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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_sum_fft_complex_3d (c06px)

## Purpose

nag_sum_fft_complex_3d (c06px) computes the three-dimensional discrete Fourier transform of a trivariate sequence of complex data values (using complex data type).

## Syntax

[x, ifail] = c06px(direct, n1, n2, n3, x)
[x, ifail] = nag_sum_fft_complex_3d(direct, n1, n2, n3, x)

## Description

nag_sum_fft_complex_3d (c06px) computes the three-dimensional discrete Fourier transform of a trivariate sequence of complex data values z j1 j2 j3 ${z}_{{j}_{1}{j}_{2}{j}_{3}}$, for j1 = 0,1,,n11$\mathit{j1}=0,1,\dots ,{n}_{1}-1$, j2 = 0,1,,n21$\mathit{j2}=0,1,\dots ,{n}_{2}-1$ and j3 = 0,1,,n31$\mathit{j3}=0,1,\dots ,{n}_{3}-1$.
The discrete Fourier transform is here defined by
 n1 − 1 n2 − 1 n3 − 1 ẑ k1 k2 k3 = 1/(sqrt( n1 n2 n3 )) ∑ ∑ ∑ z j1 j2 j3 × exp( ± 2πi(( j1 k1 )/(n1) + ( j2 k2 )/(n2) + ( j3 k3 )/(n3))), j1 = 0 j2 = 0 j3 = 0
$z^ k1 k2 k3 = 1 n1 n2 n3 ∑ j1=0 n1-1 ∑ j2=0 n2-1 ∑ j3=0 n3-1 z j1 j2 j3 × exp( ±2πi ( j1 k1 n1 + j2 k2 n2 + j3 k3 n3 ) ) ,$
where k1 = 0,1,,n11 ${k}_{1}=0,1,\dots ,{n}_{1}-1$, k2 = 0,1,,n21 ${k}_{2}=0,1,\dots ,{n}_{2}-1$ and k3 = 0,1,,n31 ${k}_{3}=0,1,\dots ,{n}_{3}-1$.
(Note the scale factor of 1/(sqrt( n1 n2 n3 )) $\frac{1}{\sqrt{{n}_{1}{n}_{2}{n}_{3}}}$ in this definition.) The minus sign is taken in the argument of the exponential within the summation when the forward transform is required, and the plus sign is taken when the backward transform is required.
A call of nag_sum_fft_complex_3d (c06px) with direct = 'F'${\mathbf{direct}}=\text{'F'}$ followed by a call with direct = 'B'${\mathbf{direct}}=\text{'B'}$ will restore the original data.
This function performs multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm (see Brigham (1974)).

## 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:     direct – string (length ≥ 1)
If the forward transform as defined in Section [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: direct = 'F'${\mathbf{direct}}=\text{'F'}$ or 'B'$\text{'B'}$.
2:     n1 – int64int32nag_int scalar
n1${n}_{1}$, the first dimension of the transform.
Constraint: n11${\mathbf{n1}}\ge 1$.
3:     n2 – int64int32nag_int scalar
n2${n}_{2}$, the second dimension of the transform.
Constraint: n21${\mathbf{n2}}\ge 1$.
4:     n3 – int64int32nag_int scalar
n3${n}_{3}$, the third dimension of the transform.
Constraint: n31${\mathbf{n3}}\ge 1$.
5:     x( n1 × n2 × n3 ${\mathbf{n1}}×{\mathbf{n2}}×{\mathbf{n3}}$) – complex array
The complex data values. Data values are stored in x using column-major ordering for storing multidimensional arrays; that is, z j1 j2 j3 ${z}_{{j}_{1}{j}_{2}{j}_{3}}$ is stored in x( 1 + j1 + n1 j2 + n1 n2 j3 )${\mathbf{x}}\left(1+{j}_{1}+{n}_{1}{j}_{2}+{n}_{1}{n}_{2}{j}_{3}\right)$.

