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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_sum_fft_qtrcosine (c06rh)

## Purpose

nag_sum_fft_qtrcosine (c06rh) computes the discrete quarter-wave Fourier cosine transforms of $m$ sequences of real data values. The elements of each sequence and its transform are stored contiguously.

## Syntax

[x, ifail] = c06rh(idir, n, x, 'm', m)
[x, ifail] = nag_sum_fft_qtrcosine(idir, n, x, 'm', m)

## 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_qtrcosine (c06rh) simultaneously calculates the quarter-wave Fourier cosine transforms of all the sequences defined by
 $x^ k p = 1n 12 x0p + ∑ j=1 n-1 xjp × cos j 2k+1 π2n , if ​ idir=1 ,$
or its inverse
 $xkp = 2n ∑ j=0 n-1 x^ j p × cos 2j+1 k π2n , if ​ idir=-1 ,$
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_qtrcosine (c06rh) with ${\mathbf{idir}}=1$ followed by a call with ${\mathbf{idir}}=-1$ will restore the original data.
The two transforms are also known as type-III DCT and type-II DCT, respectively.
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:     $\mathrm{idir}$int64int32nag_int scalar
Indicates the transform, as defined in Description, to be computed.
${\mathbf{idir}}=1$
Forward transform.
${\mathbf{idir}}=-1$
Inverse transform.
Constraint: ${\mathbf{idir}}=1$ or $-1$.
2:     $\mathrm{n}$int64int32nag_int scalar
$n$, the number of real values in each sequence.
Constraint: ${\mathbf{n}}\ge 1$.
3:     $\mathrm{x}\left(0:{\mathbf{n}}-1,{\mathbf{m}}\right)$ – double array
The data values of the $\mathit{p}$th sequence to be transformed, denoted by ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, must be stored in ${\mathbf{x}}\left(j,p\right)$.

### Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the second dimension of the array x.
$m$, the number of sequences to be transformed.
Constraint: ${\mathbf{m}}\ge 1$.

### Output Parameters

1:     $\mathrm{x}\left(0:{\mathbf{n}}-1,{\mathbf{m}}\right)$ – double array
The $n$ components of the $\mathit{p}$th quarter-wave cosine transform, denoted by ${\stackrel{^}{x}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, are stored in ${\mathbf{x}}\left(k,p\right)$, overwriting the corresponding original 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:
${\mathbf{ifail}}=1$
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=3$
Constraint: ${\mathbf{idir}}=-1$ or $1$.
${\mathbf{ifail}}=4$
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.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
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_qtrcosine (c06rh) is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. nag_sum_fft_qtrcosine (c06rh) 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 is internally allocated by this function. The total amount of memory allocated is $\mathit{O}\left(n\right)$.

## Example

This example reads in sequences of real data values and prints their quarter-wave cosine transforms as computed by nag_sum_fft_qtrcosine (c06rh) with ${\mathbf{idir}}=1$. It then calls the function again with ${\mathbf{idir}}=-1$ and prints the results which may be compared with the original data.
```function c06rh_example

fprintf('c06rh example results\n\n');

% Discrete quarter-wave cosine 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] = c06rh(idir, n, x);
disp('X under discrete quarter-wave cosine transform');
disp(x);

idir = -idir;
[x, ifail] = c06rh(idir, n, x);
disp('X reconstructed by inverse quarter-wave cosine transform');
disp(x);

```
```c06rh example results

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

X reconstructed by inverse quarter-wave cosine 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

```