Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_sum_fft_realherm_1d_multi_row (c06pp)

## Purpose

nag_sum_fft_realherm_1d_multi_row (c06pp) computes the discrete Fourier transforms of $m$ sequences, each containing $n$ real data values or a Hermitian complex sequence stored in a complex storage format.

## Syntax

[x, ifail] = c06pp(direct, m, n, x)
[x, ifail] = nag_sum_fft_realherm_1d_multi_row(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_realherm_1d_multi_row (c06pp) simultaneously calculates the Fourier transforms of all the sequences defined by
 $z^ k p = 1n ∑ j=0 n-1 xjp × exp -i 2πjkn , k= 0, 1, …, n-1 ​ and ​ p= 1, 2, …, m.$
The transformed values ${\stackrel{^}{z}}_{k}^{p}$ are complex, but for each value of $p$ the ${\stackrel{^}{z}}_{k}^{p}$ form a Hermitian sequence (i.e., ${\stackrel{^}{z}}_{n-k}^{p}$ is the complex conjugate of ${\stackrel{^}{z}}_{k}^{p}$), so they are completely determined by $mn$ real numbers (since ${\stackrel{^}{z}}_{0}^{p}$ is real, as is ${\stackrel{^}{z}}_{n/2}^{p}$ for $n$ even).
Alternatively, given $m$ Hermitian sequences of $n$ complex data values ${z}_{j}^{p}$, this function simultaneously calculates their inverse (backward) discrete Fourier transforms defined by
 $x^kp = 1n ∑ j=0 n-1 zjp × exp i 2πjkn , k=0,1,…,n-1 ​ and ​ p=1,2,…,m .$
The transformed values ${\stackrel{^}{x}}_{k}^{p}$ are real.
(Note the scale factor $\frac{1}{\sqrt{n}}$ in the above definition.)
A call of nag_sum_fft_realherm_1d_multi_row (c06pp) with ${\mathbf{direct}}=\text{'F'}$ followed by a call with ${\mathbf{direct}}=\text{'B'}$ will restore the original data.
The function uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, which is described in Temperton (1983). Special coding is provided for the factors $2$, $3$, $4$ and $5$.

## References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
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 or complex 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 $p$th sequence to be transformed are denoted by ${x}_{\mathit{j}}^{p}$, for $\mathit{j}=0,1,\dots ,n-1$, then:
• if ${\mathbf{direct}}=\text{'F'}$, ${\mathbf{x}}\left(\mathit{j}×{\mathbf{m}}+\mathit{p}\right)$ must contain ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$;
• if ${\mathbf{direct}}=\text{'B'}$, ${\mathbf{x}}\left(2×\mathit{k}×{\mathbf{m}}+\mathit{p}\right)$ and ${\mathbf{x}}\left(\left(2×\mathit{k}+1\right)×{\mathbf{m}}+\mathit{p}\right)$ must contain the real and imaginary parts respectively of ${\stackrel{^}{z}}_{k}^{p}$, for $\mathit{k}=0,1,\dots ,n/2$ and $\mathit{p}=1,2,\dots ,m$. (Note that for the sequence ${\stackrel{^}{z}}_{k}^{p}$ to be Hermitian, the imaginary part of ${\stackrel{^}{z}}_{0}^{p}$, and of ${\stackrel{^}{z}}_{n/2}^{p}$ for $n$ even, must be zero.)

None.

### Output Parameters

1:     $\mathrm{x}\left({\mathbf{m}}×\left({\mathbf{n}}+2\right)\right)$ – double array
• if ${\mathbf{direct}}=\text{'F'}$ and x is declared with bounds $\left(1:{\mathbf{m}},0:{\mathbf{n}}+1\right)$ then ${\mathbf{x}}\left(\mathit{p},2×\mathit{k}\right)$ and ${\mathbf{x}}\left(\mathit{p},2×\mathit{k}+1\right)$ will contain the real and imaginary parts respectively of ${\stackrel{^}{z}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=0,1,\dots ,n/2$ and $\mathit{p}=1,2,\dots ,m$;
• if ${\mathbf{direct}}=\text{'B'}$ and x is declared with bounds $\left(1:{\mathbf{m}},0:{\mathbf{n}}+1\right)$ then ${\mathbf{x}}\left(\mathit{p},\mathit{j}\right)$ will contain ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$.
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_realherm_1d_multi_row (c06pp) is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. nag_sum_fft_realherm_1d_multi_row (c06pp) 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 discrete Fourier transforms (as computed by nag_sum_fft_realherm_1d_multi_row (c06pp) with ${\mathbf{direct}}=\text{'F'}$), after expanding them from complex Hermitian form into a full complex sequences. Inverse transforms are then calculated by calling nag_sum_fft_realherm_1d_multi_row (c06pp) with ${\mathbf{direct}}=\text{'B'}$ showing that the original sequences are restored.
```function c06pp_example

fprintf('c06pp example results\n\n');

% 3 real sequences stored as rows
m = int64(3);
n = int64(6);
x = [0.3854   0.6772   0.1138   0.6751   0.6362  0.1424  0  0;
0.5417   0.2983   0.1181   0.7255   0.8638  0.8723  0  0;
0.9172   0.0644   0.6037   0.6430   0.0428  0.4815  0  0];
disp('Original data values:');
disp(x(:,1:n));

% Transform to get Hermitian sequences
direct = 'F';
[xt, ifail] = c06pp(direct, m, n, x);
zt = xt(:,1:2:n+1) + i*xt(:,2:2:n+2);
title = 'Discrete Fourier transforms in complex Hermitian format:';
[ifail] = x04da('General','Non-unit', zt, title);

for j = 1:m
zt(j,1:n) = nag_herm2complex(n,xt(j,:));
end
title = 'Discrete Fourier transforms in full complex format:';
disp(' ');
[ifail] = x04da('General','Non-unit', zt, title);

% Restore data by back transform
direct = 'B';
[xr, ifail] = c06pp(direct, m, n, xt);
disp(' ');
disp('Original data as restored by inverse transform:');
disp(xr(:,1:n));

function [z] = nag_herm2complex(n,x);
z(1) = complex(x(1));
for j = 2:floor(double(n)/2) + 1
z(j) = x(2*j-1) + i*x(2*j);
z(n-j+2) = x(2*j-1) - i*x(2*j);
end
```
```c06pp example results

Original data values:
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

Discrete Fourier transforms in complex Hermitian format:
1          2          3          4
1      1.0737    -0.1041     0.1126    -0.1467
0.0000    -0.0044    -0.3738     0.0000

2      1.3961    -0.0365     0.0780    -0.1521
0.0000     0.4666    -0.0607     0.0000

3      1.1237     0.0914     0.3936     0.1530
0.0000    -0.0508     0.3458     0.0000

Discrete Fourier transforms in full complex format:
1          2          3          4          5          6
1      1.0737    -0.1041     0.1126    -0.1467     0.1126    -0.1041
0.0000    -0.0044    -0.3738     0.0000     0.3738     0.0044

2      1.3961    -0.0365     0.0780    -0.1521     0.0780    -0.0365
0.0000     0.4666    -0.0607     0.0000     0.0607    -0.4666

3      1.1237     0.0914     0.3936     0.1530     0.3936     0.0914
0.0000    -0.0508     0.3458     0.0000    -0.3458     0.0508

Original data as restored by inverse 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

```