NAG Library Routine Document

Integer, Intent (In)	::	n1, n2, n3
Integer, Intent (Inout)	::	ifail
Complex (Kind=nag_wp), Intent (Inout)	::	x(n1n2n3), work(*)
Character (1), Intent (In)	::	direct

C Header Interface

#include <nagmk26.h>

void	c06pxf_ (const char direct, const Integer n1, const Integer n2, const Integer n3, Complex x[], Complex work[], Integer *ifail, const Charlen length_direct)

3

Description

c06pxf 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 (\pm 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

and

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.) The minus sign is taken in the argument of the exponential within the summation when the forward transform is required, and the plus sign is taken when the backward transform is required.

A call of c06pxf with

direct ='F'

followed by a call with

direct ='B'

will restore the original data.

This routine performs multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm (see Brigham (1974)).

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: $direct$ – Character(1)Input: On entry: if the forward transform as defined in Section 3 is to be computed, direct must be set equal to 'F'.
If the backward transform is to be computed, direct must be set equal to 'B'.

Constraint: $direct ='F'$ or $'B'$ .
2: $n1$ – IntegerInput: On entry: $n_{1}$ , the first dimension of the transform.

Constraint: $n1 \geq 1$ .
3: $n2$ – IntegerInput: On entry: $n_{2}$ , the second dimension of the transform.

Constraint: $n2 \geq 1$ .
4: $n3$ – IntegerInput: On entry: $n_{3}$ , the third dimension of the transform.

Constraint: $n3 \geq 1$ .
5: $x (n1 \times n2 \times n3)$ – Complex (Kind=nag_wp) arrayInput/Output: On entry: the complex data values. Data values are stored in x using column-major ordering for storing multidimensional arrays; that is, $z_{j_{1} j_{2} j_{3}}$ is stored in $x (1 + j_{1} + n_{1} j_{2} + n_{1} n_{2} j_{3})$ .

On exit: the corresponding elements of the computed transform.
6: $work (*)$ – Complex (Kind=nag_wp) arrayWorkspace: Note: the dimension of the array work must be at least $n1 \times n2 \times n3 + n1 + n2 + n3 + 45$ .

The workspace requirements as documented for c06pxf may be an overestimate in some implementations.

On exit: the real part of $work (1)$ contains the minimum workspace required for the current values of n1, n2 and n3 with this implementation.
7: $ifail$ – 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: $n3 \geq 1$ .

$ifail = 4$: On entry, $direct = 〈value〉$ .
Constraint: $direct ='F'$ or $'B'$ .

$ifail = 8$: An internal error has occurred in this routine. Check the routine call and any array sizes. If the call is correct then please contact NAG for assistance.

$ifail = - 99$: 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

c06pxf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

c06pxf 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}

. c06pxf 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

c06pxf (fft_complex_3d)

▸▿ 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

2

Specification