NAG Library Routine Document
C06PJF computes the multidimensional discrete Fourier transform of a multivariate sequence of complex data values.
||NDIM, ND(NDIM), N, LWORK, IFAIL
C06PJF computes the multidimensional discrete Fourier transform of a multidimensional sequence of complex data values , where , and so on. Thus the individual dimensions are , and the total number of data values is .
The discrete Fourier transform is here defined (e.g., for
. 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
The extension to higher dimensions is obvious. (Note the scale factor of in this definition.)
A call of C06PJF with followed by a call with 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 varying most rapidly).
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)
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
- 1: DIRECT – CHARACTER(1)Input
: if the forward transform as defined in Section 3
is to be computed, then DIRECT
must be set equal to 'F'.
If the backward transform is to be computed then DIRECT
must be set equal to 'B'.
- 2: NDIM – INTEGERInput
On entry: , the number of dimensions (or variables) in the multivariate data.
- 3: ND(NDIM) – INTEGER arrayInput
: the elements of ND
must contain the dimensions of the NDIM
variables; that is,
must contain the dimension of the
- , for ;
- must have less than prime factors (counting repetitions), for .
- 4: N – INTEGERInput
On entry: , the total number of data values.
must equal the product of the first NDIM
elements of the array ND
- 5: X(N) – COMPLEX (KIND=nag_wp) arrayInput/Output
: the complex data values. Data values are stored in X
using column-major ordering for storing multidimensional arrays; that is,
is stored in
On exit: the corresponding elements of the computed transform.
- 6: WORK(LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
The workspace requirements as documented for C06PJF may be an overestimate in some implementations.
: the real part of
contains the minimum workspace required for the current value of N
with this implementation.
- 7: LWORK – INTEGERInput
: the dimension of the array WORK
as declared in the (sub)program from which C06PJF is called.
- 8: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
|On entry,|| or .|
|On entry,||at least one of the first NDIM elements of ND is less than .|
|On entry,||N does not equal the product of the first NDIM elements of ND.|
|On entry,||LWORK is too small. The minimum amount of workspace required is returned in .|
|On entry,|| has more than prime factors for some .|
An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.
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).
The time taken is approximately proportional to , but also depends on the factorization of the individual dimensions . C06PJF is faster if the only prime factors are , or ; and fastest of all if they are powers of .
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.
9.1 Program Text
Program Text (c06pjfe.f90)
9.2 Program Data
Program Data (c06pjfe.d)
9.3 Program Results
Program Results (c06pjfe.r)