NAG CL Interface
c06rfc (fft_​cosine)

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1 Purpose

c06rfc computes the discrete Fourier cosine transforms of m sequences of real data values. The elements of each sequence and its transform are stored contiguously.

2 Specification

#include <nag.h>
void  c06rfc (Integer m, Integer n, double x[], NagError *fail)
The function may be called by the names: c06rfc or nag_sum_fft_cosine.

3 Description

Given m sequences of n+1 real data values xjp , for j=0,1,,n and p=1,2,,m, c06rfc simultaneously calculates the Fourier cosine transforms of all the sequences defined by
x^ k p = 2n (12x0p+ j=1 n-1 xjp×cos(jkπn)+12(-1)kxnp) ,   k= 0, 1, , n ​ and ​ p= 1, 2, , m .  
(Note the scale factor 2n 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 2, 3, 4 and 5.

4 References

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

5 Arguments

1: m Integer Input
On entry: m, the number of sequences to be transformed.
Constraint: m1.
2: n Integer Input
On entry: one less than the number of real values in each sequence, i.e., the number of values in each sequence is n+1.
Constraint: n1.
3: x[(n+1)×m] double Input/Output
On entry: the m data sequences to be transformed. The (n+1) data values of the pth sequence to be transformed, denoted by xjp, for j=0,1,,n and p=1,2,,m, must be stored in x[(p-1)×(n+1)+j].
On exit: the m Fourier cosine transforms, overwriting the corresponding original sequences. The (n+1) components of the pth Fourier cosine transform, denoted by x^kp, for k=0,1,,n and p=1,2,,m, are stored in x[(p-1)×(n+1)+k].
4: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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 value had an illegal value.
On entry, m=value.
Constraint: m1.
On entry, n=value.
Constraint: n1.
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.

7 Accuracy

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

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.

9 Further Comments

The time taken by c06rfc is approximately proportional to nm log(n), but also depends on the factors of n. c06rfc is fastest if the only prime factors of n are 2, 3 and 5, and is particularly slow if n is a large prime, or has large prime factors. This function internally allocates a workspace of order O(n) double values.

10 Example

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.

10.1 Program Text

Program Text (c06rfce.c)

10.2 Program Data

Program Data (c06rfce.d)

10.3 Program Results

Program Results (c06rfce.r)