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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_sum_fft_real_sine (c06ha)

## Purpose

nag_sum_fft_real_sine (c06ha) computes the discrete Fourier sine transforms of m$m$ sequences of real data values. This function is designed to be particularly efficient on vector processors.
Note: this function is scheduled to be withdrawn, please see c06ha in Advice on Replacement Calls for Withdrawn/Superseded Routines..

## Syntax

[x, trig, ifail] = c06ha(m, n, x, init, trig)
[x, trig, ifail] = nag_sum_fft_real_sine(m, n, x, init, trig)

## Description

Given m$m$ sequences of n1 $n-1$ real data values xjp ${x}_{\mathit{j}}^{\mathit{p}}$, for j = 1,2,,n1$\mathit{j}=1,2,\dots ,n-1$ and p = 1,2,,m$\mathit{p}=1,2,\dots ,m$, nag_sum_fft_real_sine (c06ha) simultaneously calculates the Fourier sine transforms of all the sequences defined by
 n − 1 x̂kp = sqrt(2/n) ∑ xjp × sin(jkπ/n),  k = 1,2, … ,n − 1​ and ​p = 1,2, … ,m. j = 1
$x^ k p = 2n ∑ j=1 n-1 xjp × sin( jk πn ) , k= 1,2,…,n-1 ​ and ​ p= 1,2,…,m .$
(Note the scale factor sqrt(2/n) $\sqrt{\frac{2}{n}}$ in this definition.)
The Fourier sine transform defined above is its own inverse, and two consecutive calls of this function with the same data 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)). (See the C06 Chapter Introduction.)
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$2$, 3$3$, 4$4$, 5$5$ and 6$6$. This function is designed to be particularly efficient on vector processors, and it becomes especially fast as m$m$, the number of transforms to be computed in parallel, increases.

## 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:     m – int64int32nag_int scalar
m$m$, the number of sequences to be transformed.
Constraint: m1${\mathbf{m}}\ge 1$.
2:     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 n1$n-1$.
Constraint: n1${\mathbf{n}}\ge 1$.
3:     x( m × n ${\mathbf{m}}×{\mathbf{n}}$) – double array
The data must be stored in x as if in a two-dimensional array of dimension (1 : m,1 : n)$\left(1:{\mathbf{m}},1:{\mathbf{n}}\right)$; each of the m$m$ sequences is stored in a row of the array. In other words, if the n1$n-1$ data values of the p$p$th sequence to be transformed are denoted by xjp${x}_{\mathit{j}}^{\mathit{p}}$, for j = 1,2,,n1$\mathit{j}=1,2,\dots ,n-1$ and p = 1,2,,m$\mathit{p}=1,2,\dots ,m$, then the first m(n1)$m\left(n-1\right)$ elements of the array x must contain the values
 x11 , x12 , … , x1m , x21 , x22 , … , x2m , … , xn − 11 , xn − 12 , … , xn − 1m . $x11 , x12 ,…, x1m , x21 , x22 ,…, x2m ,…, x n-1 1 , x n-1 2 ,…, x n-1 m .$
The n$n$th element of each row xnp${x}_{n}^{\mathit{p}}$, for p = 1,2,,m$\mathit{p}=1,2,\dots ,m$, is required as workspace. These m$m$ elements may contain arbitrary values on entry, and are set to zero by the function.
4:     init – string (length ≥ 1)
Indicates whether trigonometric coefficients are to be calculated.
init = 'I'${\mathbf{init}}=\text{'I'}$
Calculate the required trigonometric coefficients for the given value of n$n$, and store in the array trig.
init = 'S'${\mathbf{init}}=\text{'S'}$ or 'R'$\text{'R'}$
The required trigonometric coefficients are assumed to have been calculated and stored in the array trig in a prior call to one of nag_sum_fft_real_sine (c06ha), nag_sum_fft_real_cosine (c06hb), nag_sum_fft_real_qtrsine (c06hc) or nag_sum_fft_real_qtrcosine (c06hd). The function performs a simple check that the current value of n$n$ is consistent with the values stored in trig.
Constraint: init = 'I'${\mathbf{init}}=\text{'I'}$, 'S'$\text{'S'}$ or 'R'$\text{'R'}$.
5:     trig( 2 × n $2×{\mathbf{n}}$) – double array
If init = 'S'${\mathbf{init}}=\text{'S'}$ or 'R'$\text{'R'}$, trig must contain the required trigonometric coefficients calculated in a previous call of the function. Otherwise trig need not be set.

