c06 Chapter Contents
c06 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_fft_3d (c06pxc)

## 1  Purpose

nag_fft_3d (c06pxc) computes the three-dimensional discrete Fourier transform of a trivariate sequence of complex data values (using complex data type).

## 2  Specification

 #include #include
 void nag_fft_3d (Nag_TransformDirection direct, Integer n1, Integer n2, Integer n3, Complex x[], NagError *fail)

## 3  Description

nag_fft_3d (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 nag_fft_3d (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)).

## 4  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

## 5  Arguments

1:     directNag_TransformDirectionInput
On entry: if the forward transform as defined in Section 3 is to be computed, then direct must be set equal to $\mathrm{Nag_ForwardTransform}$.
If the backward transform is to be computed then direct must be set equal to $\mathrm{Nag_BackwardTransform}$.
Constraint: ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$ or $\mathrm{Nag_BackwardTransform}$.
2:     n1IntegerInput
On entry: ${n}_{1}$, the first dimension of the transform.
Constraint: ${\mathbf{n1}}\ge 1$.
3:     n2IntegerInput
On entry: ${n}_{2}$, the second dimension of the transform.
Constraint: ${\mathbf{n2}}\ge 1$.
4:     n3IntegerInput
On entry: ${n}_{3}$, the third dimension of the transform.
Constraint: ${\mathbf{n3}}\ge 1$.
5:     x[${\mathbf{n1}}×{\mathbf{n2}}×{\mathbf{n3}}$]ComplexInput/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:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
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.

## 7  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).

## 8  Parallelism and Performance

nag_fft_3d (c06pxc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_fft_3d (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.

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 ${n}_{3}$. nag_fft_3d (c06pxc) is faster if the only prime factors are $2$, $3$ or $5$; and fastest of all if they are powers of $2$.

## 10  Example

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.

### 10.1  Program Text

Program Text (c06pxce.c)

### 10.2  Program Data

Program Data (c06pxce.d)

### 10.3  Program Results

Program Results (c06pxce.r)