NAG Library Routine Document
c06pjf (fft_complex_multid)
1
Purpose
c06pjf computes the multidimensional discrete Fourier transform of a multivariate sequence of complex data values.
2
Specification
Fortran Interface
Integer, Intent (In)  ::  ndim, nd(ndim), n, lwork  Integer, Intent (Inout)  ::  ifail  Complex (Kind=nag_wp), Intent (Inout)  ::  x(n)  Complex (Kind=nag_wp), Intent (Out)  ::  work(lwork)  Character (1), Intent (In)  ::  direct 

3
Description
c06pjf computes the multidimensional discrete Fourier transform of a multidimensional sequence of complex data values ${z}_{{j}_{1}{j}_{2}\dots {j}_{m}}$, where ${j}_{1}=0,1,\dots ,{n}_{1}1\text{, \hspace{1em}}{j}_{2}=0,1,\dots ,{n}_{2}1$, and so on. Thus the individual dimensions are ${n}_{1},{n}_{2},\dots ,{n}_{m}$, and the total number of data values is $n={n}_{1}\times {n}_{2}\times \cdots \times {n}_{m}$.
The discrete Fourier transform is here defined (e.g., for
$m=2$) by:
where
${k}_{1}=0,1,\dots ,{n}_{1}1$ and
${k}_{2}=0,1,\dots ,{n}_{2}1$. The plus or minus sign in the argument of the exponential terms in the above definition determine the direction of the transform: a minus sign defines the
forward direction and a plus sign defines the
backward direction.
The extension to higher dimensions is obvious. (Note the scale factor of $\frac{1}{\sqrt{n}}$ in this definition.)
A call of c06pjf with ${\mathbf{direct}}=\text{'F'}$ followed by a call with ${\mathbf{direct}}=\text{'B'}$ will restore the original data.
The data values must be supplied in a onedimensional array using columnmajor storage ordering of multidimensional data (i.e., with the first subscript ${j}_{1}$ varying most rapidly).
This routine
calls
c06prf to perform onedimensional discrete Fourier transforms. Hence, the routine
uses a variant of the fast Fourier transform (FFT) algorithm (see
Brigham (1974)) known as the Stockham selfsorting algorithm, which is described in
Temperton (1983).
4
References
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Selfsorting mixedradix fast Fourier transforms J. Comput. Phys. 52 1–23
5
Arguments
 1: $\mathbf{direct}$ – 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:
${\mathbf{direct}}=\text{'F'}$ or $\text{'B'}$.
 2: $\mathbf{ndim}$ – IntegerInput

On entry: $m$, the number of dimensions (or variables) in the multivariate data.
Constraint:
${\mathbf{ndim}}\ge 1$.
 3: $\mathbf{nd}\left({\mathbf{ndim}}\right)$ – Integer arrayInput

On entry: the elements of
nd must contain the dimensions of the
ndim variables; that is,
${\mathbf{nd}}\left(i\right)$ must contain the dimension of the
$i$th variable.
Constraint:
${\mathbf{nd}}\left(\mathit{i}\right)\ge 1$, for $\mathit{i}=1,2,\dots ,{\mathbf{ndim}}$.
 4: $\mathbf{n}$ – IntegerInput

On entry: $n$, the total number of data values.
Constraint:
n must equal the product of the first
ndim elements of the array
nd.
 5: $\mathbf{x}\left({\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayInput/Output

On entry: the complex data values. Data values are stored in
x using columnmajor ordering for storing multidimensional arrays; that is,
${z}_{{j}_{1}{j}_{2}\cdots {j}_{m}}$ is stored in
${\mathbf{x}}\left(1+{j}_{1}+{n}_{1}{j}_{2}+{n}_{1}{n}_{2}{j}_{3}+\cdots \right)$.
On exit: the corresponding elements of the computed transform.
 6: $\mathbf{work}\left({\mathbf{lwork}}\right)$ – Complex (Kind=nag_wp) arrayWorkspace

The workspace requirements as documented for c06pjf may be an overestimate in some implementations.
On exit: the real part of
${\mathbf{work}}\left(1\right)$ contains the minimum workspace required for the current value of
n with this implementation.
 7: $\mathbf{lwork}$ – IntegerInput

On entry: the dimension of the array
work as declared in the (sub)program from which
c06pjf is called.
Suggested value:
${\mathbf{lwork}}\ge {\mathbf{n}}+3\times \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{nd}}\left(i\right)\right)+15$, where $i=1,2,\dots ,{\mathbf{ndim}}$.
 8: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{ or}1$. 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
$1\text{ or}1$ 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 $\mathbf{1}\text{ or}\mathbf{1}$ 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).
6
Error 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,  ${\mathbf{ndim}}<1$. 
 ${\mathbf{ifail}}=2$

On entry,  ${\mathbf{direct}}\ne \text{'F'}$ or $\text{'B'}$. 
 ${\mathbf{ifail}}=3$

On entry,  at least one of the first ndim elements of nd is less than $1$. 
 ${\mathbf{ifail}}=4$

On entry,  n does not equal the product of the first ndim elements of nd. 
 ${\mathbf{ifail}}=5$

On entry,  lwork is too small. The minimum amount of workspace required is returned in ${\mathbf{work}}\left(1\right)$. 
 ${\mathbf{ifail}}=7$

An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.
 ${\mathbf{ifail}}=99$
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.
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
c06pjf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pjf 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 implementationspecific information.
The time taken is approximately proportional to $n\times \mathrm{log}\left(n\right)$, but also depends on the factorization of the individual dimensions ${\mathbf{nd}}\left(i\right)$. c06pjf is faster if the only prime factors are $2$, $3$ or $5$; and fastest of all if they are powers of $2$.
10
Example
This example reads in a bivariate sequence of complex data values and prints the twodimensional Fourier transform. It then performs an inverse transform and prints the sequence so obtained, which may be compared to the original data values.
10.1
Program Text
Program Text (c06pjfe.f90)
10.2
Program Data
Program Data (c06pjfe.d)
10.3
Program Results
Program Results (c06pjfe.r)