The routine may be called by the names c06pjf or nagf_sum_fft_complex_multid.
3Description
c06pjf computes the multidimensional discrete Fourier transform of a multidimensional sequence of complex data values ${z}_{{j}_{1}{j}_{2}\dots {j}_{m}}$, where ${j}_{1}=0,1,\dots ,{n}_{1}-1\text{, \hspace{1em}}{j}_{2}=0,1,\dots ,{n}_{2}-1$, and so on. Thus the individual dimensions are ${n}_{1},{n}_{2},\dots ,{n}_{m}$, and the total number of data values is $n={n}_{1}\times {n}_{2}\times \cdots \times {n}_{m}$.
The discrete Fourier transform is here defined (e.g., for $m=2$) by:
where ${k}_{1}=0,1,\dots ,{n}_{1}-1$ and ${k}_{2}=0,1,\dots ,{n}_{2}-1$. The plus or minus sign in the argument of the exponential terms in the above definition determine the direction of the transform: a minus sign defines the forward direction and a plus sign defines the backward direction.
The extension to higher dimensions is obvious. (Note the scale factor of $\frac{1}{\sqrt{n}}$ in this definition.)
A call of c06pjf with ${\mathbf{direct}}=\text{'F'}$ followed by a call with ${\mathbf{direct}}=\text{'B'}$ will restore the original data.
The data values must be supplied in a one-dimensional array using column-major storage ordering of multidimensional data (i.e., with the first subscript ${j}_{1}$ varying most rapidly).
This routine
calls c06prf to perform one-dimensional discrete Fourier transforms. Hence, 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).
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: $\mathbf{ndim}$ – IntegerInput
On entry: $m$, the number of dimensions (or variables) in the multivariate data.
On entry: the elements of nd must contain the dimensions of the ndim variables; that is, ${\mathbf{nd}}\left(i\right)$ must contain the dimension of the $i$th variable.
Constraint:
${\mathbf{nd}}\left(\mathit{i}\right)\ge 1$, for $\mathit{i}=1,2,\dots ,{\mathbf{ndim}}$.
4: $\mathbf{n}$ – IntegerInput
On entry: $n$, the total number of data values.
Constraint:
n must equal the product of the first ndim elements of the array nd.
On entry: the complex data values. Data values are stored in x using column-major ordering for storing multidimensional arrays; that is, ${z}_{{j}_{1}{j}_{2}\cdots {j}_{m}}$ is stored in ${\mathbf{x}}\left(1+{j}_{1}+{n}_{1}{j}_{2}+{n}_{1}{n}_{2}{j}_{3}+\cdots \right)$.
On exit: the corresponding elements of the computed transform.
The workspace requirements as documented for c06pjf may be an overestimate in some implementations.
On exit: the real part of ${\mathbf{work}}\left(1\right)$ contains the minimum workspace required for the current value of n with this implementation.
7: $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which c06pjf is called.
Suggested value:
${\mathbf{lwork}}\ge {\mathbf{n}}+3\times \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{nd}}\left(i\right)\right)+15$, where $i=1,2,\dots ,{\mathbf{ndim}}$.
8: $\mathbf{ifail}$ – 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, ${\mathbf{ndim}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ndim}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{direct}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{direct}}=\text{'F'}$ or $\text{'B'}$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{nd}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nd}}\left(i\right)\ge 1$, for all $i$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$, product of nd elements is $\u27e8\mathit{\text{value}}\u27e9$.
Constraint: n must equal the product of the dimensions held in array nd.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{lwork}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: lwork must be at least $\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=7$
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
c06pjf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pjf 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 the individual dimensions ${\mathbf{nd}}\left(i\right)$. c06pjf is faster if the only prime factors are $2$, $3$ or $5$; and fastest of all if they are powers of $2$.
10Example
This example reads in a bivariate sequence of complex data values and prints the two-dimensional Fourier transform. It then performs an inverse transform and prints the sequence so obtained, which may be compared to the original data values.