The function may be called by the names: c06rfc or nag_sum_fft_cosine.
Given sequences of real data values , for and , c06rfc 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 c06rfc will restore the original data.
The transform calculated by this function 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 function 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: – doubleInput/Output
On entry: the data sequences to be transformed. The data values of the th sequence to be transformed, denoted by
, for and , must be stored in .
On exit: the Fourier cosine transforms, overwriting the corresponding original sequences. The components of the th Fourier cosine transform, denoted by
, for and , are stored in .
4: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, .
On entry, .
An internal error has occurred in this function.
Check the function call and any array sizes.
If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL 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 CL 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
c06rfc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The time taken by c06rfc is approximately proportional to , but also depends on the factors of . c06rfc 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 function internally allocates a workspace of order double values.
This example reads in sequences of real data values and prints their Fourier cosine transforms (as computed by c06rfc). It then calls c06rfc again and prints the results which may be compared with the original sequence.