# NAG Library Routine Document

## 1Purpose

c06pxf computes the three-dimensional discrete Fourier transform of a trivariate sequence of complex data values (using complex data type).

## 2Specification

Fortran Interface
 Subroutine c06pxf ( n1, n2, n3, x, work,
 Integer, Intent (In) :: n1, n2, n3 Integer, Intent (Inout) :: ifail Complex (Kind=nag_wp), Intent (Inout) :: x(n1*n2*n3), work(*) Character (1), Intent (In) :: direct
C Header Interface
#include nagmk26.h
 void c06pxf_ (const char *direct, const Integer *n1, const Integer *n2, const Integer *n3, Complex x[], Complex work[], Integer *ifail, const Charlen length_direct)

## 3Description

c06pxf 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 c06pxf with ${\mathbf{direct}}=\text{'F'}$ followed by a call with ${\mathbf{direct}}=\text{'B'}$ will restore the original data.
This routine performs multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm (see Brigham (1974)).

## 4References

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:     $\mathbf{direct}$ – Character(1)Input
On entry: if the forward transform as defined in Section 3 is to be computed, direct must be set equal to 'F'.
If the backward transform is to be computed, direct must be set equal to 'B'.
Constraint: ${\mathbf{direct}}=\text{'F'}$ or $\text{'B'}$.
2:     $\mathbf{n1}$ – IntegerInput
On entry: ${n}_{1}$, the first dimension of the transform.
Constraint: ${\mathbf{n1}}\ge 1$.
3:     $\mathbf{n2}$ – IntegerInput
On entry: ${n}_{2}$, the second dimension of the transform.
Constraint: ${\mathbf{n2}}\ge 1$.
4:     $\mathbf{n3}$ – IntegerInput
On entry: ${n}_{3}$, the third dimension of the transform.
Constraint: ${\mathbf{n3}}\ge 1$.
5:     $\mathbf{x}\left({\mathbf{n1}}×{\mathbf{n2}}×{\mathbf{n3}}\right)$ – 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, ${z}_{{j}_{1}{j}_{2}{j}_{3}}$ is stored in ${\mathbf{x}}\left(1+{j}_{1}+{n}_{1}{j}_{2}+{n}_{1}{n}_{2}{j}_{3}\right)$.
On exit: the corresponding elements of the computed transform.
6:     $\mathbf{work}\left(*\right)$ – Complex (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array work must be at least ${\mathbf{n1}}×{\mathbf{n2}}×{\mathbf{n3}}+{\mathbf{n1}}+{\mathbf{n2}}+{\mathbf{n3}}+45$.
The workspace requirements as documented for c06pxf 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 values of n1, n2 and n3 with this implementation.
7:     $\mathbf{ifail}$ – 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, ${\mathbf{n3}}<1$.
${\mathbf{ifail}}=4$
 On entry, ${\mathbf{direct}}\ne \text{'F'}$ or $\text{'B'}$.
${\mathbf{ifail}}=8$
An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
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

c06pxf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pxf 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.

## 9Further Comments

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 ${n}_{3}$. c06pxf 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 (c06pxfe.f90)

### 10.2Program Data

Program Data (c06pxfe.d)

### 10.3Program Results

Program Results (c06pxfe.r)

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