NAG Library Routine Document

C06FUF

+− 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

C06FUF computes the two-dimensional discrete Fourier transform of a bivariate sequence of complex data values. This routine is designed to be particularly efficient on vector processors.

2 Specification

SUBROUTINE C06FUF (

M, N, X, Y, INIT, TRIGM, TRIGN, WORK, IFAIL)

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

3 Description

C06FUF computes the two-dimensional discrete Fourier transform of a bivariate sequence of complex data values

z_{j_{1} j_{2}}

, for

j_{1} = 0, 1, \dots, m - 1

and

j_{2} = 0, 1, \dots, n - 1

The discrete Fourier transform is here defined by

{\hat{z}}_{k_{1} k_{2}} = \frac{1}{\sqrt{m n}} \sum_{j_{1} = 0}^{m - 1} \sum_{j_{2} = 0}^{n - 1} z_{j_{1} j_{2}} \times \exp (- 2 π i (\frac{j_{1} k_{1}}{m} + \frac{j_{2} k_{2}}{n})),

where

k_{1} = 0, 1, \dots, m - 1

k_{2} = 0, 1, \dots, n - 1

(Note the scale factor of

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

in this definition.)

To compute the inverse discrete Fourier transform, defined with

\exp (+ 2 π i (\dots))

in the above formula instead of

\exp (- 2 π i (\dots))

, this routine should be preceded and followed by calls of C06GCF to form the complex conjugates of the data values and the transform.

This routine calls C06FRF to perform multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm in Brigham (1974). It is designed to be particularly efficient on vector processors.

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 length of the first dimension of

Z

. Consider the matrix

Z

with elements

Z_{i j} = z_{i + 1 j + 1}

, where

z

is the bivariate sequence defined in Section 3, then

m

is the number of rows of

Z

Constraint:

M \geq 1

2: N – INTEGERInput

On entry:

n

, the length of the second dimension of

Z

. Consider the matrix

Z

with elements

Z_{i j} = z_{i + 1 j + 1}

, where

z

is the bivariate sequence defined in Section 3, then

n

is the number of columns of

Z

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 values must be stored in arrays X and Y respectively. If X and Y are regarded as two-dimensional arrays of dimension

(0 : M - 1, 0 : N - 1)

, then

X (j_{1}, j_{2})

and

Y (j_{1}, j_{2})

must contain the real and imaginary parts of

z_{j_{1} j_{2}}

On exit: the real and imaginary parts respectively of the corresponding elements of the computed transform.

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 values of $m$ and $n$ , and store in the corresponding arrays TRIGM and TRIGN.
$INIT ='S'$ or $'R'$: The required trigonometric coefficients are assumed to have been calculated and stored in the arrays TRIGM and TRIGN in a prior call to C06FUF. The routine performs a simple check that the current values of $m$ and $n$ are consistent with the corresponding values stored in TRIGM and TRIGN.

Constraint:

INIT ='I'

'S'

'R'

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

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

On entry: if

INIT ='S'

'R'

, TRIGM and TRIGN must contain the required coefficients calculated in a previous call of the routine. Otherwise TRIGM and TRIGN need not be set.

m = n

the same array may be supplied for TRIGM and TRIGN.

On exit: TRIGM and TRIGN contain the required coefficients (computed by the routine if

INIT ='I'

8: WORK( $2 \times M \times N$ ) – REAL (KIND=nag_wp) arrayWorkspace

9: 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 at least one of the arrays TRIGM and TRIGN is inconsistent with the current value of M or N.

$IFAIL = 6$: 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 is approximately proportional to

m n \times \log (m n)

, but also depends on the factorization of the individual dimensions

m

and

n

. C06FUF is faster if the only prime factors are

2

3

5

; and fastest of all if they are powers of

2

9 Example

This example reads in a bivariate sequence of complex data values and prints the two-dimensional Fourier transform. It then performs an inverse transform and prints the sequence so obtained, which may be compared to the original data values.

NAG Library Routine DocumentC06FUF

+− 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

C06FUF