None.

work

### Output Parameters

1:     x( m × n ${\mathbf{m}}×{\mathbf{n}}$) – double array
The m$m$ Fourier transforms stored as if in a two-dimensional array of dimension (1 : m,1 : n)$\left(1:{\mathbf{m}},1:{\mathbf{n}}\right)$. Each of the m$m$ transforms is stored in a row of the array, overwriting the corresponding original sequence. If the n1$n-1$ components of the p$\mathit{p}$th Fourier sine transform are denoted by kp${\stackrel{^}{x}}_{\mathit{k}}^{\mathit{p}}$, for k = 1,2,,n1$\mathit{k}=1,2,\dots ,n-1$ and p = 1,2,,m$\mathit{p}=1,2,\dots ,m$, then the mn$mn$ elements of the array x contain the values
 x̂11 , x̂12 , … , x̂1m , x̂21 , x̂22 , … , x̂2m , … , x̂n − 11 , x̂n − 12 , … , x̂n − 1m , 0 , 0 , … , 0   (m times) .
If n = 1$n=1$, the m$m$ elements of x are set to zero.
2:     trig( 2 × n $2×{\mathbf{n}}$) – double array
Contains the required coefficients (computed by the function if init = 'I'${\mathbf{init}}=\text{'I'}$).
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, m < 1${\mathbf{m}}<1$.
ifail = 2${\mathbf{ifail}}=2$
 On entry, n < 1${\mathbf{n}}<1$.
ifail = 3${\mathbf{ifail}}=3$
 On entry, init ≠ 'I'${\mathbf{init}}\ne \text{'I'}$, 'S'$\text{'S'}$ or 'R'$\text{'R'}$.
ifail = 4${\mathbf{ifail}}=4$
Not used at this Mark.
ifail = 5${\mathbf{ifail}}=5$
 On entry, init = 'S'${\mathbf{init}}=\text{'S'}$ or 'R'$\text{'R'}$, but the array trig and the current value of n are inconsistent.
ifail = 6${\mathbf{ifail}}=6$
An unexpected error has occurred in an internal call. Check all function calls and array dimensions. Seek expert help.

## 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_sine (c06ha) is approximately proportional to nm log(n)$nm\mathrm{log}\left(n\right)$, but also depends on the factors of n$n$. nag_sum_fft_real_sine (c06ha) is fastest if the only prime factors of n$n$ are 2$2$, 3$3$ and 5$5$, and is particularly slow if n$n$ is a large prime, or has large prime factors.

## Example

```function nag_sum_fft_real_sine_example
m = int64(3);
n = int64(6);
x = [0.6772;
0.2983;
0.0644;
0.1138;
0.1181;
0.6037;
0.6751;
0.7255;
0.643;
0.6362;
0.8638;
0.0428;
0.1424;
0.8723;
0.4815;
0;
0;
0];
init = 'Initial';
trig = zeros(12,1);
[xOut, trigOut, ifail] = nag_sum_fft_real_sine(m, n, x, init, trig)
```
```

xOut =

1.0014
1.2477
0.8521
0.0062
-0.6599
0.0719
0.0834
0.2570
-0.0561
0.5286
0.0858
-0.4890
0.2514
0.2658
0.2056
0
0
0

trigOut =

1
1
1
1
1
6
0
0
0
0
0
6

ifail =

0

```
```function c06ha_example
m = int64(3);
n = int64(6);
x = [0.6772;
0.2983;
0.0644;
0.1138;
0.1181;
0.6037;
0.6751;
0.7255;
0.643;
0.6362;
0.8638;
0.0428;
0.1424;
0.8723;
0.4815;
0;
0;
0];
init = 'Initial';
trig = zeros(12,1);
[xOut, trigOut, ifail] = c06ha(m, n, x, init, trig)
```
```

xOut =

1.0014
1.2477
0.8521
0.0062
-0.6599
0.0719
0.0834
0.2570
-0.0561
0.5286
0.0858
-0.4890
0.2514
0.2658
0.2056
0
0
0

trigOut =

1
1
1
1
1
6
0
0
0
0
0
6

ifail =

0

```