The function may be called by the names: c06rhc or nag_sum_fft_qtrcosine.
Given sequences of real data values , for and , c06rhc simultaneously calculates the quarter-wave Fourier cosine transforms of all the sequences defined by
or its inverse
where and .
(Note the scale factor in this definition.)
A call of c06rhc with followed by a call with will restore the original data.
The two transforms are also known as type-III DCT and type-II DCT, respectively.
The transform calculated by this function 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)).
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: – Nag_TransformDirectionInput
On entry: indicates the transform, as defined in Section 3, to be computed.
2: – IntegerInput
On entry: , the number of sequences to be transformed.
3: – IntegerInput
On entry: , the number of real values in each sequence.
4: – 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 quarter-wave cosine transforms, overwriting the corresponding original sequences. The components of the th quarter-wave cosine transform, denoted by
, for and , are stored in .
5: – 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
c06rhc 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 c06rhc is approximately proportional to , but also depends on the factors of . c06rhc 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 quarter-wave cosine transforms as computed by c06rhc with . It then calls the function again with and prints the results which may be compared with the original data.