NAG Library Function Document

nag_sum_fft_qtrcosine (c06rhc)


    1  Purpose
    7  Accuracy


nag_sum_fft_qtrcosine (c06rhc) computes the discrete quarter-wave Fourier cosine transforms of m sequences of real data values. The elements of each sequence and its transform are stored contiguously.


#include <nag.h>
#include <nagc06.h>
void  nag_sum_fft_qtrcosine (Nag_TransformDirection direct, Integer m, Integer n, double x[], NagError *fail)


Given m sequences of n real data values xjp , for j=0,1,,n-1 and p=1,2,,m, nag_sum_fft_qtrcosine (c06rhc) simultaneously calculates the quarter-wave Fourier cosine transforms of all the sequences defined by
x^ k p = 1n 12 x0p + j=1 n-1 xjp × cos j 2k+1 π2n ,   if ​ direct=Nag_ForwardTransform ,  
or its inverse
xkp = 2n j=0 n-1 x^ j p × cos 2j+1 k π2n ,   if ​ direct=Nag_BackwardTransform ,  
where k=0,1,,n-1 and p=1,2,,m.
(Note the scale factor 1n  in this definition.)
A call of nag_sum_fft_qtrcosine (c06rhc) with direct=Nag_ForwardTransform followed by a call with direct=Nag_BackwardTransform 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 2, 3, 4 and 5.


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:     direct Nag_TransformDirectionInput
On entry: indicates the transform, as defined in Section 3, to be computed.
Forward transform.
Inverse transform.
Constraint: direct=Nag_ForwardTransform or Nag_BackwardTransform.
2:     m IntegerInput
On entry: m, the number of sequences to be transformed.
Constraint: m1.
3:     n IntegerInput
On entry: n, the number of real values in each sequence.
Constraint: n1.
4:     x[n×m] doubleInput/Output
On entry: the m data sequences to be transformed. The data values of the pth sequence to be transformed, denoted by xjp, for j=0,1,,n-1 and p=1,2,,m, must be stored in x[p-1×n+j].
On exit: the m quarter-wave cosine transforms, overwriting the corresponding original sequences. The n components of the pth quarter-wave cosine transform, denoted by x^kp, for k=0,1,,n-1 and p=1,2,,m, are stored in x[p-1×n+k].
5:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation 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.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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).

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 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.

Further Comments

The time taken by nag_sum_fft_qtrcosine (c06rhc) is approximately proportional to nm logn, but also depends on the factors of n. nag_sum_fft_qtrcosine (c06rhc) 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 On 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 direct=Nag_ForwardTransform. It then calls the function again with direct=Nag_BackwardTransform and prints the results which may be compared with the original data.

Program Text

Program Text (c06rhce.c)

Program Data

Program Data (c06rhce.d)

Program Results

Program Results (c06rhce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017