None.

work

### Output Parameters

1:     x( n1 × n2 × n3 ${\mathbf{n1}}×{\mathbf{n2}}×{\mathbf{n3}}$) – complex array
The corresponding elements of the computed transform.
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, n1 < 1${\mathbf{n1}}<1$.
ifail = 2${\mathbf{ifail}}=2$
 On entry, n2 < 1${\mathbf{n2}}<1$.
ifail = 3${\mathbf{ifail}}=3$
 On entry, n3 < 1${\mathbf{n3}}<1$.
ifail = 4${\mathbf{ifail}}=4$
 On entry, direct ≠ 'F'${\mathbf{direct}}\ne \text{'F'}$ or 'B'$\text{'B'}$.
ifail = 5${\mathbf{ifail}}=5$
 On entry, n1 has more than 30$30$ prime factors.
ifail = 6${\mathbf{ifail}}=6$
 On entry, n2 has more than 30$30$ prime factors.
ifail = 7${\mathbf{ifail}}=7$
 On entry, n3 has more than 30$30$ prime factors.
ifail = 8${\mathbf{ifail}}=8$
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 is approximately proportional to n1 n2 n3 × log(n1n2n3) ${n}_{1}{n}_{2}{n}_{3}×\mathrm{log}\left({n}_{1}{n}_{2}{n}_{3}\right)$, but also depends on the factorization of the individual dimensions n1 ${n}_{1}$, n2 ${n}_{2}$ and n3 ${n}_{3}$. nag_sum_fft_complex_3d (c06px) is faster if the only prime factors are 2$2$, 3$3$ or 5$5$; and fastest of all if they are powers of 2$2$.

## Example

```function nag_sum_fft_complex_3d_example
direct = 'F';
n1 = int64(2);
n2 = int64(3);
n3 = int64(4);
x = [1;
0.5 + 0.5i;
0.994 - 0.111i;
0.494 + 0.111i;
0.903 - 0.43i;
0.403 + 0.43i;
0.999 - 0.04i;
0.499 + 0.04i;
0.989 - 0.151i;
0.489 + 0.151i;
0.885 - 0.466i;
0.385 + 0.466i;
0.987 - 0.159i;
0.487 + 0.159i;
0.963 - 0.268i;
0.463 + 0.268i;
0.823 - 0.568i;
0.323 + 0.568i;
0.936 - 0.352i;
0.436 + 0.352i;
0.891 - 0.454i;
0.391 + 0.454i;
0.694 - 0.72i;
0.194 + 0.72i];
[xOut, ifail] = nag_sum_fft_complex_3d(direct, n1, n2, n3, x)
```
```

xOut =

3.2921 + 0.1021i
1.2247 - 1.6203i
0.1433 - 0.0860i
0.4243 + 0.3197i
0.1433 + 0.2902i
-0.4243 + 0.3197i
0.0506 - 0.0416i
0.3548 + 0.0833i
0.0155 + 0.1527i
0.0204 - 0.1147i
-0.0502 + 0.1180i
0.0070 - 0.0800i
0.1127 + 0.1021i
0.0000 + 0.1621i
-0.0245 + 0.1268i
0.0134 - 0.0914i
-0.0245 + 0.0773i
-0.0134 - 0.0914i
0.0506 + 0.2458i
-0.3548 + 0.0833i
-0.0502 + 0.0861i
-0.0070 - 0.0800i
0.0155 + 0.0515i
-0.0204 - 0.1147i

ifail =

0

```
```function c06px_example
direct = 'F';
n1 = int64(2);
n2 = int64(3);
n3 = int64(4);
x = [1;
0.5 + 0.5i;
0.994 - 0.111i;
0.494 + 0.111i;
0.903 - 0.43i;
0.403 + 0.43i;
0.999 - 0.04i;
0.499 + 0.04i;
0.989 - 0.151i;
0.489 + 0.151i;
0.885 - 0.466i;
0.385 + 0.466i;
0.987 - 0.159i;
0.487 + 0.159i;
0.963 - 0.268i;
0.463 + 0.268i;
0.823 - 0.568i;
0.323 + 0.568i;
0.936 - 0.352i;
0.436 + 0.352i;
0.891 - 0.454i;
0.391 + 0.454i;
0.694 - 0.72i;
0.194 + 0.72i];
[xOut, ifail] = c06px(direct, n1, n2, n3, x)
```
```

xOut =

3.2921 + 0.1021i
1.2247 - 1.6203i
0.1433 - 0.0860i
0.4243 + 0.3197i
0.1433 + 0.2902i
-0.4243 + 0.3197i
0.0506 - 0.0416i
0.3548 + 0.0833i
0.0155 + 0.1527i
0.0204 - 0.1147i
-0.0502 + 0.1180i
0.0070 - 0.0800i
0.1127 + 0.1021i
0.0000 + 0.1621i
-0.0245 + 0.1268i
0.0134 - 0.0914i
-0.0245 + 0.0773i
-0.0134 - 0.0914i
0.0506 + 0.2458i
-0.3548 + 0.0833i
-0.0502 + 0.0861i
-0.0070 - 0.0800i
0.0155 + 0.0515i
-0.0204 - 0.1147i

ifail =

0

```