NAG Library Routine Document

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

Fortran Interface

Subroutine c06fxf (

n1, n2, n3, x, y, init, trign1, trign2, trign3, work, ifail)

Integer, Intent (In)	::	n1, n2, n3
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (Inout)	::	x(n1n2n3), y(n1n2n3), trign1(1), trign2(1), trign3(1)
Real (Kind=nag_wp), Intent (Out)	::	work(1)
Character (1), Intent (In)	::	init

C Header Interface

#include <nagmk26.h>

void	c06fxf_ (const Integer n1, const Integer n2, const Integer n3, double x[], double y[], const char init, double trign1[], double trign2[], double trign3[], double work[], Integer *ifail, const Charlen length_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

Arguments

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 $(0 : n1 - 1, 0 : n2 - 1, 0 : n3 - 1)$ , $x (j_{1}, j_{2}, j_{3})$ 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
7: $trign1 (1)$ – Real (Kind=nag_wp) arrayInput/Output
8: $trign2 (1)$ – Real (Kind=nag_wp) arrayInput/Output
9: $trign3 (1)$ – Real (Kind=nag_wp) arrayInput/Output
10: $work (1)$ – Real (Kind=nag_wp) arrayOutput: These arguments are no longer accessed by c06fxf.
11: $ifail$ – 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, $n1 = 〈value〉$ .
Constraint: $n1 \geq 1$ .

$ifail = 2$: On entry, $n2 = 〈value〉$ .
Constraint: $n2 \geq 1$ .

$ifail = 3$: On entry, $n3 = 〈value〉$ .
Constraint: $n3 \geq 1$ .

$ifail = - 99$: 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

c06fxf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

c06fxf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

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 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

10

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 Document

c06fxf (fft_complex_3d_sep)

▸▿ 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