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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_sum_fft_sine (c06re)

## Purpose

nag_sum_fft_sine (c06re) computes the discrete Fourier sine transforms of $m$ sequences of real data values. The elements of each sequence and its transform are stored contiguously.

## Syntax

[x, ifail] = c06re(n, x, 'm', m)
[x, ifail] = nag_sum_fft_sine(n, x, 'm', m)

## Description

Given $m$ sequences of $n-1$ real data values ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=1,2,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, nag_sum_fft_sine (c06re) simultaneously calculates the Fourier sine transforms of all the sequences defined by
 $x^ k p = 2n ∑ j=1 n-1 xjp × sin jk πn , k= 1, 2, …, n-1 ​ and ​ p= 1, 2, …, m .$
(Note the scale factor $\sqrt{\frac{2}{n}}$ in this definition.)
This transform is also known as type-I DST.
Since the Fourier sine transform defined above is its own inverse, two consecutive calls of nag_sum_fft_sine (c06re) 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 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:     $\mathrm{n}$int64int32nag_int scalar
One more than the number of real values in each sequence, i.e., the number of values in each sequence is $n-1$.
Constraint: ${\mathbf{n}}\ge 1$.
2:     $\mathrm{x}\left({\mathbf{n}}-1,{\mathbf{m}}\right)$ – double array
The data values of the $\mathit{p}$th sequence to be transformed, denoted by ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=1,2,\dots ,n-1$ 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: $1$
$m$, the number of sequences to be transformed.
Constraint: ${\mathbf{m}}\ge 1$.

### Output Parameters

1:     $\mathrm{x}\left({\mathbf{n}}-1,{\mathbf{m}}\right)$ – double array
The $\left(n-1\right)$ components of the $\mathit{p}$th Fourier sine transform, denoted by ${\stackrel{^}{x}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=1,2,\dots ,n-1$ 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: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=3$
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_sine (c06re) is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. nag_sum_fft_sine (c06re) 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 of order $\mathit{O}\left(n\right)$ is internally allocated by this function.

## Example

This example reads in sequences of real data values and prints their Fourier sine transforms (as computed by nag_sum_fft_sine (c06re)). It then calls nag_sum_fft_sine (c06re) again and prints the results which may be compared with the original sequence.
```function c06re_example

fprintf('c06re example results\n\n');

% Discrete sine transform of 3 sequences of length 5
m = int64(3);
n = int64(6);
x = [0.6772  0.6037 0.8638;
0.2983  0.6751 0.0428;
0.0644  0.7255 0.1424;
0.1138  0.6430 0.8723;
0.1181  0.6362 0.4815];

[x, ifail] = c06re(n,x);
disp('X under discrete sine transform:');
disp(x);

% Reconstruct using same transform
[x, ifail] = c06re(n,x);
disp('X reconstructed under second sine transform:');
disp(x);

```
```c06re example results

X under discrete sine transform:
0.4728    1.4358    0.9281
0.3718   -0.0002   -0.2236
0.4220    0.2970    0.6945
0.1873   -0.0323    0.6059
0.0607    0.1177    0.0130

X reconstructed under second sine transform:
0.6772    0.6037    0.8638
0.2983    0.6751    0.0428
0.0644    0.7255    0.1424
0.1138    0.6430    0.8723
0.1181    0.6362    0.4815

```