c06rff computes the discrete Fourier cosine transforms of $m$ sequences of real data values. The elements of each sequence and its transform are stored contiguously.
The routine may be called by the names c06rff or nagf_sum_fft_cosine.
3Description
Given $m$ sequences of $n+1$ real data values ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, c06rff simultaneously calculates the Fourier cosine transforms of all the sequences defined by
(Note the scale factor $\sqrt{\frac{2}{n}}$ in this definition.)
This transform is also known as type-I DCT.
Since the Fourier cosine transform defined above is its own inverse, two consecutive calls of c06rff will restore the original data.
The transform calculated by this routine can be used to solve Poisson's equation when the derivative of the solution is specified at both left and right boundaries (see Swarztrauber (1977)).
The routine uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, described in Temperton (1983), together with pre- and post-processing stages described in Swarztrauber (1982). Special coding is provided for the factors $2$, $3$, $4$ and $5$.
4References
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Swarztrauber P N (1977) The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle SIAM Rev.19(3) 490–501
Swarztrauber P N (1982) Vectorizing the FFT's Parallel Computation (ed G Rodrique) 51–83 Academic Press
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: one less than the number of real values in each sequence, i.e., the number of values in each sequence is $n+1$.
Constraint:
${\mathbf{n}}\ge 1$.
3: $\mathbf{x}(0:{\mathbf{n}},{\mathbf{m}})$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the data values of the $\mathit{p}$th sequence to be transformed, denoted by
${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, must be stored in ${\mathbf{x}}(j,p)$.
On exit: the $(n+1)$ components of the $\mathit{p}$th Fourier cosine transform, denoted by
${\hat{x}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=0,1,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, are stored in ${\mathbf{x}}(k,p)$, overwriting the corresponding original values.
4: $\mathbf{ifail}$ – 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: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=3$
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.
c06rff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 c06rff is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. c06rff 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. Workspace of order $\mathit{O}\left(n\right)$ is internally allocated by this routine.
10Example
This example reads in sequences of real data values and prints their Fourier cosine transforms (as computed by c06rff). It then calls c06rff again and prints the results which may be compared with the original sequence.