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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_sum_fft_qtrcosine (c06rh)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_sum_fft_qtrcosine (c06rh) 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.


[x, ifail] = c06rh(idir, n, x, 'm', m)
[x, ifail] = nag_sum_fft_qtrcosine(idir, n, x, 'm', m)


Given m sequences of n real data values xjp , for j=0,1,,n-1 and p=1,2,,m, nag_sum_fft_qtrcosine (c06rh) 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 ​ idir=1 ,  
or its inverse
xkp = 2n j=0 n-1 x^ j p × cos 2j+1 k π2n ,   if ​ idir=-1 ,  
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 (c06rh) with idir=1 followed by a call with idir=-1 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


Compulsory Input Parameters

1:     idir int64int32nag_int scalar
Indicates the transform, as defined in Description, to be computed.
Forward transform.
Inverse transform.
Constraint: idir=1 or -1.
2:     n int64int32nag_int scalar
n, the number of real values in each sequence.
Constraint: n1.
3:     x0:n-1m – double array
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 xjp.

Optional Input Parameters

1:     m int64int32nag_int scalar
Default: the second dimension of the array x.
m, the number of sequences to be transformed.
Constraint: m1.

Output Parameters

1:     x0:n-1m – double array
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 xkp, overwriting the corresponding original values.
2:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
Constraint: m1.
Constraint: n1.
Constraint: idir=-1 or 1.
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 unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


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).

Further Comments

The time taken by nag_sum_fft_qtrcosine (c06rh) is approximately proportional to nm logn, but also depends on the factors of n. nag_sum_fft_qtrcosine (c06rh) 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 is internally allocated by this function. The total amount of memory allocated is On.


This example reads in sequences of real data values and prints their quarter-wave cosine transforms as computed by nag_sum_fft_qtrcosine (c06rh) with idir=1. It then calls the function again with idir=-1 and prints the results which may be compared with the original data.
function c06rh_example

fprintf('c06rh example results\n\n');

% Discrete quarter-wave cosine transform of 3 sequences of length 6
m = int64(3);
n = int64(6);
x = [ 0.3854   0.5417   0.9172; 
      0.6772   0.2983   0.0644; 
      0.1138   0.1181   0.6037; 
      0.6751   0.7255   0.6430; 
      0.6362   0.8638   0.0428; 
      0.1424   0.8723   0.4815];

idir = int64(1);
[x, ifail] = c06rh(idir, n, x);
disp('X under discrete quarter-wave cosine transform');

idir = -idir;
[x, ifail] = c06rh(idir, n, x);
disp('X reconstructed by inverse quarter-wave cosine transform');

c06rh example results

X under discrete quarter-wave cosine transform
    0.7257    0.7479    0.6713
   -0.2216   -0.6172   -0.1363
    0.1011    0.4112   -0.0064
    0.2355    0.0791   -0.0285
   -0.1406    0.1331    0.4758
   -0.2282   -0.0906    0.1475

X reconstructed by inverse quarter-wave cosine transform
    0.3854    0.5417    0.9172
    0.6772    0.2983    0.0644
    0.1138    0.1181    0.6037
    0.6751    0.7255    0.6430
    0.6362    0.8638    0.0428
    0.1424    0.8723    0.4815

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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