The routine may be called by the names c06pvf or nagf_sum_fft_real_2d.
3Description
c06pvf computes the two-dimensional discrete Fourier transform of a bivariate sequence of real data values ${x}_{{j}_{1}{j}_{2}}$, for ${j}_{1}=0,1,\dots ,m-1$ and ${j}_{2}=0,1,\dots ,n-1$.
where ${k}_{1}=0,1,\dots ,m-1$ and ${k}_{2}=0,1,\dots ,n-1$. (Note the scale factor of $\frac{1}{\sqrt{mn}}$ in this definition.)
The transformed values ${\hat{z}}_{{k}_{1}{k}_{2}}$ are complex. Because of conjugate symmetry (i.e., ${\hat{z}}_{{k}_{1}{k}_{2}}$ is the complex conjugate of ${\hat{z}}_{(m-{k}_{1})(n-{k}_{2})}$), only slightly more than half of the Fourier coefficients need to be stored in the output.
A call of c06pvf followed by a call of c06pwf will restore the original data.
This routine calls c06pqfandc06prf to perform multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm in Brigham (1974) and Temperton (1983).
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 first dimension of the transform.
Constraint:
${\mathbf{m}}\ge 1$.
2: $\mathbf{n}$ – IntegerInput
On entry: $n$, the second dimension of the transform.
Constraint:
${\mathbf{n}}\ge 1$.
3: $\mathbf{x}\left({\mathbf{m}}\times {\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the real input dataset $x$, where
${x}_{{j}_{1}{j}_{2}}$ is stored in ${\mathbf{x}}\left({j}_{2}\times m+{j}_{1}\right)$, for ${j}_{1}=0,1,\dots ,m-1$ and ${j}_{2}=0,1,\dots ,n-1$. That is, if x is regarded as a two-dimensional array of dimension $(0:{\mathbf{m}}-1,0:{\mathbf{n}}-1)$, ${\mathbf{x}}({j}_{1},{j}_{2})$ must contain ${x}_{{j}_{1}{j}_{2}}$.
On exit: the complex output dataset $\hat{z}$, where
${\hat{z}}_{{k}_{1}{k}_{2}}$ is stored in ${\mathbf{y}}\left({k}_{2}\times (m/2+1)+{k}_{1}\right)$, for ${k}_{1}=0,1,\dots ,m/2$ and ${k}_{2}=0,1,\dots ,n-1$. That is, if y is regarded as a two-dimensional array of dimension $(0:{\mathbf{m}}/2,0:{\mathbf{n}}-1)$, ${\mathbf{y}}({k}_{1},{k}_{2})$ contains ${\hat{z}}_{{k}_{1}{k}_{2}}$. Note the first dimension is cut roughly by half to remove the redundant information due to conjugate symmetry.
5: $\mathbf{ifail}$ – 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:
${\mathbf{ifail}}=1$
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 forward transform using c06pvf and a backward transform using c06pwf, and comparing the results with the original sequence (in exact arithmetic they would be identical).
8Parallelism and Performance
c06pvf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pvf 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 c06pvf is approximately proportional to $mn\mathrm{log}\left(mn\right)$, but also depends on the factors of $m$ and $n$. c06pvf is fastest if the only prime factors of $m$ and $n$ are $2$, $3$ and $5$, and is particularly slow if $m$ or $n$ is a large prime, or has large prime factors.
Workspace is internally allocated by c06pvf. The total size of these arrays is approximately proportional to $mn$.
10Example
This example reads in a bivariate sequence of real data values and prints their discrete Fourier transforms as computed by c06pvf. Inverse transforms are then calculated by calling c06pwf showing that the original sequences are restored.