NAG Library Routine Document
C06PQF computes the discrete Fourier transforms of sequences, each containing real data values or a Hermitian complex sequence stored column-wise in a complex storage format.
||N, M, IFAIL
real data values
, C06PQF simultaneously calculates the Fourier transforms of all the sequences defined by
The transformed values are complex, but for each value of the form a Hermitian sequence (i.e., is the complex conjugate of ), so they are completely determined by real numbers (since is real, as is for even).
Hermitian sequences of
complex data values
, this routine simultaneously calculates their inverse (backward
) discrete Fourier transforms defined by
The transformed values
(Note the scale factor in the above definition.)
A call of C06PQF with followed by a call with will restore the original data.
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)
. Special coding is provided for the factors
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
- 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: N – INTEGERInput
On entry: , the number of real or complex values in each sequence.
- 3: M – INTEGERInput
On entry: , the number of sequences to be transformed.
- 4: X() – REAL (KIND=nag_wp) arrayInput/Output
: the data must be stored in X
as if in a two-dimensional array of dimension
; each of the
sequences is stored in a column
of the array. In other words, if the data values of the
th sequence to be transformed are denoted by
- if ,
must contain , for and ;
if , and must contain the real and imaginary parts respectively of , for and . (Note that for the sequence to be Hermitian, the imaginary part of , and of for even, must be zero.)
if and X is declared with bounds then and will contain the real and imaginary parts respectively of , for and ;
if and X is declared with bounds then will contain , for and .
- 5: WORK() – REAL (KIND=nag_wp) arrayWorkspace
the dimension of the array WORK
must be at least
The workspace requirements as documented for C06PQF may be an overestimate in some implementations.
contains the minimum workspace required for the current values of M
with this implementation.
- 6: 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 .|
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 by C06PQF is approximately proportional to , but also depends on the factors of . C06PQF is fastest if the only prime factors of are , and , and is particularly slow if is a large prime, or has large prime factors.
This example reads in sequences of real data values and prints their discrete Fourier transforms (as computed by C06PQF with ), after expanding them from complex Hermitian form into a full complex sequences.
Inverse transforms are then calculated by calling C06PQF with showing that the original sequences are restored.
9.1 Program Text
Program Text (c06pqfe.f90)
9.2 Program Data
Program Data (c06pqfe.d)
9.3 Program Results
Program Results (c06pqfe.r)