NAG FL Interface
c06pff (fft_complex_multid_1d)
1
Purpose
c06pff computes the discrete Fourier transform of one variable in a multivariate sequence of complex data values.
2
Specification
Fortran Interface
Integer, Intent (In) 
:: 
ndim, l, 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 

C++ Header Interface
#include <nag.h> extern "C" {
}

The routine may be called by the names c06pff or nagf_sum_fft_complex_multid_1d.
3
Description
c06pff computes the discrete Fourier transform of one variable (the $l$th say) in a multivariate sequence of complex data values ${z}_{{j}_{1}{j}_{2}\cdots {j}_{m}}$, 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 routine computes
$n/{n}_{l}$ onedimensional transforms defined by
where
${k}_{l}=0,1,\dots ,{n}_{l}1$. 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.
(Note the scale factor of $\frac{1}{\sqrt{{n}_{l}}}$ in this definition.)
A call of c06pff 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 complex 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:
$\mathbf{ndim}$ – Integer
Input

On entry: $m$, the number of dimensions (or variables) in the multivariate data.
Constraint:
${\mathbf{ndim}}\ge 1$.

3:
$\mathbf{l}$ – Integer
Input

On entry: $l$, the index of the variable (or dimension) on which the discrete Fourier transform is to be performed.
Constraint:
$1\le {\mathbf{l}}\le {\mathbf{ndim}}$.

4:
$\mathbf{nd}\left({\mathbf{ndim}}\right)$ – Integer array
Input

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 $\mathit{i}=1,2,\dots ,{\mathbf{ndim}}$.

5:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the total number of data values.
Constraint:
n must equal the product of the first
ndim elements of the array
nd.

6:
$\mathbf{x}\left({\mathbf{n}}\right)$ – Complex (Kind=nag_wp) array
Input/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.

7:
$\mathbf{work}\left({\mathbf{lwork}}\right)$ – Complex (Kind=nag_wp) array
Workspace

The workspace requirements as documented for c06pff 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.

8:
$\mathbf{lwork}$ – Integer
Input

On entry: the dimension of the array
work as declared in the (sub)program from which
c06pff is called.
Suggested value:
${\mathbf{lwork}}\ge {\mathbf{n}}+{\mathbf{nd}}\left({\mathbf{l}}\right)+15$.

9:
$\mathbf{ifail}$ – Integer
Input/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 4 in the Introduction to the NAG Library FL Interface 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, ${\mathbf{ndim}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ndim}}\ge 1$.
 ${\mathbf{ifail}}=2$

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

On entry, ${\mathbf{direct}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{direct}}=\text{'F'}$ or $\text{'B'}$.
 ${\mathbf{ifail}}=4$

On entry, ${\mathbf{nd}}\left(\u2329\mathit{\text{value}}\u232a\right)=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nd}}\left(i\right)\ge 1$, for all $i$.
 ${\mathbf{ifail}}=5$

On entry,
${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$, product of
nd elements is
$\u2329\mathit{\text{value}}\u232a$.
Constraint:
n must equal the product of the dimensions held in array
nd.
 ${\mathbf{ifail}}=6$

On entry,
${\mathbf{lwork}}=\u2329\mathit{\text{value}}\u232a$.
Constraint:
lwork must be at least
$\u2329\mathit{\text{value}}\u232a$.
 ${\mathbf{ifail}}=8$

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.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 7 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 9 in the Introduction to the NAG Library FL Interface 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
c06pff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pff 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 $n\times \mathrm{log}{n}_{l}$, but also depends on the factorization of ${n}_{l}$. c06pff is faster if the only prime factors of ${n}_{l}$ are $2$, $3$ or $5$; and fastest of all if ${n}_{l}$ is a power of $2$.
10
Example
This example reads in a bivariate sequence of complex data values and prints the discrete Fourier transform of the second variable. It then performs an inverse transform and prints the sequence so obtained, which may be compared with the original data values.
10.1
Program Text
10.2
Program Data
10.3
Program Results