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NAG Toolbox: nag_sum_fft_qtrsine (c06rg)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_sum_fft_qtrsine (c06rg) computes the discrete quarter-wave Fourier sine transforms of m sequences of real data values. The elements of each sequence and its transform are stored contiguously.

Syntax

[x, ifail] = c06rg(idir, x, 'm', m, 'n', n)
[x, ifail] = nag_sum_fft_qtrsine(idir, x, 'm', m, 'n', n)

Description

Given m sequences of n real data values xjp , for j=1,2,,n and p=1,2,,m, nag_sum_fft_qtrsine (c06rg) simultaneously calculates the quarter-wave Fourier sine transforms of all the sequences defined by
x^ k p = 1n j=1 n-1 xjp × sin j 2k-1 π2n + 12 -1 k-1 xnp ,   if ​ idir=1 ,  
or its inverse
xkp = 2n j=1 n x^ j p × sin 2j-1 k π2n ,   if ​ idir=-1 ,  
where k=1,2,,n and p=1,2,,m.
(Note the scale factor 1n  in this definition.)
A call of nag_sum_fft_qtrsine (c06rg) with idir=1 followed by a call with idir=-1 will restore the original data.
The two transforms are also known as type-III DST and type-II DST, respectively.
The transform calculated by this function can be used to solve Poisson's equation when the solution is specified at the left boundary, and the derivative of 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.

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

Parameters

Compulsory Input Parameters

1:     idir int64int32nag_int scalar
Indicates the transform, as defined in Description, to be computed.
idir=1
Forward transform.
idir=-1
Inverse transform.
Constraint: idir=1 or -1.
2:     xnm – double array
The data values of the pth sequence to be transformed, denoted by xjp, for j=1,2,,n 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.
2:     n int64int32nag_int scalar
Default: the first dimension of the array x.
n, the number of real values in each sequence.
Constraint: n1.

Output Parameters

1:     xnm – double array
The n components of the pth quarter-wave sine transform, denoted by x^kp, for k=1,2,,n 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:
   ifail=1
Constraint: m1.
   ifail=2
Constraint: n1.
   ifail=3
Constraint: idir=-1 or 1.
   ifail=4
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.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

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).

Further Comments

The time taken by nag_sum_fft_qtrsine (c06rg) is approximately proportional to nm logn, but also depends on the factors of n. nag_sum_fft_qtrsine (c06rg) 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 On is internally allocated by this function.

Example

This example reads in sequences of real data values and prints their quarter-wave sine transforms as computed by nag_sum_fft_qtrsine (c06rg) 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 c06rg_example


fprintf('c06rg example results\n\n');

% Discrete quarter-wave sine 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] = c06rg(idir,x);
disp('X under discrete quarter-wave sine transform:');
disp(x);

% Reconstruct using same transform
idir = -idir;
[x, ifail] = c06rg(idir,x);
disp('X reconstructed by inverse quarter-wave sine transform:');
disp(x);


c06rg example results

X under discrete quarter-wave sine transform:
    0.7304    0.9274    0.6268
    0.2078   -0.1152    0.3547
    0.1150    0.2532    0.0760
    0.2577    0.2883    0.3078
   -0.2869   -0.0026    0.4987
   -0.0815   -0.0635   -0.0507

X reconstructed by inverse quarter-wave sine 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


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