences of

n

complex data values

z_{j}^{p}

, for

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

and

p = 1, 2, \dots, m

, C06FRF simultaneously calculates the Fourier transforms of all the sequences defined by

{\hat{z}}_{k}^{p} = \frac{1}{\sqrt{n}} \sum_{j = 0}^{n - 1} z_{j}^{p} \times \exp (- i \frac{2 π j k}{n}), k = 0, 1, \dots, n - 1 ​ and ​ p = 1, 2, \dots, m .

(Note the scale factor

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

in this definition.)

The discrete Fourier transform is sometimes defined using a positive sign in the exponential term

{\hat{z}}_{k}^{p} = \frac{1}{\sqrt{n}} \sum_{j = 0}^{n - 1} z_{j}^{p} \times \exp (+ i \frac{2 π j k}{n}) .

To compute this form, this routine should be preceded and followed by a call of C06GCF to form the complex conjugates of the

z_{j}^{p}

and the

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

The routine uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, which is described in Temperton (1983). Special code 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

Temperton C (1983) Self-sorting mixed-radix fast Fourier transforms J. Comput. Phys. 52 1–23

5 Parameters

1: M – INTEGERInput

On entry:

m

, the number of sequences to be transformed.

Constraint:

M \geq 1

2: N – INTEGERInput

On entry:

n

, the number of complex values in each sequence.

Constraint:

N \geq 1

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

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

On entry: the real and imaginary parts of the complex data must be stored in X and Y respectively 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 each array. In other words, if the real parts of the

p

th sequence to be transformed are denoted by

x_{j}^{p}

, for

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

, 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: X and Y are overwritten by the real and imaginary parts of the complex transforms.

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 C06FPF, C06FQF or C06FRF. 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 that have been previously calculated. Otherwise TRIG need not be set.

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

INIT ='I'

7: WORK( $2 \times M \times N$ ) – 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.

$IFAIL = 6$: 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 C06FRF is approximately proportional to

n m \log n

, but also depends on the factors of

n

. C06FRF 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 complex data values and prints their discrete Fourier transforms (as computed by C06FRF). Inverse transforms are then calculated using C06FRF and C06GCF and printed out, showing that the original sequences are restored.

NAG Library Routine DocumentC06FRF

+− 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

C06FRF