NAG Library Routine Document
C06FPF computes the discrete Fourier transforms of sequences, each containing real data values. This routine is designed to be particularly efficient on vector processors.
||M, N, IFAIL
||X(M*N), TRIG(2*N), WORK(M*N)
real data values
, C06FPF simultaneously calculates the Fourier transforms of all the sequences defined by
(Note the scale factor
in this definition.)
The transformed values
are complex, but for each value of
form a Hermitian sequence (i.e.,
is the complex conjugate of
), so they are completely determined by
real numbers (see also the C06 Chapter Introduction
The discrete Fourier transform is sometimes defined using a positive sign in the exponential term:
To compute this form, this routine should be followed by forming the complex conjugates of the
; that is
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
. This routine is designed to be particularly efficient on vector processors, and it becomes especially fast as
, the number of transforms to be computed in parallel, increases.
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: M – INTEGERInput
On entry: , the number of sequences to be transformed.
- 2: N – INTEGERInput
On entry: , the number of real values in each sequence.
- 3: 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 row
of the array. In other words, if the data values of the
th sequence to be transformed are denoted by
, then the
elements of the array X
must contain the values
discrete Fourier transforms stored as if in a two-dimensional array of dimension
. Each of the
transforms is stored in a row
of the array in Hermitian form, overwriting the corresponding original sequence. If the
components of the discrete Fourier transform
are written as
, then for
is contained in
, and for
is contained in
. (See also Section 2.1.2
in the C06 Chapter Introduction.)
- 4: INIT – CHARACTER(1)Input
: indicates whether trigonometric coefficients are to be calculated.
- Calculate the required trigonometric coefficients for the given value of , and store in the array TRIG.
- The required trigonometric coefficients are assumed to have been calculated and stored in the array TRIG in a prior call to one of C06FPF, C06FQF or C06FRF. The routine performs a simple check that the current value of is consistent with the values stored in TRIG.
, or .
- 5: TRIG() – REAL (KIND=nag_wp) arrayInput/Output
must contain the required trigonometric coefficients that have been previously calculated. Otherwise TRIG
need not be set.
On exit: contains the required coefficients (computed by the routine if ).
- 6: WORK() – REAL (KIND=nag_wp) arrayWorkspace
- 7: 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 .|
Not used at this Mark.
|On entry,|| or , but the array TRIG and the current value of N are inconsistent.|
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 C06FPF is approximately proportional to , but also depends on the factors of . C06FPF 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 C06FPF). The Fourier transforms are expanded into full complex form using and printed. Inverse transforms are then calculated by conjugating and calling C06FQF
showing that the original sequences are restored.
9.1 Program Text
Program Text (c06fpfe.f90)
9.2 Program Data
Program Data (c06fpfe.d)
9.3 Program Results
Program Results (c06fpfe.r)