c06paf calculates the discrete Fourier transform of a sequence of $n$ real data values or of a Hermitian sequence of $n$ complex data values stored in compact form in a real array.
The routine may be called by the names c06paf or nagf_sum_fft_realherm_1d.
3Description
Given a sequence of $n$ real data values ${x}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$, c06paf calculates their discrete Fourier transform (in the forward direction) defined by
The transformed values ${\hat{z}}_{k}$ are complex, but they form a Hermitian sequence (i.e., ${\hat{z}}_{n-k}$ is the complex conjugate of ${\hat{z}}_{k}$), so they are completely determined by $n$ real numbers (since ${\hat{z}}_{0}$ is real, as is ${\hat{z}}_{n/2}$ for $n$ even).
Alternatively, given a Hermitian sequence of $n$ complex data values ${z}_{j}$, this routine calculates their inverse (backward) discrete Fourier transform defined by
(Note the scale factor of $\frac{1}{\sqrt{n}}$ in the above definitions.)
A call of c06paf with ${\mathbf{direct}}=\text{'F'}$ followed by a call with ${\mathbf{direct}}=\text{'B'}$ will restore the original data.
c06paf 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).
The same functionality is available using the forward and backward transform routine pair: c06pvfandc06pwf on setting ${\mathbf{n}}=1$. This pair use a different storage solution; real data is stored in a real array, while Hermitian data (the first unconjugated half) is stored in a complex array.
4References
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Self-sorting mixed-radix fast Fourier transforms J. Comput. Phys.52 1–23
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: $\mathbf{x}\left({\mathbf{n}}+2\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if x is declared with bounds $(0:{\mathbf{n}}+1)$ in the subroutine from which c06paf is called:
if ${\mathbf{direct}}=\text{'F'}$,
${\mathbf{x}}\left(\mathit{j}\right)$ must contain ${x}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$;
if ${\mathbf{direct}}=\text{'B'}$, ${\mathbf{x}}\left(2\times \mathit{k}\right)$ and ${\mathbf{x}}\left(2\times \mathit{k}+1\right)$ must contain the real and imaginary parts respectively of ${z}_{\mathit{k}}$, for $\mathit{k}=0,1,\dots ,n/2$. (Note that for the sequence ${z}_{k}$ to be Hermitian, the imaginary part of ${z}_{0}$, and of ${z}_{n/2}$ for $n$ even, must be zero.)
On exit:
if ${\mathbf{direct}}=\text{'F'}$ and x is declared with bounds $(0:{\mathbf{n}}+1)$,
${\mathbf{x}}\left(2\times \mathit{k}\right)$ and ${\mathbf{x}}\left(2\times \mathit{k}+1\right)$ will contain the real and imaginary parts respectively of ${\hat{z}}_{\mathit{k}}$, for $\mathit{k}=0,1,\dots ,n/2$;
if ${\mathbf{direct}}=\text{'B'}$ and x is declared with bounds $(0:{\mathbf{n}}+1)$,
${\mathbf{x}}\left(\mathit{j}\right)$ will contain ${\hat{x}}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$.
3: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of data values.
Constraint:
${\mathbf{n}}\ge 1$.
4: $\mathbf{work}\left(*\right)$ – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array work
must be at least
$3\times {\mathbf{n}}+100$.
The workspace requirements as documented for c06paf may be an overestimate in some implementations.
On exit: ${\mathbf{work}}\left(1\right)$ contains the minimum workspace required for the current value of n with this implementation.
5: $\mathbf{ifail}$ – 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, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{direct}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{direct}}=\text{'F'}$ or $\text{'B'}$.
${\mathbf{ifail}}=3$
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.
c06paf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06paf 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 is approximately proportional to $n\times \mathrm{log}\left(n\right)$, but also depends on the factorization of $n$. c06paf is faster if the only prime factors of $n$ are $2$, $3$ or $5$; and fastest of all if $n$ is a power of $2$.
10Example
This example reads in a sequence of real data values and prints their discrete Fourier transform (as computed by c06paf with ${\mathbf{direct}}=\text{'F'}$), after expanding it from complex Hermitian form into a full complex sequence. It then performs an inverse transform using c06paf with ${\mathbf{direct}}=\text{'B'}$, and prints the sequence so obtained alongside the original data values.