NAG Library Routine Document
c06pxf (fft_complex_3d)
1
Purpose
c06pxf computes the threedimensional discrete Fourier transform of a trivariate sequence of complex data values (using complex data type).
2
Specification
Fortran Interface
Integer, Intent (In)  ::  n1, n2, n3  Integer, Intent (Inout)  ::  ifail  Complex (Kind=nag_wp), Intent (Inout)  ::  x(n1*n2*n3), work(*)  Character (1), Intent (In)  ::  direct 

3
Description
c06pxf computes the threedimensional 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
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 ${\mathbf{direct}}=\text{'F'}$ followed by a call with ${\mathbf{direct}}=\text{'B'}$ will restore the original data.
This routine performs multiple onedimensional 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) 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: $\mathbf{n1}$ – IntegerInput

On entry: ${n}_{1}$, the first dimension of the transform.
Constraint:
${\mathbf{n1}}\ge 1$.
 3: $\mathbf{n2}$ – IntegerInput

On entry: ${n}_{2}$, the second dimension of the transform.
Constraint:
${\mathbf{n2}}\ge 1$.
 4: $\mathbf{n3}$ – IntegerInput

On entry: ${n}_{3}$, the third dimension of the transform.
Constraint:
${\mathbf{n3}}\ge 1$.
 5: $\mathbf{x}\left({\mathbf{n1}}\times {\mathbf{n2}}\times {\mathbf{n3}}\right)$ – 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,
${z}_{{j}_{1}{j}_{2}{j}_{3}}$ is stored in
${\mathbf{x}}\left(1+{j}_{1}+{n}_{1}{j}_{2}+{n}_{1}{n}_{2}{j}_{3}\right)$.
On exit: the corresponding elements of the computed transform.
 6: $\mathbf{work}\left(*\right)$ – Complex (Kind=nag_wp) arrayWorkspace

Note: the dimension of the array
work
must be at least
${\mathbf{n1}}\times {\mathbf{n2}}\times {\mathbf{n3}}+{\mathbf{n1}}+{\mathbf{n2}}+{\mathbf{n3}}+45$.
The workspace requirements as documented for c06pxf 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 values of
n1,
n2 and
n3 with this implementation.
 7: $\mathbf{ifail}$ – 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,  ${\mathbf{n1}}<1$. 
 ${\mathbf{ifail}}=2$

On entry,  ${\mathbf{n2}}<1$. 
 ${\mathbf{ifail}}=3$

On entry,  ${\mathbf{n3}}<1$. 
 ${\mathbf{ifail}}=4$

On entry,  ${\mathbf{direct}}\ne \text{'F'}$ or $\text{'B'}$. 
 ${\mathbf{ifail}}=8$

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
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 implementationspecific information.
The time taken is approximately proportional to ${n}_{1}{n}_{2}{n}_{3}\times \mathrm{log}\left({n}_{1}{n}_{2}{n}_{3}\right)$, 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 $2$, $3$ or $5$; 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 threedimensional 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 (c06pxfe.f90)
10.2
Program Data
Program Data (c06pxfe.d)
10.3
Program Results
Program Results (c06pxfe.r)