The routine may be called by the names c06rff or nagf_sum_fft_cosine.
Given sequences of real data values , for and , c06rff simultaneously calculates the Fourier cosine transforms of all the sequences defined by
(Note the scale factor in this definition.)
This transform is also known as type-I DCT.
Since the Fourier cosine transform defined above is its own inverse, two consecutive calls of c06rff 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 both left and right boundaries (see Swarztrauber (1977)).
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 , , and .
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: – IntegerInput
On entry: , the number of sequences to be transformed.
2: – IntegerInput
On entry: one less than the number of real values in each sequence, i.e., the number of values in each sequence is .
3: – Real (Kind=nag_wp) arrayInput/Output
On entry: the data values of the th sequence to be transformed, denoted by
, for and , must be stored in .
On exit: the components of the th Fourier cosine transform, denoted by
, for and , are stored in , overwriting the corresponding original values.
4: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, .
On entry, .
An internal error has occurred in this routine.
Check the routine call and any array sizes.
If the call is correct then please contact NAG for assistance.
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
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).
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
c06rff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The time taken by c06rff is approximately proportional to , but also depends on the factors of . c06rff is fastest if the only prime factors of are , and , and is particularly slow if is a large prime, or has large prime factors. Workspace of order is internally allocated by this routine.
This example reads in sequences of real data values and prints their Fourier cosine transforms (as computed by c06rff). It then calls c06rff again and prints the results which may be compared with the original sequence.