C06 Chapter Contents
C06 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentC06PJF

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

C06PJF computes the multidimensional discrete Fourier transform of a multivariate sequence of complex data values.

## 2  Specification

 SUBROUTINE C06PJF ( DIRECT, NDIM, ND, N, X, WORK, LWORK, IFAIL)
 INTEGER NDIM, ND(NDIM), N, LWORK, IFAIL COMPLEX (KIND=nag_wp) X(N), WORK(LWORK) CHARACTER(1) DIRECT

## 3  Description

C06PJF 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 C06PJF with ${\mathbf{DIRECT}}=\text{'F'}$ followed by a call with ${\mathbf{DIRECT}}=\text{'B'}$ 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 routine calls C06PRF to perform one-dimensional discrete Fourier transforms. Hence, the routine 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  Parameters

1:     DIRECT – CHARACTER(1)Input
On entry: if the forward transform as defined in Section 3 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: ${\mathbf{DIRECT}}=\text{'F'}$ or $\text{'B'}$.
2:     NDIM – INTEGERInput
On entry: $m$, the number of dimensions (or variables) in the multivariate data.
Constraint: ${\mathbf{NDIM}}\ge 1$.
3:     ND(NDIM) – INTEGER arrayInput
On entry: the elements of ND must contain the dimensions of the NDIM variables; that is, ${\mathbf{ND}}\left(i\right)$ must contain the dimension of the $i$th variable.
Constraints:
• ${\mathbf{ND}}\left(\mathit{i}\right)\ge 1$, for $\mathit{i}=1,2,\dots ,{\mathbf{NDIM}}$;
• ${\mathbf{ND}}\left(\mathit{i}\right)$ must have less than $31$ prime factors (counting repetitions), for $\mathit{i}=1,2,\dots ,{\mathbf{NDIM}}$.
4:     N – INTEGERInput
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) – COMPLEX (KIND=nag_wp) arrayInput/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 ${\mathbf{X}}\left(1+{j}_{1}+{n}_{1}{j}_{2}+{n}_{1}{n}_{2}{j}_{3}+\cdots \right)$.
On exit: the corresponding elements of the computed transform.
6:     WORK(LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
The workspace requirements as documented for C06PJF may be an overestimate in some implementations.
On exit: the real part of ${\mathbf{WORK}}\left(1\right)$ contains the minimum workspace required for the current value of N with this implementation.
7:     LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which C06PJF is called.
Suggested value: ${\mathbf{LWORK}}\ge {\mathbf{N}}+3×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ND}}\left(i\right)\right)+15$, where $i=1,2,\dots ,{\mathbf{NDIM}}$
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, ${\mathbf{DIRECT}}\ne \text{'F'}$ or $\text{'B'}$.
${\mathbf{IFAIL}}=3$
 On entry, at least one of the first NDIM elements of ND is less than $1$.
${\mathbf{IFAIL}}=4$
 On entry, N does not equal the product of the first NDIM elements of ND.
${\mathbf{IFAIL}}=5$
 On entry, LWORK is too small. The minimum amount of workspace required is returned in ${\mathbf{WORK}}\left(1\right)$.
${\mathbf{IFAIL}}=6$
 On entry, ${\mathbf{ND}}\left(i\right)$ has more than $30$ prime factors for some $i$.
${\mathbf{IFAIL}}=7$
An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.

## 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 ${\mathbf{ND}}\left(i\right)$. C06PJF 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 (c06pjfe.f90)

### 9.2  Program Data

Program Data (c06pjfe.d)

### 9.3  Program Results

Program Results (c06pjfe.r)