# NAG Library Routine Document

## 1Purpose

c06fxf computes the three-dimensional discrete Fourier transform of a trivariate sequence of complex data values. This routine is designed to be particularly efficient on vector processors.

## 2Specification

Fortran Interface
 Subroutine c06fxf ( n1, n2, n3, x, y, init, work,
 Integer, Intent (In) :: n1, n2, n3 Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Inout) :: x(n1*n2*n3), y(n1*n2*n3), trign1(1), trign2(1), trign3(1) Real (Kind=nag_wp), Intent (Out) :: work(1) Character (1), Intent (In) :: init
#include nagmk26.h
 void c06fxf_ (const Integer *n1, const Integer *n2, const Integer *n3, double x[], double y[], const char *init, double trign1[], double trign2[], double trign3[], double work[], Integer *ifail, const Charlen length_init)

## 3Description

c06fxf 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
 $z^ k1 k2 k3 = 1 n1 n2 n3 ∑ j1=0 n1-1 ∑ j2=0 n2-1 ∑ j3=0 n3-1 z j1 j2 j3 × exp -2πi j1k1 n1 + j2k2 n2 + j3k3 n3 ,$
where ${k}_{1}=0,1,\dots ,{n}_{1}-1$, ${k}_{2}=0,1,\dots ,{n}_{2}-1$, ${k}_{3}=0,1,\dots ,{n}_{3}-1$.
(Note the scale factor of $\frac{1}{\sqrt{{n}_{1}{n}_{2}{n}_{3}}}$ in this definition.)
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 forming the complex conjugates of the data values and the transform.
This routine performs, for each dimension, multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm (see Brigham (1974)). It is designed to be particularly efficient on vector processors.
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:     $\mathbf{n1}$ – IntegerInput
On entry: ${n}_{1}$, the first dimension of the transform.
Constraint: ${\mathbf{n1}}\ge 1$.
2:     $\mathbf{n2}$ – IntegerInput
On entry: ${n}_{2}$, the second dimension of the transform.
Constraint: ${\mathbf{n2}}\ge 1$.
3:     $\mathbf{n3}$ – IntegerInput
On entry: ${n}_{3}$, the third dimension of the transform.
Constraint: ${\mathbf{n3}}\ge 1$.
4:     $\mathbf{x}\left({\mathbf{n1}}×{\mathbf{n2}}×{\mathbf{n3}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
5:     $\mathbf{y}\left({\mathbf{n1}}×{\mathbf{n2}}×{\mathbf{n3}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the real and imaginary parts of the complex data values must be stored in arrays x and y respectively. If x and y are regarded as three-dimensional arrays of dimension $\left(0:{\mathbf{n1}}-1,0:{\mathbf{n2}}-1,0:{\mathbf{n3}}-1\right)$, ${\mathbf{x}}\left({j}_{1},{j}_{2},{j}_{3}\right)$ and ${\mathbf{y}}\left({j}_{1},{j}_{2},{j}_{3}\right)$ must contain the real and imaginary parts of ${z}_{{j}_{1}{j}_{2}{j}_{3}}$.
On exit: the real and imaginary parts respectively of the corresponding elements of the computed transform.
6:     $\mathbf{init}$ – Character(1)Input
7:     $\mathbf{trign1}\left(1\right)$ – Real (Kind=nag_wp) arrayInput/Output
8:     $\mathbf{trign2}\left(1\right)$ – Real (Kind=nag_wp) arrayInput/Output
9:     $\mathbf{trign3}\left(1\right)$ – Real (Kind=nag_wp) arrayInput/Output
10:   $\mathbf{work}\left(1\right)$ – Real (Kind=nag_wp) arrayOutput
These arguments are no longer accessed by c06fxf.
11:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, 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, ${\mathbf{n1}}<1$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{n2}}<1$.
${\mathbf{ifail}}=3$
 On entry, ${\mathbf{n3}}<1$.
${\mathbf{ifail}}=4$
Not used at this Mark.
${\mathbf{ifail}}=5$
Not used at this Mark.
${\mathbf{ifail}}=6$
Not used at this Mark.
${\mathbf{ifail}}=7$
An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation 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

c06fxf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06fxf 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 routine. 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 ${n}_{3}$. c06fxf 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 (c06fxfe.f90)

### 10.2Program Data

Program Data (c06fxfe.d)

### 10.3Program Results

Program Results (c06fxfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017