NAG Library Routine Document
C06HDF computes the discrete quarter-wave Fourier cosine transforms of sequences of 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
, C06HDF simultaneously calculates the quarter-wave Fourier cosine transforms of all the sequences defined by
or its inverse
(Note the scale factor in this definition.)
A call of C06HDF with followed by a call with will restore the original data.
The transform calculated by this routine can be used to solve Poisson's equation when the derivative of the solution is specified at the left boundary, and the solution is specified at the right boundary (see Swarztrauber (1977)
). (See the C06 Chapter Introduction
The routine uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)
) known as the Stockham self-sorting algorithm, described in Temperton (1983)
, together with pre- and post-processing stages described in Swarztrauber (1982)
. 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
Swarztrauber P N (1977) The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle SIAM Rev. 19(3) 490–501
Swarztrauber P N (1982) Vectorizing the FFT's Parallel Computation (ed G Rodrique) 51–83 Academic Press
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: M – INTEGERInput
On entry: , the number of sequences to be transformed.
- 3: N – INTEGERInput
On entry: , the number of real values in each sequence.
- 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 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
quarter-wave cosine transforms stored as if in a two-dimensional array of dimension
. Each of the
transforms is stored in a row
of the array, overwriting the corresponding original sequence.
components of the
th quarter-wave cosine transform are denoted by
, then the
elements of the array X
contain the values
- 5: 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 C06HAF, C06HBF, C06HCF or C06HDF. The routine performs a simple check that the current value of is consistent with the values stored in TRIG.
, or .
- 6: TRIG() – REAL (KIND=nag_wp) arrayInput/Output
must contain the required trigonometric coefficients calculated in a previous call of the routine. Otherwise TRIG
need not be set.
On exit: contains the required coefficients (computed by the routine if ).
- 7: WORK() – REAL (KIND=nag_wp) arrayWorkspace
- 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 .|
Not used at this Mark.
|On entry,|| or , but the array TRIG and the current value of N are inconsistent.|
|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 C06HDF is approximately proportional to , but also depends on the factors of . C06HDF 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 quarter-wave cosine transforms as computed by C06HDF with . It then calls the routine again with and prints the results which may be compared with the original data.
9.1 Program Text
Program Text (c06hdfe.f90)
9.2 Program Data
Program Data (c06hdfe.d)
9.3 Program Results
Program Results (c06hdfe.r)