c06pqf computes the discrete Fourier transforms of $m$ sequences, each containing $n$ real data values or a Hermitian complex sequence stored column-wise in a complex storage format.
The routine may be called by the names c06pqf or nagf_sum_fft_realherm_1d_multi_col.
3Description
Given $m$ sequences of $n$ real data values ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ 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., ${\hat{z}}_{n-k}^{p}$ 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 self-sorting algorithm, which is described in Temperton (1983). Special coding is provided for the factors $2$, $3$, $4$ and $5$.
4References
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys.52 340–350
5Arguments
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: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of real or complex values in each sequence.
Constraint:
${\mathbf{n}}\ge 1$.
3: $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of sequences to be transformed.
Constraint:
${\mathbf{m}}\ge 1$.
4: $\mathbf{x}\left(({\mathbf{n}}+2)\times {\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the $m$ real or Hermitian data sequences to be transformed.
if ${\mathbf{direct}}=\text{'F'}$, the $m$ real data sequences,
${x}^{\mathit{p}}=({x}_{1}^{\mathit{p}},{x}_{2}^{\mathit{p}},\dots ,{x}_{n}^{\mathit{p}})$, for $\mathit{p}=1,2,\dots ,m$, should be stored sequentially in x, with a stride of $n+2$ between sequences.
if ${\mathbf{direct}}=\text{'B'}$, the $m$ Hermitian data sequences,
${\hat{z}}^{\mathit{p}}=({\hat{z}}_{1}^{\mathit{p}},{\hat{z}}_{2}^{\mathit{p}},\dots ,{\hat{z}}_{n/2+1}^{\mathit{p}})=(\mathrm{Re}\left({\hat{z}}_{1}^{\mathit{p}}\right),\mathrm{Im}\left({\hat{z}}_{1}^{\mathit{p}}\right),\mathrm{Re}\left({\hat{z}}_{2}^{\mathit{p}}\right),\mathrm{Im}\left({\hat{z}}_{2}^{\mathit{p}}\right),\dots ,\mathrm{Re}\left({\hat{z}}_{n/2+1}^{\mathit{p}}\right),\mathrm{Im}\left({\hat{z}}_{n/2+1}^{\mathit{p}}\right))$, for $\mathit{p}=1,2,\dots ,m$, should be stored sequentially in x, with a stride of $n+2$ between sequences.
In other words:
if ${\mathbf{direct}}=\text{'F'}$,
${\mathbf{x}}\left((\mathit{p}-1)\times ({\mathbf{n}}+2)+\mathit{j}\right)$ must contain ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=1,2,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$;
if ${\mathbf{direct}}=\text{'B'}$, ${\mathbf{x}}\left((\mathit{p}-1)\times ({\mathbf{n}}+2)+(2\times \mathit{k}-1)\right)$ and ${\mathbf{x}}\left((\mathit{p}-1)\times ({\mathbf{n}}+2)+(2\times \mathit{k})\right)$ must contain the real and imaginary parts respectively of ${\hat{z}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=1,2,\dots ,n/2+1$ and $\mathit{p}=1,2,\dots ,m$. (Note that for the sequence ${\hat{z}}_{k}^{p}$ to be Hermitian, the imaginary part of ${\hat{z}}_{1}^{p}$, and of ${\hat{z}}_{n/2+1}^{p}$ for $n$ even, must be zero.)
On exit:
if ${\mathbf{direct}}=\text{'F'}$ then the $m$ sequences,
${\hat{z}}^{\mathit{p}}$, for $\mathit{p}=1,2,\dots ,m$ stored as described on entry for ${\mathbf{direct}}=\text{'B'}$
if ${\mathbf{direct}}=\text{'B'}$ then the $m$ sequences,
${x}^{\mathit{p}}$, for $\mathit{p}=1,2,\dots ,m$ stored as described on entry for ${\mathbf{direct}}=\text{'F'}$
5: $\mathbf{work}\left(*\right)$ – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array work
must be at least
$({\mathbf{m}}+2)\times {\mathbf{n}}+15$.
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: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, 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, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{direct}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{direct}}=\text{'F'}$ or $\text{'B'}$.
${\mathbf{ifail}}=4$
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 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.
7Accuracy
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).
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
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 implementation-specific information.
9Further Comments
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.
10Example
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 full complex sequences.
Inverse transforms are then calculated by calling c06pqf with ${\mathbf{direct}}=\text{'B'}$ showing that the original sequences are restored.