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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$m$ 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$m$ sequences of n + 1 $n+1$ real data values xjp ${x}_{\mathit{j}}^{\mathit{p}}$, for j = 0,1,,n$\mathit{j}=0,1,\dots ,n$ and p = 1,2,,m$\mathit{p}=1,2,\dots ,m$, nag_sum_fft_real_cosine_simple (c06rb) simultaneously calculates the Fourier cosine transforms of all the sequences defined by
kp = sqrt(2/n)
 ( n − 1 ) (1/2)x0p + ∑ xjp × cos(jkπ/n) + (1/2)( − 1)kxnp j = 1
,   k = 0, 1, , n ​ and ​ p = 1, 2, , m .
$x^ k p = 2n ( 12 x0p + ∑ j=1 n-1 xjp × cos( jk πn ) + 12 (-1)k xnp ) , k= 0, 1, …, n ​ and ​ p= 1, 2, …, m .$
(Note the scale factor sqrt(2/n) $\sqrt{\frac{2}{n}}$ 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$2$, 3$3$, 4$4$ and 5$5$.

## 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:     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 less than the number of real values in each sequence, i.e., the number of values in each sequence is n + 1$n+1$.
Constraint: n1${\mathbf{n}}\ge 1$.
3:     x( m × (n + 3) ${\mathbf{m}}×\left({\mathbf{n}}+3\right)$) – double array
the data must be stored in x as if in a two-dimensional array of dimension (1 : m,0 : n + 2)$\left(1:{\mathbf{m}},0:{\mathbf{n}}+2\right)$; each of the m$m$ sequences is stored in a row of the array. In other words, if the (n + 1)$\left(n+1\right)$ data values of the p$\mathit{p}$th sequence to be transformed are denoted by xjp${x}_{\mathit{j}}^{\mathit{p}}$, for j = 0,1,,n$\mathit{j}=0,1,\dots ,n$ and p = 1,2,,m$\mathit{p}=1,2,\dots ,m$, then the first m(n + 1)$m\left(n+1\right)$ elements of the array x must contain the values
 x01 , x02 , … , x0m , x11 , x12 , … , x1m , … , xn1 , xn2 , … , xnm . $x01 , x02 ,…, x0m , x11 , x12 ,…, x1m ,…, xn1 , xn2 ,…, xnm .$
The (n + 2)$\left(n+2\right)$th and (n + 3)$\left(n+3\right)$th elements of each row xn + 2p , xn + 3p${x}_{n+2}^{\mathit{p}},{x}_{n+3}^{\mathit{p}}$, for p = 1,2,,m$\mathit{p}=1,2,\dots ,m$, are required as workspace. These 2m$2m$ elements may contain arbitrary values as they are set to zero by the function.

None.

work

### Output Parameters

1:     x( m × (n + 3) ${\mathbf{m}}×\left({\mathbf{n}}+3\right)$) – double array
the m$m$ Fourier cosine transforms stored as if in a two-dimensional array of dimension (1 : m,0 : n + 2)$\left(1:{\mathbf{m}},0:{\mathbf{n}}+2\right)$. Each of the m$m$ transforms is stored in a row of the array, overwriting the corresponding original data. If the (n + 1)$\left(n+1\right)$ components of the p$\mathit{p}$th Fourier cosine transform are denoted by kp${\stackrel{^}{x}}_{\mathit{k}}^{\mathit{p}}$, for k = 0,1,,n$\mathit{k}=0,1,\dots ,n$ and p = 1,2,,m$\mathit{p}=1,2,\dots ,m$, then the m(n + 3)$m\left(n+3\right)$ elements of the array x contain the values
 x̂01 , x̂02 , … , x̂0m , x̂11 , x̂12 , … , x̂1m , … , x̂n1 , x̂n2 , … , x̂nm , 0 , 0 , … ,0  (2m times) .
2:     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${\mathbf{ifail}}=3$
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_cosine_simple (c06rb) 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_cosine_simple (c06rb) 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_cosine_simple_example
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;
0.643;
0.6362;
0.8638;
0.0428;
0.1424;
0.8723;
0.4815;
0.9562;
0.4936;
0.2057;
0;
0;
0;
0;
0;
0];
[xOut, ifail] = nag_sum_fft_real_cosine_simple(m, n, x)
```
```

xOut =

1.6833
1.9605
1.3838
-0.0482
-0.4884
0.1588
0.0176
-0.0655
-0.0761
0.1368
0.4444
-0.1184
0.3240
0.0964
0.3512
-0.5830
0.0856
0.5759
-0.0427
-0.2289
0.0110
0
0
0
-0.0482
-0.4884
0.1588

ifail =

0

```
```function c06rb_example
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;
0.643;
0.6362;
0.8638;
0.0428;
0.1424;
0.8723;
0.4815;
0.9562;
0.4936;
0.2057;
0;
0;
0;
0;
0;
0];
[xOut, ifail] = c06rb(m, n, x)
```
```

xOut =

1.6833
1.9605
1.3838
-0.0482
-0.4884
0.1588
0.0176
-0.0655
-0.0761
0.1368
0.4444
-0.1184
0.3240
0.0964
0.3512
-0.5830
0.0856
0.5759
-0.0427
-0.2289
0.0110
0
0
0
-0.0482
-0.4884
0.1588

ifail =

0

```