C06 Chapter Contents
C06 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentC06FJF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

C06FJF computes the multidimensional discrete Fourier transform of a multivariate sequence of complex data values.

## 2  Specification

 SUBROUTINE C06FJF ( NDIM, ND, N, X, Y, WORK, LWORK, IFAIL)
 INTEGER NDIM, ND(NDIM), N, LWORK, IFAIL REAL (KIND=nag_wp) X(N), Y(N), WORK(LWORK)

## 3  Description

C06FJF 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 ${k}_{1}=0,1,\dots ,{n}_{1}-1$, ${k}_{2}=0,1,\dots ,{n}_{2}-1$.
The extension to higher dimensions is obvious. (Note the scale factor of $\frac{1}{\sqrt{n}}$ in this definition.)
To compute the inverse discrete Fourier transform, defined with $\mathrm{exp}\left(+2\pi i\left(\dots \right)\right)$ in the above formula instead of $\mathrm{exp}\left(-2\pi i\left(\dots \right)\right)$, this routine should be preceded and followed by the complex conjugation of the data values and the transform (by negating the imaginary parts stored in $y$).
The data values must be supplied in a pair of one-dimensional arrays (real and imaginary parts separately), in accordance with the Fortran convention for storing multidimensional data (i.e., with the first subscript ${j}_{1}$ varying most rapidly).
This routine calls C06FCF to perform one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm in Brigham (1974), and hence there are some restrictions on the values of the ${n}_{i}$ (see Section 5).

## 4  References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall

## 5  Parameters

1:     NDIM – INTEGERInput
On entry: $m$, the number of dimensions (or variables) in the multivariate data.
Constraint: ${\mathbf{NDIM}}\ge 1$.
2:     ND(NDIM) – INTEGER arrayInput
On entry: ${\mathbf{ND}}\left(\mathit{i}\right)$ must contain ${n}_{\mathit{i}}$ (the dimension of the $\mathit{i}$th variable) , for $\mathit{i}=1,2,\dots ,m$. The largest prime factor of each ${\mathbf{ND}}\left(i\right)$ must not exceed $19$, and the total number of prime factors of ${\mathbf{ND}}\left(i\right)$, counting repetitions, must not exceed $20$.
Constraint: ${\mathbf{ND}}\left(\mathit{i}\right)\ge 1$, for $\mathit{i}=1,2,\dots ,{\mathbf{NDIM}}$.
3:     N – INTEGERInput
On entry: $n$, the total number of data values.
Constraint: ${\mathbf{N}}={\mathbf{ND}}\left(1\right)×{\mathbf{ND}}\left(2\right)×\cdots ×{\mathbf{ND}}\left({\mathbf{NDIM}}\right)$.
4:     X(N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: ${\mathbf{X}}\left(1+{j}_{1}+{n}_{1}{j}_{2}+{n}_{1}{n}_{2}{j}_{3}+\dots \right)$ must contain the real part of the complex data value ${z}_{{j}_{1}{j}_{2}\dots {j}_{m}}$, for $0\le {j}_{1}\le {n}_{1}-1,0\le {j}_{2}\le {n}_{2}-1,\dots \text{}$; i.e., the values are stored in consecutive elements of the array according to the Fortran convention for storing multidimensional arrays.
On exit: the real parts of the corresponding elements of the computed transform.
5:     Y(N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: the imaginary parts of the complex data values, stored in the same way as the real parts in the array X.
On exit: the imaginary parts of the corresponding elements of the computed transform.
6:     WORK(LWORK) – REAL (KIND=nag_wp) arrayWorkspace
7:     LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which C06FJF is called.
Constraint: ${\mathbf{LWORK}}\ge 3×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left\{{\mathbf{ND}}\left(i\right)\right\}$.
8:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
 On entry, ${\mathbf{NDIM}}<1$.
${\mathbf{IFAIL}}=2$
 On entry, ${\mathbf{N}}\ne {\mathbf{ND}}\left(1\right)×{\mathbf{ND}}\left(2\right)×\cdots ×{\mathbf{ND}}\left({\mathbf{NDIM}}\right)$.
${\mathbf{IFAIL}}=10×l+1$
At least one of the prime factors of ${\mathbf{ND}}\left(l\right)$ is greater than $19$.
${\mathbf{IFAIL}}=10×l+2$
${\mathbf{ND}}\left(l\right)$ has more than $20$ prime factors.
${\mathbf{IFAIL}}=10×l+3$
 On entry, ${\mathbf{ND}}\left(l\right)<1$.
${\mathbf{IFAIL}}=10×l+4$
 On entry, ${\mathbf{LWORK}}<3×{\mathbf{ND}}\left(l\right)$.

## 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).

The time taken is approximately proportional to $n×\mathrm{log}n$, but also depends on the factorization of the individual dimensions ${\mathbf{ND}}\left(i\right)$. C06FJF is faster if the only prime factors are $2$, $3$ or $5$; and fastest of all if they are powers of $2$.

## 9  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.

### 9.1  Program Text

Program Text (c06fjfe.f90)

### 9.2  Program Data

Program Data (c06fjfe.d)

### 9.3  Program Results

Program Results (c06fjfe.r)