The routine may be called by the names c06pff or nagf_sum_fft_complex_multid_1d.
3Description
c06pff computes the discrete Fourier transform of one variable (the $l$th say) in a multivariate sequence of complex data values ${z}_{{j}_{1}{j}_{2}\cdots {j}_{m}}$, 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 routine computes $n/{n}_{l}$ one-dimensional transforms defined by
where ${k}_{l}=0,1,\dots ,{n}_{l}-1$. 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.
(Note the scale factor of $\frac{1}{\sqrt{{n}_{l}}}$ in this definition.)
A call of c06pff 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 complex 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.
Constraint:
${\mathbf{ndim}}\ge 1$.
3: $\mathbf{l}$ – IntegerInput
On entry: $l$, the index of the variable (or dimension) on which the discrete Fourier transform is to be performed.
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 $\mathit{i}=1,2,\dots ,{\mathbf{ndim}}$.
5: $\mathbf{n}$ – 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.
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, ${\mathbf{l}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: $1\le {\mathbf{l}}\le {\mathbf{ndim}}$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{direct}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{direct}}=\text{'F'}$ or $\text{'B'}$.
${\mathbf{ifail}}=4$
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 $i$.
${\mathbf{ifail}}=5$
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.
${\mathbf{ifail}}=6$
On entry, ${\mathbf{lwork}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: lwork must be at least $\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=8$
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.
c06pff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pff 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 $n\times \mathrm{log}{n}_{l}$, but also depends on the factorization of ${n}_{l}$. c06pff is faster if the only prime factors of ${n}_{l}$ are $2$, $3$ or $5$; and fastest of all if ${n}_{l}$ is a power of $2$.
10Example
This example reads in a bivariate sequence of complex data values and prints the discrete Fourier transform of the second variable. It then performs an inverse transform and prints the sequence so obtained, which may be compared with the original data values.