NAG Library Routine Document
C06PQF
1 Purpose
C06PQF computes the discrete Fourier transforms of $m$ sequences, each containing $n$ real data values or a Hermitian complex sequence stored columnwise in a complex storage format.
2 Specification
INTEGER 
N, M, IFAIL 
REAL (KIND=nag_wp) 
X((N+2)*M), WORK(*) 
CHARACTER(1) 
DIRECT 

3 Description
Given
$m$ sequences of
$n$ real data values
${x}_{\mathit{j}}^{\mathit{p}}$, for
$\mathit{j}=0,1,\dots ,n1$ and
$\mathit{p}=1,2,\dots ,m$, C06PQF simultaneously calculates the Fourier transforms of all the sequences defined by
The transformed values ${\hat{z}}_{k}^{p}$ are complex, but for each value of $p$ the ${\hat{z}}_{k}^{p}$ form a Hermitian sequence (i.e., ${\hat{z}}_{nk}^{p}$ is the complex conjugate of ${\hat{z}}_{k}^{p}$), so they are completely determined by $mn$ real numbers (since ${\hat{z}}_{0}^{p}$ is real, as is ${\hat{z}}_{n/2}^{p}$ for $n$ even).
Alternatively, given
$m$ Hermitian sequences of
$n$ complex data values
${z}_{j}^{p}$, this routine simultaneously calculates their inverse (
backward) discrete Fourier transforms defined by
The transformed values
${\hat{x}}_{k}^{p}$ are real.
(Note the scale factor $\frac{1}{\sqrt{n}}$ in the above definition.)
A call of C06PQF with ${\mathbf{DIRECT}}=\text{'F'}$ followed by a call with ${\mathbf{DIRECT}}=\text{'B'}$ will restore the original data.
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). Special coding is provided for the factors
$2$,
$3$,
$4$ and
$5$.
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: $\mathrm{DIRECT}$ – CHARACTER(1)Input

On entry: if the forward transform as defined in
Section 3 is to be computed, then
DIRECT must be set equal to 'F'.
If the backward transform is to be computed then
DIRECT must be set equal to 'B'.
Constraint:
${\mathbf{DIRECT}}=\text{'F'}$ or $\text{'B'}$.
 2: $\mathrm{N}$ – INTEGERInput

On entry: $n$, the number of real or complex values in each sequence.
Constraint:
${\mathbf{N}}\ge 1$.
 3: $\mathrm{M}$ – INTEGERInput

On entry: $m$, the number of sequences to be transformed.
Constraint:
${\mathbf{M}}\ge 1$.
 4: $\mathrm{X}\left(\left({\mathbf{N}}+2\right)\times {\mathbf{M}}\right)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry: the data must be stored in
X as if in a twodimensional array of dimension
$\left(0:{\mathbf{N}}+1,1:{\mathbf{M}}\right)$; each of the
$m$ sequences is stored in a
column of the array. In other words, if the data values of the
$p$th sequence to be transformed are denoted by
${x}_{\mathit{j}}^{p}$, for
$\mathit{j}=0,1,\dots ,n1$, then:
 if ${\mathbf{DIRECT}}=\text{'F'}$,
${\mathbf{X}}\left(\left(\mathit{p}1\right)\times \left({\mathbf{N}}+2\right)+\mathit{j}\right)$ must contain ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n1$ and $\mathit{p}=1,2,\dots ,m$;

if ${\mathbf{DIRECT}}=\text{'B'}$, ${\mathbf{X}}\left(\left(\mathit{p}1\right)\times \left({\mathbf{N}}+2\right)+2\times \mathit{k}\right)$ and ${\mathbf{X}}\left(\left(\mathit{p}1\right)\times \left({\mathbf{N}}+2\right)+2\times \mathit{k}+1\right)$ must contain the real and imaginary parts respectively of ${\hat{z}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=0,1,\dots ,n/2$ and $\mathit{p}=1,2,\dots ,m$. (Note that for the sequence ${\hat{z}}_{k}^{p}$ to be Hermitian, the imaginary part of ${\hat{z}}_{0}^{p}$, and of ${\hat{z}}_{n/2}^{p}$ for $n$ even, must be zero.)
On exit:

if ${\mathbf{DIRECT}}=\text{'F'}$ and X is declared with bounds $\left(0:{\mathbf{N}}+1,1:{\mathbf{M}}\right)$ then ${\mathbf{X}}\left(2\times \mathit{k},\mathit{p}\right)$ and ${\mathbf{X}}\left(2\times \mathit{k}+1,\mathit{p}\right)$ will contain the real and imaginary parts respectively of ${\hat{z}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=0,1,\dots ,n/2$ and $\mathit{p}=1,2,\dots ,m$;

if ${\mathbf{DIRECT}}=\text{'B'}$ and X is declared with bounds $\left(0:{\mathbf{N}}+1,1:{\mathbf{M}}\right)$ then ${\mathbf{X}}\left(\mathit{j},\mathit{p}\right)$ will contain ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n1$ and $\mathit{p}=1,2,\dots ,m$.
 5: $\mathrm{WORK}\left(*\right)$ – REAL (KIND=nag_wp) arrayWorkspace

Note: the dimension of the array
WORK
must be at least
$\left({\mathbf{M}}+2\right)\times {\mathbf{N}}+15$.
The workspace requirements as documented for C06PQF may be an overestimate in some implementations.
On exit:
${\mathbf{WORK}}\left(1\right)$ contains the minimum workspace required for the current values of
M and
N with this implementation.
 6: $\mathrm{IFAIL}$ – 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,  ${\mathbf{M}}<1$. 
 ${\mathbf{IFAIL}}=2$

On entry,  ${\mathbf{N}}<1$. 
 ${\mathbf{IFAIL}}=3$

On entry,  ${\mathbf{DIRECT}}\ne \text{'F'}$ or $\text{'B'}$. 
 ${\mathbf{IFAIL}}=4$

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
C06PQF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
C06PQF 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 C06PQF is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. C06PQF is fastest if the only prime factors of $n$ are $2$, $3$ and $5$, and is particularly slow if $n$ is a large prime, or has large prime factors.
10 Example
This example reads in sequences of real data values and prints their discrete Fourier transforms (as computed by C06PQF with ${\mathbf{DIRECT}}=\text{'F'}$), after expanding them from complex Hermitian form into a full complex sequences.
Inverse transforms are then calculated by calling C06PQF with ${\mathbf{DIRECT}}=\text{'B'}$ showing that the original sequences are restored.
10.1 Program Text
Program Text (c06pqfe.f90)
10.2 Program Data
Program Data (c06pqfe.d)
10.3 Program Results
Program Results (c06pqfe.r)