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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_sum_fft_real_qtrcosine_simple (c06rd)

## Purpose

nag_sum_fft_real_qtrcosine_simple (c06rd) computes the discrete quarter-wave Fourier cosine transforms of $m$ sequences of real data values.

## Syntax

[x, ifail] = c06rd(direct, m, n, x)
[x, ifail] = nag_sum_fft_real_qtrcosine_simple(direct, m, n, x)

## Description

Given $m$ sequences of $n$ real data values ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, nag_sum_fft_real_qtrcosine_simple (c06rd) simultaneously calculates the quarter-wave Fourier cosine transforms of all the sequences defined by
 $x^kp = 1n 12 x0p + ∑ j=1 n-1 xjp × cos j 2k-1 π2n , if ​ direct='F' ,$
or its inverse
 $xkp = 2n ∑ j= 0 n- 1 x^ j p × cos 2j- 1 k π2n , if ​ direct='B' ,$
where $k=0,1,\dots ,n-1$ and $p=1,2,\dots ,m$.
(Note the scale factor $\frac{1}{\sqrt{n}}$ in this definition.)
A call of nag_sum_fft_real_qtrcosine_simple (c06rd) with ${\mathbf{direct}}=\text{'F'}$ followed by a call with ${\mathbf{direct}}=\text{'B'}$ 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 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 $5$.

## 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:     $\mathrm{direct}$ – 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: ${\mathbf{direct}}=\text{'F'}$ or $\text{'B'}$.
2:     $\mathrm{m}$int64int32nag_int scalar
$m$, the number of sequences to be transformed.
Constraint: ${\mathbf{m}}\ge 1$.
3:     $\mathrm{n}$int64int32nag_int scalar
$n$, the number of real values in each sequence.
Constraint: ${\mathbf{n}}\ge 1$.
4:     $\mathrm{x}\left({\mathbf{m}}×\left({\mathbf{n}}+2\right)\right)$ – double array
the data must be stored in x as if in a two-dimensional array of dimension $\left(1:{\mathbf{m}},0:{\mathbf{n}}+1\right)$; each of the $m$ sequences is stored in a row of the array. In other words, if the data values of the $\mathit{p}$th sequence to be transformed are denoted by ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, then the first $mn$ elements of the array x must contain the values
 $x01 , x02 ,…, x0m , x11 , x12 ,…, x1m ,…, x n-1 1 , x n-1 2 ,…, x n-1 m .$
The $\left(n+1\right)$th and $\left(n+2\right)$th elements of each row ${x}_{n}^{\mathit{p}},{x}_{n+1}^{\mathit{p}}$, 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:     $\mathrm{x}\left({\mathbf{m}}×\left({\mathbf{n}}+2\right)\right)$ – double array
the $m$ quarter-wave cosine transforms stored as if in a two-dimensional array of dimension $\left(1:{\mathbf{m}},0:{\mathbf{n}}+1\right)$. Each of the $m$ transforms is stored in a row of the array, overwriting the corresponding original sequence. If the $n$ components of the $\mathit{p}$th quarter-wave cosine transform are denoted by ${\stackrel{^}{x}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, then the $m\left(n+2\right)$ 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, ${\mathbf{n}}<1$.
${\mathbf{ifail}}=3$
 On entry, ${\mathbf{direct}}\ne \text{'F'}$ or $\text{'B'}$.
${\mathbf{ifail}}=4$
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_qtrcosine_simple (c06rd) is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. nag_sum_fft_real_qtrcosine_simple (c06rd) 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 cosine transforms as computed by nag_sum_fft_real_qtrcosine_simple (c06rd) with ${\mathbf{direct}}=\text{'F'}$. It then calls the function again with ${\mathbf{direct}}=\text{'B'}$ and prints the results which may be compared with the original data.
```function c06rd_example

fprintf('c06rd example results\n\n');

% Discrete quarter-wave cosine transform of 3 sequences of length 6
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];

direct = 'Forward';
[xt, ifail] = c06rd(direct, m, n, x);
disp('X under discrete quarter-wave cosine transform');
disp(reshape(xt(1:m*n),m,n));

direct = 'Backward';
[xr, ifail] = c06rd(direct, m, n, xt);
disp('X reconstructed by inverse quarter-wave cosine transform');
y = reshape(xr(1:m*n),m,n);
disp(y);

```
```c06rd example results

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

X reconstructed by inverse quarter-wave cosine 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

```