The routine may be called by the names c06fbf or nagf_sum_fft_hermitian_1d_rfmt.
3Description
Given a Hermitian sequence of $n$ complex data values ${z}_{\mathit{j}}$ (i.e., a sequence such that ${z}_{0}$ is real and ${z}_{n-\mathit{j}}$ is the complex conjugate of ${z}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n-1$), c06fbf calculates their discrete Fourier transform defined by
(Note the scale factor of $\frac{1}{\sqrt{n}}$ in this definition.) The transformed values ${\hat{x}}_{k}$ are purely real (see also the C06 Chapter Introduction).
To compute the inverse discrete Fourier transform defined by
this routine should be preceded by forming the complex conjugates of the ${\hat{z}}_{k}$; that is, $x\left(\mathit{k}\right)=-x\left(\mathit{k}\right)$, for $\mathit{k}=n/2+2,\dots ,n$.
c06fbf uses the fast Fourier transform (FFT) algorithm (see Brigham (1974)).
4References
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
5Arguments
1: $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the sequence to be transformed stored in Hermitian form. If the data values ${z}_{j}$ are written as ${x}_{j}+i{y}_{j}$, and if x is declared with bounds $(0:{\mathbf{n}}-1)$ in the subroutine from which c06fbf is called, then for $0\le j\le n/2$, ${x}_{j}$ is contained in ${\mathbf{x}}\left(j\right)$, and for $1\le j\le (n-1)/2$, ${y}_{j}$ is contained in ${\mathbf{x}}\left(n-j\right)$. (See also Section 2.1.2 in the C06 Chapter Introduction.)
On exit: the components of the discrete Fourier transform ${\hat{x}}_{k}$. If x is declared with bounds $(0:{\mathbf{n}}-1)$ in the subroutine from which c06fbf is called,
${\hat{x}}_{\mathit{k}}$ is stored in ${\mathbf{x}}\left(\mathit{k}\right)$, for $\mathit{k}=0,1,\dots ,n-1$.
2: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of data values.
Constraint:
${\mathbf{n}}>1$.
3: $\mathbf{work}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
4: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1$ 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 $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ 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 $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=3$
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}>1$.
${\mathbf{ifail}}=-99$
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
c06fbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06fbf 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 $n$. c06fbf 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 which is assumed to be a Hermitian sequence of complex data values stored in Hermitian form. The input sequence is expanded into a full complex sequence and printed alongside the original sequence. The discrete Fourier transform (as computed by c06fbf) is printed out. It then performs an inverse transform using c06faf and conjugation, and prints the sequence so obtained alongside the original data values.