NAG Library Function Document
nag_sum_fft_cosine (c06rfc) computes the discrete Fourier cosine transforms of sequences of real data values. The elements of each sequence and its transform are stored contiguously.
||nag_sum_fft_cosine (Integer m,
real data values
, nag_sum_fft_cosine (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 nag_sum_fft_cosine (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
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
m – IntegerInput
On entry: , the number of sequences to be transformed.
n – IntegerInput
On entry: one less than the number of real values in each sequence, i.e., the number of values in each sequence is .
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 Fourier cosine transforms, overwriting the corresponding original sequences. The components of the th Fourier 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_cosine (c06rfc) 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_cosine (c06rfc) is approximately proportional to , but also depends on the factors of . nag_sum_fft_cosine (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 nag_sum_fft_cosine (c06rfc)). It then calls nag_sum_fft_cosine (c06rfc) again and prints the results which may be compared with the original sequence.
10.1 Program Text
Program Text (c06rfce.c)
10.2 Program Data
Program Data (c06rfce.d)
10.3 Program Results
Program Results (c06rfce.r)