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NAG Toolbox: nag_sum_fft_real_qtrsine_simple (c06rc)
Purpose
nag_sum_fft_real_qtrsine_simple (c06rc) computes the discrete quarter-wave Fourier sine transforms of sequences of real data values.
Syntax
Description
Given
sequences of
real data values
, for
and
,
nag_sum_fft_real_qtrsine_simple (c06rc) simultaneously calculates the quarter-wave Fourier sine transforms of all the sequences defined by
or its inverse
where
and
.
(Note the scale factor in this definition.)
A call of nag_sum_fft_real_qtrsine_simple (c06rc) with followed by a call with 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)).
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
,
,
and
.
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:
– 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:
or .
- 2:
– int64int32nag_int scalar
-
, the number of sequences to be transformed.
Constraint:
.
- 3:
– int64int32nag_int scalar
-
, the number of real values in each sequence.
Constraint:
.
- 4:
– double array
-
the data must be stored in
x as if in a two-dimensional array of dimension
; each of the
sequences is stored in a
row of the array.
In other words, if the data values of the
th sequence to be transformed are denoted by
, for
and
, then the first
elements of the array
x must contain the values
The
th and
th elements of each row
, for
, are required as workspace. These
elements may contain arbitrary values as they are set to zero by the function.
Optional Input Parameters
None.
Output Parameters
- 1:
– double array
-
the
quarter-wave sine transforms stored as if in a two-dimensional array of dimension
. Each of the
transforms is stored in a
row of the array, overwriting the corresponding original sequence.
If the
components of the
th quarter-wave sine transform are denoted by
, for
and
, then the
elements of the array
x contain the values
- 2:
– int64int32nag_int scalar
unless the function detects an error (see
Error Indicators and Warnings).
Error Indicators and Warnings
Errors or warnings detected by the function:
-
-
-
-
-
-
On entry, | or . |
-
-
An unexpected error has occurred in an internal call. Check all function calls and array dimensions. Seek expert help.
-
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.
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_simple (c06rc) is approximately proportional to , but also depends on the factors of . nag_sum_fft_real_qtrsine_simple (c06rc) is fastest if the only prime factors of are , and , and is particularly slow if 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_simple (c06rc) with . It then calls the function again with and prints the results which may be compared with the original data.
Open in the MATLAB editor:
c06rc_example
function c06rc_example
fprintf('c06rc example results\n\n');
direct = 'Forward';
m = int64(3);
n = int64(6);
x = zeros(m,(n+2));
x(1:m,1:n) = [ 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];
[xt, ifail] = c06rc(direct, m, n, x);
disp('X under discrete quarter-wave sine transform:');
disp(reshape(xt(1:m*n),m,n));
direct = 'Backward';
[xr, ifail] = c06rc(direct, m, n, xt);
disp('X reconstructed by inverse quarter-wave sine transform:');
y = reshape(xr(1:m*n),m,n);
disp(y);
c06rc example results
X under discrete quarter-wave sine transform:
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
X reconstructed by inverse quarter-wave sine 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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