NAG Library Routine Document

1Purpose

c06fjf computes the multidimensional discrete Fourier transform of a multivariate sequence of complex data values.

2Specification

Fortran Interface
 Subroutine c06fjf ( ndim, nd, n, x, y, work,
 Integer, Intent (In) :: ndim, nd(ndim), n, lwork Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Inout) :: x(n), y(n) Real (Kind=nag_wp), Intent (Out) :: work(lwork)
#include <nagmk26.h>
 void c06fjf_ (const Integer *ndim, const Integer nd[], const Integer *n, double x[], double y[], double work[], const Integer *lwork, Integer *ifail)

3Description

c06fjf computes the multidimensional discrete Fourier transform of a multidimensional sequence of complex data values ${z}_{{j}_{1}{j}_{2}\dots {j}_{m}}$, where ${j}_{1}=0,1,\dots ,{n}_{1}-1\text{, }{j}_{2}=0,1,\dots ,{n}_{2}-1$, and so on. Thus the individual dimensions are ${n}_{1},{n}_{2},\dots ,{n}_{m}$, and the total number of data values is $n={n}_{1}×{n}_{2}×\cdots ×{n}_{m}$.
The discrete Fourier transform is here defined (e.g., for $m=2$) by:
 $z^ k1 , k2 = 1n ∑ j1=0 n1-1 ∑ j2=0 n2-1 z j1j2 × exp -2πi j1k1 n1 + j2k2 n2 ,$
where ${k}_{1}=0,1,\dots ,{n}_{1}-1$, ${k}_{2}=0,1,\dots ,{n}_{2}-1$.
The extension to higher dimensions is obvious. (Note the scale factor of $\frac{1}{\sqrt{n}}$ in this definition.)
To compute the inverse discrete Fourier transform, defined with $\mathrm{exp}\left(+2\pi i\left(\dots \right)\right)$ in the above formula instead of $\mathrm{exp}\left(-2\pi i\left(\dots \right)\right)$, this routine should be preceded and followed by the complex conjugation of the data values and the transform (by negating the imaginary parts stored in $y$).
The data values must be supplied in a pair of one-dimensional arrays (real and imaginary parts separately), in accordance with the Fortran convention for storing multidimensional data (i.e., with the first subscript ${j}_{1}$ varying most rapidly).
This routine calls c06fcf to perform one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm in Brigham (1974).

4References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall

5Arguments

1:     $\mathbf{ndim}$ – IntegerInput
On entry: $m$, the number of dimensions (or variables) in the multivariate data.
Constraint: ${\mathbf{ndim}}\ge 1$.
2:     $\mathbf{nd}\left({\mathbf{ndim}}\right)$ – Integer arrayInput
On entry: ${\mathbf{nd}}\left(\mathit{i}\right)$ must contain ${n}_{\mathit{i}}$ (the dimension of the $\mathit{i}$th variable), for $\mathit{i}=1,2,\dots ,m$.
Constraint: ${\mathbf{nd}}\left(\mathit{i}\right)\ge 1$, for $\mathit{i}=1,2,\dots ,{\mathbf{ndim}}$.
3:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the total number of data values.
Constraint: ${\mathbf{n}}={\mathbf{nd}}\left(1\right)×{\mathbf{nd}}\left(2\right)×\cdots ×{\mathbf{nd}}\left({\mathbf{ndim}}\right)$.
4:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: ${\mathbf{x}}\left(1+{j}_{1}+{n}_{1}{j}_{2}+{n}_{1}{n}_{2}{j}_{3}+\dots \right)$ must contain the real part of the complex data value ${z}_{{j}_{1}{j}_{2}\dots {j}_{m}}$, for $0\le {j}_{1}\le {n}_{1}-1,0\le {j}_{2}\le {n}_{2}-1,\dots \text{}$; i.e., the values are stored in consecutive elements of the array according to the Fortran convention for storing multidimensional arrays.
On exit: the real parts of the corresponding elements of the computed transform.
5:     $\mathbf{y}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the imaginary parts of the complex data values, stored in the same way as the real parts in the array x.
On exit: the imaginary parts of the corresponding elements of the computed transform.
6:     $\mathbf{work}\left({\mathbf{lwork}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
7:     $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which c06fjf is called.
Constraint: ${\mathbf{lwork}}\ge 3×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left\{{\mathbf{nd}}\left(i\right)\right\}$.
8:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . 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  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  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{ndim}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ndim}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}={\mathbf{nd}}\left(1\right)×{\mathbf{nd}}\left(2\right)×\cdots ×{\mathbf{nd}}\left({\mathbf{ndim}}\right)$.
${\mathbf{ifail}}=10×l+3$
On entry, $l=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nd}}\left(l\right)\ge 1$.
${\mathbf{ifail}}=10×l+4$
On entry, ${\mathbf{lwork}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nd}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lwork}}\ge 3×{\mathbf{nd}}\left(l\right)$.
${\mathbf{ifail}}=-99$
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.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7Accuracy

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).

8Parallelism and Performance

c06fjf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06fjf 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.

The time taken is approximately proportional to $n×\mathrm{log}\left(n\right)$, but also depends on the factorization of the individual dimensions ${\mathbf{nd}}\left(i\right)$. c06fjf is faster if the only prime factors are $2$, $3$ or $5$; and fastest of all if they are powers of $2$.

10Example

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.

10.1Program Text

Program Text (c06fjfe.f90)

10.2Program Data

Program Data (c06fjfe.d)

10.3Program Results

Program Results (c06fjfe.r)