# NAG CL Interfacec06pxc (fft_​complex_​3d)

## 1Purpose

c06pxc computes the three-dimensional discrete Fourier transform of a trivariate sequence of complex data values (using complex data type).

## 2Specification

 #include
 void c06pxc (Nag_TransformDirection direct, Integer n1, Integer n2, Integer n3, Complex x[], NagError *fail)
The function may be called by the names: c06pxc, nag_sum_fft_complex_3d or nag_fft_3d.

## 3Description

c06pxc computes the three-dimensional discrete Fourier transform of a trivariate sequence of complex data values ${z}_{{j}_{1}{j}_{2}{j}_{3}}$, for ${j}_{1}=0,1,\dots ,{n}_{1}-1$, ${j}_{2}=0,1,\dots ,{n}_{2}-1$ and ${j}_{3}=0,1,\dots ,{n}_{3}-1$.
The discrete Fourier transform is here defined by
 $z^ k1 k2 k3 = 1 n1 n2 n3 ∑ j1=0 n1-1 ∑ j2=0 n2-1 ∑ j3=0 n3-1 z j1 j2 j3 × exp ±2πi j1 k1 n1 + j2 k2 n2 + j3 k3 n3 ,$
where ${k}_{1}=0,1,\dots ,{n}_{1}-1$, ${k}_{2}=0,1,\dots ,{n}_{2}-1$ and ${k}_{3}=0,1,\dots ,{n}_{3}-1$.
(Note the scale factor of $\frac{1}{\sqrt{{n}_{1}{n}_{2}{n}_{3}}}$ 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 c06pxc with ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$ followed by a call with ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$ will restore the original data.
This function performs multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm (see Brigham (1974)).

## 4References

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

## 5Arguments

1: $\mathbf{direct}$Nag_TransformDirection Input
On entry: if the forward transform as defined in Section 3 is to be computed, direct must be set equal to $\mathrm{Nag_ForwardTransform}$.
If the backward transform is to be computed, direct must be set equal to $\mathrm{Nag_BackwardTransform}$.
Constraint: ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$ or $\mathrm{Nag_BackwardTransform}$.
2: $\mathbf{n1}$Integer Input
On entry: ${n}_{1}$, the first dimension of the transform.
Constraint: ${\mathbf{n1}}\ge 1$.
3: $\mathbf{n2}$Integer Input
On entry: ${n}_{2}$, the second dimension of the transform.
Constraint: ${\mathbf{n2}}\ge 1$.
4: $\mathbf{n3}$Integer Input
On entry: ${n}_{3}$, the third dimension of the transform.
Constraint: ${\mathbf{n3}}\ge 1$.
5: $\mathbf{x}\left[{\mathbf{n1}}×{\mathbf{n2}}×{\mathbf{n3}}\right]$Complex Input/Output
On entry: the complex data values. Data values are stored in x using column-major ordering for storing multidimensional arrays; that is, ${z}_{{j}_{1}{j}_{2}{j}_{3}}$ is stored in ${\mathbf{x}}\left[{j}_{1}+{n}_{1}{j}_{2}+{n}_{1}{n}_{2}{j}_{3}\right]$.
On exit: the corresponding elements of the computed transform.
6: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n1}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n1}}\ge 1$.
On entry, ${\mathbf{n2}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n2}}\ge 1$.
On entry, ${\mathbf{n3}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n3}}\ge 1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 7Accuracy

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).

## 8Parallelism and Performance

c06pxc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pxc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The time taken is approximately proportional to ${n}_{1}{n}_{2}{n}_{3}×\mathrm{log}\left({n}_{1}{n}_{2}{n}_{3}\right)$, but also depends on the factorization of the individual dimensions ${n}_{1}$, ${n}_{2}$ and ${n}_{3}$. c06pxc is faster if the only prime factors are $2$, $3$ or $5$; and fastest of all if they are powers of $2$.

## 10Example

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

### 10.1Program Text

Program Text (c06pxce.c)

### 10.2Program Data

Program Data (c06pxce.d)

### 10.3Program Results

Program Results (c06pxce.r)