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c06 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_fft_multid_full (c06pjc)

## 1  Purpose

nag_fft_multid_full (c06pjc) computes the multidimensional discrete Fourier transform of a multivariate sequence of complex data values.

## 2  Specification

 #include #include
 void nag_fft_multid_full (Nag_TransformDirection direct, Integer ndim, const Integer nd[], Integer n, Complex x[], NagError *fail)

## 3  Description

nag_fft_multid_full (c06pjc) 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_fft_multid_full (c06pjc) with ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$ followed by a call with ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$ 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 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).

## 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:     ndimIntegerInput
On entry: $m$, the number of dimensions (or variables) in the multivariate data.
Constraint: ${\mathbf{ndim}}\ge 1$.
3:     nd[ndim]const IntegerInput
On entry: the elements of nd must contain the dimensions of the ndim variables; that is, ${\mathbf{nd}}\left[i-1\right]$ must contain the dimension of the $i$th variable.
Constraint: ${\mathbf{nd}}\left[\mathit{i}-1\right]\ge 1$, for $\mathit{i}=1,2,\dots ,{\mathbf{ndim}}$.
4:     nIntegerInput
On entry: $n$, the total number of data values.
Constraint: n must equal the product of the first ndim elements of the array nd.
5:     x[n]ComplexInput/Output
On entry: the complex data values. Data values are stored in x using column-major ordering for storing multidimensional arrays; that is, ${z}_{{j}_{1}{j}_{2}\cdots {j}_{m}}$ is stored in ${\mathbf{x}}\left[{j}_{1}+{n}_{1}{j}_{2}+{n}_{1}{n}_{2}{j}_{3}+\cdots \right]$.
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, ${\mathbf{ndim}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ndim}}\ge 1$.
NE_INT_2
n must equal the product of the dimensions held in array nd: ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$, product of nd elements is $⟨\mathit{\text{value}}⟩$.
On entry ${\mathbf{nd}}\left[\mathit{I}-1\right]=⟨\mathit{\text{value}}⟩$ and $\mathit{I}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nd}}\left[\mathit{I}-1\right]\ge 1$.
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_multid_full (c06pjc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_fft_multid_full (c06pjc) 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 $n×\mathrm{log}\left(n\right)$, but also depends on the factorization of the individual dimensions ${\mathbf{nd}}\left[i-1\right]$. nag_fft_multid_full (c06pjc) 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 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.

### 10.1  Program Text

Program Text (c06pjce.c)

### 10.2  Program Data

Program Data (c06pjce.d)

### 10.3  Program Results

Program Results (c06pjce.r)