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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_sum_fft_real_cosine_simple (c06rb)

Purpose

nag_sum_fft_real_cosine_simple (c06rb) computes the discrete Fourier cosine transforms of $m$ sequences of real data values.

Syntax

[x, ifail] = c06rb(m, n, x)
[x, ifail] = nag_sum_fft_real_cosine_simple(m, n, x)

Description

Given $m$ sequences of $n+1$ real data values ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, nag_sum_fft_real_cosine_simple (c06rb) simultaneously calculates the Fourier cosine transforms of all the sequences defined by
 $x^ k p = 2n 12 x0p + ∑ j=1 n-1 xjp × cos jk πn + 12 -1k xnp , k= 0, 1, …, n ​ and ​ p= 1, 2, …, m .$
(Note the scale factor $\sqrt{\frac{2}{n}}$ in this definition.)
Since the Fourier cosine transform is its own inverse, two consecutive calls of nag_sum_fft_real_cosine_simple (c06rb) will restore the original data.
The transform calculated by this function can be used to solve Poisson's equation when the derivative of the solution is specified at both left and right boundaries (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:     $\mathrm{m}$int64int32nag_int scalar
$m$, the number of sequences to be transformed.
Constraint: ${\mathbf{m}}\ge 1$.
2:     $\mathrm{n}$int64int32nag_int scalar
One less than the number of real values in each sequence, i.e., the number of values in each sequence is $n+1$.
Constraint: ${\mathbf{n}}\ge 1$.
3:     $\mathrm{x}\left({\mathbf{m}}×\left({\mathbf{n}}+3\right)\right)$ – double array
the data must be stored in x as if in a two-dimensional array of dimension $\left(1:{\mathbf{m}},0:{\mathbf{n}}+2\right)$; each of the $m$ sequences is stored in a row of the array. In other words, if the $\left(n+1\right)$ data values of the $\mathit{p}$th sequence to be transformed are denoted by ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, then the first $m\left(n+1\right)$ elements of the array x must contain the values
 $x01 , x02 ,…, x0m , x11 , x12 ,…, x1m ,…, xn1 , xn2 ,…, xnm .$
The $\left(n+2\right)$th and $\left(n+3\right)$th elements of each row ${x}_{n+2}^{\mathit{p}},{x}_{n+3}^{\mathit{p}}$, for $\mathit{p}=1,2,\dots ,m$, are required as workspace. These $2m$ elements may contain arbitrary values as they are set to zero by the function.

None.

Output Parameters

1:     $\mathrm{x}\left({\mathbf{m}}×\left({\mathbf{n}}+3\right)\right)$ – double array
the $m$ Fourier cosine transforms stored as if in a two-dimensional array of dimension $\left(1:{\mathbf{m}},0:{\mathbf{n}}+2\right)$. Each of the $m$ transforms is stored in a row of the array, overwriting the corresponding original data. If the $\left(n+1\right)$ components of the $\mathit{p}$th Fourier cosine transform are denoted by ${\stackrel{^}{x}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=0,1,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, then the $m\left(n+3\right)$ elements of the array x contain the values
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{m}}<1$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{n}}<1$.
${\mathbf{ifail}}=3$
An unexpected error has occurred in an internal call. Check all function calls and array dimensions. Seek expert help.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
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).

The time taken by nag_sum_fft_real_cosine_simple (c06rb) is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. nag_sum_fft_real_cosine_simple (c06rb) 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 Fourier cosine transforms (as computed by nag_sum_fft_real_cosine_simple (c06rb)). It then calls the function again and prints the results which may be compared with the original sequence.
```function c06rb_example

fprintf('c06rb example results\n\n');

% Discrete cosine transform of 3 sequences of length 7
m = int64(3);
n = int64(6);
x = zeros(m,n+3);
x(1:m,1:(n+1)) = [0.3854 0.6772 0.1138 0.6751 0.6362 0.1424 0.9562;
0.5417 0.2983 0.1181 0.7255 0.8638 0.8723 0.4936;
0.9172 0.0644 0.6037 0.6430 0.0428 0.4815 0.2057];

xf = reshape(x,[m*(n+3),1]);
[xt, ifail] = c06rb(m, n, xf);
disp('X under discrete cosine transform:');
disp(reshape(xt(1:m*(n+1)),m,n+1));

% Reconstruct using same transform
[xr, ifail] = c06rb(m,n,xt);
disp('X reconstructed under second cosine transform:');
xr = reshape(xr(1:m*(n+1)),m,n+1);
disp(xr);

```
```c06rb example results

X under discrete cosine transform:
1.6833   -0.0482    0.0176    0.1368    0.3240   -0.5830   -0.0427
1.9605   -0.4884   -0.0655    0.4444    0.0964    0.0856   -0.2289
1.3838    0.1588   -0.0761   -0.1184    0.3512    0.5759    0.0110

X reconstructed under second cosine transform:
0.3854    0.6772    0.1138    0.6751    0.6362    0.1424    0.9562
0.5417    0.2983    0.1181    0.7255    0.8638    0.8723    0.4936
0.9172    0.0644    0.6037    0.6430    0.0428    0.4815    0.2057

```