c06fpf computes the discrete Fourier transforms of $m$ sequences, each containing $n$ real data values. This routine is designed to be particularly efficient on vector processors.
The routine may be called by the names c06fpf or nagf_sum_withdraw_fft_real_1d_multi_rfmt.
3Description
Given $m$ sequences of $n$ real data values ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, c06fpf simultaneously calculates the Fourier transforms of all the sequences defined by
(Note the scale factor $\frac{1}{\sqrt{n}}$ in this definition.)
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 (see also the C06 Chapter Introduction).
The discrete Fourier transform is sometimes defined using a positive sign in the exponential term:
To compute this form, this routine should be followed by forming the complex conjugates of the ${\hat{z}}_{k}^{p}$; that is $x\left(\mathit{k}\right)=-x\left(\mathit{k}\right)$, for $\mathit{k}=(n/2+1)\times m+1,\dots ,m\times n$.
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 $2$, $3$, $4$, $5$ and $6$. This routine is designed to be particularly efficient on vector processors, and it becomes especially fast as $m$, the number of transforms to be computed in parallel, increases.
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: $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of sequences to be transformed.
Constraint:
${\mathbf{m}}\ge 1$.
2: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of real values in each sequence.
Constraint:
${\mathbf{n}}\ge 1$.
3: $\mathbf{x}\left({\mathbf{m}}\times {\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the data must be stored in x as if in a two-dimensional array of dimension $(1:{\mathbf{m}},0:{\mathbf{n}}-1)$; each of the $m$ sequences is stored in a row of the array. In other words, if the data values of the $p$th sequence to be transformed are denoted by
${x}_{j}^{p}$, for $\mathit{j}=0,1,\dots ,n-1$, the $mn$ elements of the array x must contain the values
On exit: the $m$ discrete Fourier transforms stored as if in a two-dimensional array of dimension $(1:{\mathbf{m}},0:{\mathbf{n}}-1)$. Each of the $m$ transforms is stored in a row of the array in Hermitian form, overwriting the corresponding original sequence. If the $n$ components of the discrete Fourier transform ${\hat{z}}_{k}^{p}$ are written as ${a}_{k}^{p}+i{b}_{k}^{p}$, then for $0\le k\le n/2$, ${a}_{k}^{p}$ is contained in ${\mathbf{x}}(p,k)$, and for $1\le k\le (n-1)/2$, ${b}_{k}^{p}$ is contained in ${\mathbf{x}}(p,n-k)$. (See also Section 2.1.2 in the C06 Chapter Introduction.)
4: $\mathbf{init}$ – Character(1)Input
On entry: indicates whether trigonometric coefficients are to be calculated.
${\mathbf{init}}=\text{'I'}$
Calculate the required trigonometric coefficients for the given value of $n$, and store in the array trig.
${\mathbf{init}}=\text{'S'}$ or $\text{'R'}$
The required trigonometric coefficients are assumed to have been calculated and stored in the array trig in a prior call to one of c06fpforc06fqf. The routine performs a simple check that the current value of $n$ is consistent with the values stored in trig.
Constraint:
${\mathbf{init}}=\text{'I'}$, $\text{'S'}$ or $\text{'R'}$.
5: $\mathbf{trig}\left(2\times {\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if ${\mathbf{init}}=\text{'S'}$ or $\text{'R'}$, trig must contain the required trigonometric coefficients that have been previously calculated. Otherwise trig need not be set.
On exit: contains the required coefficients (computed by the routine if ${\mathbf{init}}=\text{'I'}$).
6: $\mathbf{work}\left({\mathbf{m}}\times {\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
7: $\mathbf{ifail}$ – 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, ${\mathbf{init}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{init}}=\text{'I'}$, $\text{'S'}$ or $\text{'R'}$.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{init}}=\u27e8\mathit{\text{value}}\u27e9$ but n and trig array incompatible.
${\mathbf{ifail}}=-99$
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
c06fpf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06fpf 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 c06fpf is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. c06fpf 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 c06fpf). The Fourier transforms are expanded into full complex form using and printed. Inverse transforms are then calculated by conjugating and calling c06fqf showing that the original sequences are restored.