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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_sum_fft_complex_multid (c06pj)

## Purpose

nag_sum_fft_complex_multid (c06pj) computes the multidimensional discrete Fourier transform of a multivariate sequence of complex data values.

## Syntax

[x, ifail] = c06pj(direct, nd, x, 'ndim', ndim, 'n', n)
[x, ifail] = nag_sum_fft_complex_multid(direct, nd, x, 'ndim', ndim, 'n', n)

## Description

nag_sum_fft_complex_multid (c06pj) computes the multidimensional discrete Fourier transform of a multidimensional sequence of complex data values ${z}_{{j}_{1}{j}_{2}\dots {j}_{m}}$, where ${j}_{1}=0,1,\dots ,{n}_{1}-1\text{, }{j}_{2}=0,1,\dots ,{n}_{2}-1$, and so on. Thus the individual dimensions are ${n}_{1},{n}_{2},\dots ,{n}_{m}$, and the total number of data values is $n={n}_{1}×{n}_{2}×\cdots ×{n}_{m}$.
The discrete Fourier transform is here defined (e.g., for $m=2$) by:
 $z^ k1 , k2 = 1n ∑ j1=0 n1-1 ∑ j2=0 n2-1 z j1j2 × exp ±2πi j1k1 n1 + j2k2 n2 ,$
where ${k}_{1}=0,1,\dots ,{n}_{1}-1$ and ${k}_{2}=0,1,\dots ,{n}_{2}-1$. The plus or minus sign in the argument of the exponential terms in the above definition determine the direction of the transform: a minus sign defines the forward direction and a plus sign defines the backward direction.
The extension to higher dimensions is obvious. (Note the scale factor of $\frac{1}{\sqrt{n}}$ in this definition.)
A call of nag_sum_fft_complex_multid (c06pj) with ${\mathbf{direct}}=\text{'F'}$ followed by a call with ${\mathbf{direct}}=\text{'B'}$ will restore the original data.
The data values must be supplied in a one-dimensional array using column-major storage ordering of multidimensional data (i.e., with the first subscript ${j}_{1}$ varying most rapidly).
This function calls nag_sum_fft_complex_1d_multi_row (c06pr) to perform one-dimensional discrete Fourier transforms. Hence, 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).

## References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Self-sorting mixed-radix fast Fourier transforms J. Comput. Phys. 52 1–23

## 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:     $\mathrm{nd}\left({\mathbf{ndim}}\right)$int64int32nag_int array
The elements of nd must contain the dimensions of the ndim variables; that is, ${\mathbf{nd}}\left(i\right)$ must contain the dimension of the $i$th variable.
Constraint: ${\mathbf{nd}}\left(\mathit{i}\right)\ge 1$, for $\mathit{i}=1,2,\dots ,{\mathbf{ndim}}$.
3:     $\mathrm{x}\left({\mathbf{n}}\right)$ – complex array
The complex data values. Data values are stored in x using column-major ordering for storing multidimensional arrays; that is, ${z}_{{j}_{1}{j}_{2}\cdots {j}_{m}}$ is stored in ${\mathbf{x}}\left(1+{j}_{1}+{n}_{1}{j}_{2}+{n}_{1}{n}_{2}{j}_{3}+\cdots \right)$.

### Optional Input Parameters

1:     $\mathrm{ndim}$int64int32nag_int scalar
Default: the dimension of the array nd.
$m$, the number of dimensions (or variables) in the multivariate data.
Constraint: ${\mathbf{ndim}}\ge 1$.
2:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the array x.
$n$, the total number of data values.
Constraint: n must equal the product of the first ndim elements of the array nd.

### Output Parameters

1:     $\mathrm{x}\left({\mathbf{n}}\right)$ – complex array
The corresponding elements of the computed transform.
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, ${\mathbf{ndim}}<1$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{direct}}\ne \text{'F'}$ or $\text{'B'}$.
${\mathbf{ifail}}=3$
 On entry, at least one of the first ndim elements of nd is less than $1$.
${\mathbf{ifail}}=4$
 On entry, n does not equal the product of the first ndim elements of nd.
${\mathbf{ifail}}=5$
 On entry, lwork is too small. The minimum amount of workspace required is returned in $\mathit{work}\left(1\right)$.
${\mathbf{ifail}}=7$
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 is approximately proportional to $n×\mathrm{log}\left(n\right)$, but also depends on the factorization of the individual dimensions ${\mathbf{nd}}\left(i\right)$. nag_sum_fft_complex_multid (c06pj) is faster if the only prime factors are $2$, $3$ or $5$; and fastest of all if they are powers of $2$.

## Example

This example reads in a bivariate sequence of complex data values and prints the two-dimensional Fourier transform. It then performs an inverse transform and prints the sequence so obtained, which may be compared to the original data values.
```function c06pj_example

fprintf('c06pj example results\n\n');

% 2D Sequence
nd = [int64(3) 5];
x = [1.000+0.000i  0.999-0.040i  0.987-0.159i  0.936-0.352i  0.802-0.597i;
0.994-0.111i  0.989-0.151i  0.963-0.268i  0.891-0.454i  0.731-0.682i;
0.903-0.430i  0.885-0.466i  0.823-0.568i  0.694-0.720i  0.467-0.884i];

% Transform x
direct = 'F';
[xt, ifail] = c06pj(direct, nd, x);

% Restore x by inverse transform
direct = 'B';
[xr, ifail] = c06pj(direct, nd, xt);

disp('Original data:');
disp(x);
disp('Discrete Fourier Transform:');
disp(reshape(xt,nd));
fprintf('Original sequence as restored by inverse transform:\n');
disp(reshape(xr,nd));

```
```c06pj example results

Original data:
1.0000 + 0.0000i   0.9990 - 0.0400i   0.9870 - 0.1590i   0.9360 - 0.3520i   0.8020 - 0.5970i
0.9940 - 0.1110i   0.9890 - 0.1510i   0.9630 - 0.2680i   0.8910 - 0.4540i   0.7310 - 0.6820i
0.9030 - 0.4300i   0.8850 - 0.4660i   0.8230 - 0.5680i   0.6940 - 0.7200i   0.4670 - 0.8840i

Discrete Fourier Transform:
3.3731 - 1.5187i   0.4814 - 0.0907i   0.2507 + 0.1776i   0.0543 + 0.3188i  -0.4194 + 0.4145i
0.4565 + 0.1368i   0.0549 + 0.0317i   0.0093 + 0.0389i  -0.0217 + 0.0356i  -0.0759 + 0.0045i
-0.1705 + 0.4927i  -0.0375 + 0.0584i  -0.0423 + 0.0082i  -0.0377 - 0.0255i  -0.0022 - 0.0829i

Original sequence as restored by inverse transform:
1.0000 + 0.0000i   0.9990 - 0.0400i   0.9870 - 0.1590i   0.9360 - 0.3520i   0.8020 - 0.5970i
0.9940 - 0.1110i   0.9890 - 0.1510i   0.9630 - 0.2680i   0.8910 - 0.4540i   0.7310 - 0.6820i
0.9030 - 0.4300i   0.8850 - 0.4660i   0.8230 - 0.5680i   0.6940 - 0.7200i   0.4670 - 0.8840i

```