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NAG Toolbox: nag_sum_withdraw_fft_real_qtrsine (c06hc)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_sum_fft_real_qtrsine (c06hc) computes the discrete quarter-wave Fourier sine transforms of m sequences of real data values. This function is designed to be particularly efficient on vector processors.
Note: this function is scheduled to be withdrawn, please see c06hc in Advice on Replacement Calls for Withdrawn/Superseded Routines..

Syntax

[x, trig, ifail] = c06hc(direct, m, n, x, init, trig)
[x, trig, ifail] = nag_sum_withdraw_fft_real_qtrsine(direct, m, n, x, init, trig)

Description

Given m sequences of n real data values xjp , for j=1,2,,n and p=1,2,,m, nag_sum_fft_real_qtrsine (c06hc) simultaneously calculates the quarter-wave Fourier sine transforms of all the sequences defined by
x^ k p = 1n j=1 n-1 xjp × sinj2k-1π2n + 12 -1 k-1 xnp ,   if ​ direct='F' ,  
or its inverse
xkp = 2n j= 1 n x^jp × sin 2j- 1 k π2n ,   if ​ direct='B' ,  
for k= 1,2,,n  and p= 1,2,,m .
(Note the scale factor 1n  in this definition.)
A call of nag_sum_fft_real_qtrsine (c06hc) with direct='F' followed by a call with direct='B' will restore the original data.
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)). (See the C06 Chapter Introduction.)
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, 5 and 6. This function is designed to be particularly efficient on vector processors, and it becomes especially fast as m, the number of transforms to be computed in parallel, increases.

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:     direct – string (length ≥ 1)
If the forward transform as defined in Description is to be computed, then direct must be set equal to 'F'.
If the backward transform is to be computed then direct must be set equal to 'B'.
Constraint: direct='F' or 'B'.
2:     m int64int32nag_int scalar
m, the number of sequences to be transformed.
Constraint: m1.
3:     n int64int32nag_int scalar
n, the number of real values in each sequence.
Constraint: n1.
4:     x m×n – double array
The data must be stored in x as if in a two-dimensional array of dimension 1:m,1:n; each of the m sequences is stored in a row of the array. In other words, if the data values of the pth sequence to be transformed are denoted by xjp, for j=1,2,,n and p=1,2,,m, then the mn elements of the array x must contain the values
x11 , x12 ,, x1m , x21 , x22 ,, x2m ,, xn1 , xn2 ,, xnm .  
5:     init – string (length ≥ 1)
Indicates whether trigonometric coefficients are to be calculated.
init='I'
Calculate the required trigonometric coefficients for the given value of n, and store in the array trig.
init='S' or 'R'
The required trigonometric coefficients are assumed to have been calculated and stored in the array trig in a prior call to one of nag_sum_fft_real_sine (c06ha), nag_sum_fft_real_cosine (c06hb), nag_sum_fft_real_qtrsine (c06hc) or nag_sum_fft_real_qtrcosine (c06hd). The function performs a simple check that the current value of n is consistent with the values stored in trig.
Constraint: init='I', 'S' or 'R'.
6:     trig 2×n – double array
If init='S' or 'R', trig must contain the required trigonometric coefficients calculated in a previous call of the function. Otherwise trig need not be set.

Optional Input Parameters

None.

Output Parameters

1:     x m×n – double array
The m quarter-wave sine transforms stored as if in a two-dimensional array of dimension 1:m,1:n. Each of the m transforms is stored in a row of the array, overwriting the corresponding original sequence. If the n components of the pth quarter-wave sine transform are denoted by x^kp, for k=1,2,,n and p=1,2,,m, then the mn elements of the array x contain the values
x^11 , x^12 ,, x^1m , x^21 , x^22 ,, x^2m ,, x^n1 , x^n2 ,, x^nm .  
2:     trig 2×n – double array
Contains the required coefficients (computed by the function if init='I').
3:     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
On entry,m<1.
   ifail=2
On entry,n<1.
   ifail=3
On entry,init'I', 'S' or 'R'.
   ifail=4
Not used at this Mark.
   ifail=5
On entry,init='S' or 'R', but the array trig and the current value of n are inconsistent.
   ifail=6
On entry,direct'F' or 'B'.
   ifail=7
An unexpected error has occurred in an internal call. Check all function calls and array dimensions. Seek expert help.
   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_real_qtrsine (c06hc) is approximately proportional to nm logn, but also depends on the factors of n. nag_sum_fft_real_qtrsine (c06hc) 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.

Example

This example reads in sequences of real data values and prints their quarter-wave sine transforms as computed by nag_sum_fft_real_qtrsine (c06hc) with direct='F'. It then calls the function again with direct='B' and prints the results which may be compared with the original data.
function c06hc_example


fprintf('c06hc example results\n\n');

m = int64(3);
n = int64(6);
x = [0.3854  0.6772  0.1138  0.6751  0.6362  0.1424;
     0.5417  0.2983  0.1181  0.7255  0.8638  0.8723;
     0.9172  0.0644  0.6037  0.6430  0.0428  0.4815];

direct = 'Forward';
init = 'Initial';
trig = zeros(2*n,1);
[xt, trig, ifail] = c06hc(direct, m, n, x, init, trig);

disp('Original data values:');
disp(x);

disp('Discrete quarter-wave Fourier sine transforms:');
disp(xt);

% Restore data by inverse transform
direct = 'Backward';
init = 'Subsequent';
[xr, trig, ifail] = c06hc(direct, m, n, xt, init, trig);

disp('Original data as restored by inverse transform:');
disp(xr);


c06hc example results

Original data values:
    0.3854    0.6772    0.1138    0.6751    0.6362    0.1424
    0.5417    0.2983    0.1181    0.7255    0.8638    0.8723
    0.9172    0.0644    0.6037    0.6430    0.0428    0.4815

Discrete quarter-wave Fourier sine transforms:
    0.7304    0.2078    0.1150    0.2577   -0.2869   -0.0815
    0.9274   -0.1152    0.2532    0.2883   -0.0026   -0.0635
    0.6268    0.3547    0.0760    0.3078    0.4987   -0.0507

Original data as restored by inverse transform:
    0.3854    0.6772    0.1138    0.6751    0.6362    0.1424
    0.5417    0.2983    0.1181    0.7255    0.8638    0.8723
    0.9172    0.0644    0.6037    0.6430    0.0428    0.4815


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Chapter Introduction
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