NAG Library Routine Document
c06pyf (fft_real_3d)
1
Purpose
c06pyf computes the threedimensional discrete Fourier transform of a trivariate sequence of real data values.
2
Specification
Fortran Interface
Integer, Intent (In)  ::  n1, n2, n3  Integer, Intent (Inout)  ::  ifail  Real (Kind=nag_wp), Intent (In)  ::  x(n1*n2*n3)  Complex (Kind=nag_wp), Intent (Out)  ::  y((n1/2+1)*n2*n3) 

C Header Interface
#include nagmk26.h
void 
c06pyf_ (const Integer *n1, const Integer *n2, const Integer *n3, const double x[], Complex y[], Integer *ifail) 

3
Description
c06pyf computes the threedimensional discrete Fourier transform of a trivariate sequence of real data values ${x}_{{j}_{1}{j}_{2}{j}_{3}}$, 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 transformed values ${\hat{z}}_{{k}_{1}{k}_{2}{k}_{3}}$ are complex. Because of conjugate symmetry (i.e., ${\hat{z}}_{{k}_{1}{k}_{2}{k}_{3}}$ is the complex conjugate of ${\hat{z}}_{\left({n}_{1}{k}_{1}\right)\left({n}_{2}{k}_{2}\right)\left({n}_{3}{k}_{3}\right)}$), only slightly more than half of the Fourier coefficients need to be stored in the output.
A call of
c06pyf followed by a call of
c06pzf will restore the original data.
This routine calls
c06pqf and
c06prf to perform multiple onedimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm in
Brigham (1974) and
Temperton (1983).
4
References
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Fast mixedradix real Fourier transforms J. Comput. Phys. 52 340–350
5
Arguments
 1: $\mathbf{n1}$ – IntegerInput

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

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

On entry: ${n}_{3}$, the third dimension of the transform.
Constraint:
${\mathbf{n3}}\ge 1$.
 4: $\mathbf{x}\left({\mathbf{n1}}\times {\mathbf{n2}}\times {\mathbf{n3}}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the real input dataset
$x$, where
${x}_{{j}_{1}{j}_{2}{j}_{3}}$ is stored in
${\mathbf{x}}\left({j}_{3}\times {n}_{1}{n}_{2}+{j}_{2}\times {n}_{1}+{j}_{1}+1\right)$, for
${j}_{1}=0,1,\dots ,{n}_{1}1$,
${j}_{2}=0,1,\dots ,{n}_{2}1$ and
${j}_{3}=0,1,\dots ,{n}_{3}1$.
That is, if
x is regarded as a threedimensional array of dimension
$\left(0:{\mathbf{n1}}1,0:{\mathbf{n2}}1,0:{\mathbf{n3}}1\right)$,
${\mathbf{x}}\left({j}_{1},{j}_{2},{j}_{3}\right)$ must contain
${x}_{{j}_{1}{j}_{2}{j}_{3}}$.
 5: $\mathbf{y}\left(\left({\mathbf{n1}}/2+1\right)\times {\mathbf{n2}}\times {\mathbf{n3}}\right)$ – Complex (Kind=nag_wp) arrayOutput

On exit: the complex output dataset
$\hat{z}$, where
${\hat{z}}_{{k}_{1}{k}_{2}{k}_{3}}$ is stored in
${\mathbf{y}}\left({k}_{3}\times \left({n}_{1}/2+1\right){n}_{2}+{k}_{2}\times \left({n}_{1}/2+1\right)+{k}_{1}+1\right)$, for
${k}_{1}=0,1,\dots ,{n}_{1}/2$,
${k}_{2}=0,1,\dots ,{n}_{2}1$ and
${k}_{3}=0,1,\dots ,{n}_{3}1$.
That is, if
y is regarded as a threedimensional array of dimension
$\left(0:{\mathbf{n1}}/2,0:{\mathbf{n2}}1,0:{\mathbf{n3}}1\right)$,
${\mathbf{y}}\left({k}_{1},{k}_{2},{k}_{3}\right)$ contains
${\hat{z}}_{{k}_{1}{k}_{2}{k}_{3}}$.
Note the first dimension is cut roughly by half to remove the redundant information due to conjugate symmetry.
 6: $\mathbf{ifail}$ – 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, ${\mathbf{n1}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n1}}\ge 1$.
 ${\mathbf{ifail}}=2$

On entry, ${\mathbf{n2}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n2}}\ge 1$.
 ${\mathbf{ifail}}=3$

On entry, ${\mathbf{n3}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n3}}\ge 1$.
 ${\mathbf{ifail}}=4$

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.
 ${\mathbf{ifail}}=99$
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 forward transform using
c06pyf and a backward transform using
c06pzf, and comparing the results with the original sequence (in exact arithmetic they would be identical).
8
Parallelism and Performance
c06pyf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pyf 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 by c06pyf is approximately proportional to ${n}_{1}{n}_{2}{n}_{3}\mathrm{log}\left({n}_{1}{n}_{2}{n}_{3}\right)$, but also depends on the factors of ${n}_{1}$, ${n}_{2}$ and ${n}_{3}$. c06pyf is fastest if the only prime factors of ${n}_{1}$, ${n}_{2}$ and ${n}_{3}$ are $2$, $3$ and $5$, and is particularly slow if one of the dimensions is a large prime, or has large prime factors.
Workspace is internally allocated by c06pyf. The total size of these arrays is approximately proportional to ${n}_{1}{n}_{2}{n}_{3}$.
10
Example
This example reads in a trivariate sequence of real data values and prints their discrete Fourier transforms as computed by
c06pyf. Inverse transforms are then calculated by calling
c06pzf showing that the original sequences are restored.
10.1
Program Text
Program Text (c06pyfe.f90)
10.2
Program Data
Program Data (c06pyfe.d)
10.3
Program Results
Program Results (c06pyfe.r)