NAG Library Routine Document
C06PCF calculates the discrete Fourier transform of a sequence of complex data values (using complex data type).
Given a sequence of
complex data values
, C06PCF calculates their (forward
) discrete Fourier transform (DFT) defined by
(Note the scale factor of
in this definition.) The minus sign is taken in the argument of the exponential within the summation when the forward transform is required, and the plus sign is taken when the backward transform is required.
A call of C06PCF with followed by a call with will restore the original data.
C06PCF 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)
is a large prime number or if
contains large prime factors, then the Fourier transform is performed using Bluestein's algorithm (see Bluestein (1968)
), which expresses the DFT as a convolution that in turn can be efficiently computed using FFTs of highly composite sizes.
Bluestein L I (1968) A linear filtering approach to the computation of the discrete Fourier transform Northeast Electronics Research and Engineering Meeting Record 10 218–219
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: X(N) – COMPLEX (KIND=nag_wp) arrayInput/Output
: if X
is declared with bounds
in the subroutine from which C06PCF is called, then
: the components of the discrete Fourier transform. If X
is declared with bounds
in the subroutine from which C06PCF is called, then for
is contained in
- 3: N – INTEGERInput
, the number of data values. The total number of prime factors of N
, counting repetitions, must not exceed
- 4: WORK() – COMPLEX (KIND=nag_wp) arrayWorkspace
the dimension of the array WORK
must be at least
The workspace requirements as documented for C06PCF 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.
- 5: 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,||N has more than prime factors.|
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 . C06PCF is faster if the only prime factors of are , or ; and fastest of all if is a power of . When the Bluestein's FFT algorithm is in use, an addtional workspace of size approximately is allocated internally.
This example reads in a sequence of complex data values and prints their discrete Fourier transform (as computed by C06PCF with ). It then performs an inverse transform using C06PCF with , and prints the sequence so obtained alongside the original data values.
9.1 Program Text
Program Text (c06pcfe.f90)
9.2 Program Data
Program Data (c06pcfe.d)
9.3 Program Results
Program Results (c06pcfe.r)