NAG FL Interface
c06rff (fft_​cosine)

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1 Purpose

c06rff 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

Fortran Interface
Subroutine c06rff ( m, n, x, ifail)
Integer, Intent (In) :: m, n
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (Inout) :: x(0:n,m)
C Header Interface
#include <nag.h>
void  c06rff_ (const Integer *m, const Integer *n, double x[], Integer *ifail)
The routine may be called by the names c06rff or nagf_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, c06rff 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 c06rff will restore the original data.
The transform calculated by this routine 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 routine 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(0:n,m) Real (Kind=nag_wp) array Input/Output
On entry: the 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(j,p).
On exit: 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(k,p), overwriting the corresponding original values.
4: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, m=value.
Constraint: m1.
On entry, n=value.
Constraint: n1.
An internal error has occurred in this routine. Check the routine call and any array sizes. If the call is correct then please contact NAG for assistance.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL 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 FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL 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

Background information to multithreading can be found in the Multithreading documentation.
c06rff 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken by c06rff is approximately proportional to nm log(n), but also depends on the factors of n. c06rff 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. Workspace of order O(n) is internally allocated by this routine.

10 Example

This example reads in sequences of real data values and prints their Fourier cosine transforms (as computed by c06rff). It then calls c06rff again and prints the results which may be compared with the original sequence.

10.1 Program Text

Program Text (c06rffe.f90)

10.2 Program Data

Program Data (c06rffe.d)

10.3 Program Results

Program Results (c06rffe.r)