NAG Library Function Document
nag_sum_fft_qtrcosine (c06rhc) computes the discrete quarter-wave Fourier cosine transforms of sequences of real data values. The elements of each sequence and its transform are stored contiguously.
||nag_sum_fft_qtrcosine (Nag_TransformDirection direct,
real data values
, nag_sum_fft_qtrcosine (c06rhc) 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 nag_sum_fft_qtrcosine (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
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
direct – Nag_TransformDirectionInput
: indicates the transform, as defined in Section 3
, to be computed.
- Forward transform.
- Inverse transform.
m – IntegerInput
On entry: , the number of sequences to be transformed.
n – IntegerInput
On entry: , the number of real values in each sequence.
x – 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 .
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
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
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).
8 Parallelism and Performance
nag_sum_fft_qtrcosine (c06rhc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the Users' Note
for your implementation for any additional implementation-specific information.
The time taken by nag_sum_fft_qtrcosine (c06rhc) is approximately proportional to , but also depends on the factors of . nag_sum_fft_qtrcosine (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 nag_sum_fft_qtrcosine (c06rhc) with . It then calls the function again with and prints the results which may be compared with the original data.
10.1 Program Text
Program Text (c06rhce.c)
10.2 Program Data
Program Data (c06rhce.d)
10.3 Program Results
Program Results (c06rhce.r)