NAG Library Routine Document

C06FXF

+− 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

C06FXF computes the three-dimensional discrete Fourier transform of a trivariate sequence of complex data values. This routine is designed to be particularly efficient on vector processors.

2 Specification

SUBROUTINE C06FXF (

N1, N2, N3, X, Y, INIT, TRIGN1, TRIGN2, TRIGN3, WORK, IFAIL)

INTEGER	N1, N2, N3, IFAIL
REAL (KIND=nag_wp)	X(N1N2N3), Y(N1N2N3), TRIGN1(2N1), TRIGN2(2N2), TRIGN3(2N3), WORK(2N1N2N3)
CHARACTER(1)	INIT

3 Description

C06FXF computes the three-dimensional discrete Fourier transform of a trivariate sequence of complex data values

z_{j_{1} j_{2} j_{3}}

, for

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

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

and

j_{3} = 0, 1, \dots, n_{3} - 1

The discrete Fourier transform is here defined by

{\hat{z}}_{k_{1} k_{2} k_{3}} = \frac{1}{\sqrt{n_{1} n_{2} n_{3}}} \sum_{j_{1} = 0}^{n_{1} - 1} \sum_{j_{2} = 0}^{n_{2} - 1} \sum_{j_{3} = 0}^{n_{3} - 1} z_{j_{1} j_{2} j_{3}} \times \exp (- 2 π i (\frac{j_{1} k_{1}}{n_{1}} + \frac{j_{2} k_{2}}{n_{2}} + \frac{j_{3} k_{3}}{n_{3}})),

where

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

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

k_{3} = 0, 1, \dots, n_{3} - 1

(Note the scale factor of

\frac{1}{\sqrt{n_{1} n_{2} n_{3}}}

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 forming the complex conjugates of the data values and the transform.

This routine performs, for each dimension, multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm (see 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: N1 – INTEGERInput

On entry:

n_{1}

, the first dimension of the transform.

Constraint:

N1 \geq 1

2: N2 – INTEGERInput

On entry:

n_{2}

, the second dimension of the transform.

Constraint:

N2 \geq 1

3: N3 – INTEGERInput

On entry:

n_{3}

, the third dimension of the transform.

Constraint:

N3 \geq 1

4: X( $N1 \times N2 \times N3$ ) – REAL (KIND=nag_wp) arrayInput/Output

5: Y( $N1 \times N2 \times N3$ ) – 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 three-dimensional arrays of dimension

(0 : N1 - 1, 0 : N2 - 1, 0 : N3 - 1)

, then

X (j_{1}, j_{2}, j_{3})

and

Y (j_{1}, j_{2}, j_{3})

must contain the real and imaginary parts of

z_{j_{1} j_{2} j_{3}}

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

6: 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 $n_{1}$ , $n_{2}$ and $n_{3}$ , and store in the corresponding arrays TRIGN1, TRIGN2 and TRIGN3.
$INIT ='S'$ or $'R'$: The required trigonometric coefficients are assumed to have been calculated and stored in the arrays TRIGN1, TRIGN2 and TRIGN3 in a prior call to C06FXF. The routine performs a simple check that the current values of $n_{1}$ , $n_{2}$ and $n_{3}$ are consistent with the corresponding values stored in TRIGN1, TRIGN2 and TRIGN3.

Constraint:

INIT ='I'

'S'

'R'

7: TRIGN1( $2 \times N1$ ) – REAL (KIND=nag_wp) arrayInput/Output

8: TRIGN2( $2 \times N2$ ) – REAL (KIND=nag_wp) arrayInput/Output

9: TRIGN3( $2 \times N3$ ) – REAL (KIND=nag_wp) arrayInput/Output

On entry: if

INIT ='S'

'R'

, TRIGN1, TRIGN2 and TRIGN3 must contain the required coefficients calculated in a previous call of the routine. Otherwise TRIGN1, TRIGN2 and TRIGN3 need not be set. If

N1 = N2

, the same array may be supplied for TRIGN1 and TRIGN2. Similar considerations apply if

N2 = N3

N1 = N3

On exit: TRIGN1, TRIGN2 and TRIGN3 contain the required coefficients (computed by the routine if

INIT ='I'

10: WORK( $2 \times N1 \times N2 \times N3$ ) – REAL (KIND=nag_wp) arrayWorkspace

11: 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,

N1 < 1

$IFAIL = 2$

On entry,

N2 < 1

$IFAIL = 3$

On entry,

N3 < 1

$IFAIL = 4$

On entry,

INIT \neq'I'

'S'

'R'

$IFAIL = 5$: Not used at this Mark.

$IFAIL = 6$

On entry,

INIT ='S'

'R'

, but at least one of the arrays TRIGN1, TRIGN2 and TRIGN3 is inconsistent with the current value of N1, N2 or N3.

$IFAIL = 7$: 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 is approximately proportional to

n_{1} n_{2} n_{3} \times \log (n_{1} n_{2} n_{3})

, but also depends on the factorization of the individual dimensions

n_{1}

n_{2}

and

n_{3}

. C06FXF 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 trivariate sequence of complex data values and prints the three-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 DocumentC06FXF

+− 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

C06FXF