C06 Chapter Contents
C06 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentC06PYF

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

C06PYF computes the three-dimensional discrete Fourier transform of a trivariate sequence of real data values.

## 2  Specification

 SUBROUTINE C06PYF ( N1, N2, N3, X, Y, IFAIL)
 INTEGER N1, N2, N3, IFAIL REAL (KIND=nag_wp) X(N1*N2*N3) COMPLEX (KIND=nag_wp) Y((N1/2+1)*N2*N3)

## 3  Description

C06PYF computes the three-dimensional discrete Fourier transform of a trivariate sequence of real data values ${x}_{{j}_{1}{j}_{2}{j}_{3}}$, 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
 $z^ k1 k2 k3 = 1 n1 n2 n3 ∑ j1=0 n1-1 ∑ j2=0 n2-1 ∑ j3=0 n3-1 x j1 j2 j3 × exp -2πi j1 k1 n1 + j2 k2 n2 + j3 k3 n3 ,$
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 transformed values ${\stackrel{^}{z}}_{{k}_{1}{k}_{2}{k}_{3}}$ are complex. Because of conjugate symmetry (i.e., ${\stackrel{^}{z}}_{{k}_{1}{k}_{2}{k}_{3}}$ is the complex conjugate of ${\stackrel{^}{z}}_{\left({n}_{1}-{k}_{1}\right){k}_{2}{k}_{3}}$), only slightly more than half of the Fourier coefficients need to be stored in the output.
A call of C06PYF followed by a call of C06PZF will restore the original data.
This routine calls C06PQF and C06PRF to perform multiple one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm in Brigham (1974) and Temperton (1983).

## 4  References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys. 52 340–350

## 5  Parameters

1:     N1 – INTEGERInput
On entry: ${n}_{1}$, the first dimension of the transform.
Constraint: ${\mathbf{N1}}\ge 1$.
2:     N2 – INTEGERInput
On entry: ${n}_{2}$, the second dimension of the transform.
Constraint: ${\mathbf{N2}}\ge 1$.
3:     N3 – INTEGERInput
On entry: ${n}_{3}$, the third dimension of the transform.
Constraint: ${\mathbf{N3}}\ge 1$.
4:     X(${\mathbf{N1}}×{\mathbf{N2}}×{\mathbf{N3}}$) – REAL (KIND=nag_wp) arrayInput
On entry: the real input dataset $x$, where ${x}_{{j}_{1}{j}_{2}{j}_{3}}$ is stored in ${\mathbf{X}}\left({j}_{3}×{n}_{1}{n}_{2}+{j}_{2}×{n}_{1}+{j}_{1}+1\right)$, for ${j}_{1}=0,1,\dots ,{n}_{1}-1$, ${j}_{2}=0,1,\dots ,{n}_{2}-1$ and ${j}_{3}=0,1,\dots ,{n}_{3}-1$. That is, if X is regarded as a three-dimensional array of dimension $\left(0:{\mathbf{N1}}-1,0:{\mathbf{N2}}-1,0:{\mathbf{N3}}-1\right)$, then ${\mathbf{X}}\left({j}_{1},{j}_{2},{j}_{3}\right)$ must contain ${x}_{{j}_{1}{j}_{2}{j}_{3}}$.
5:     Y($\left({\mathbf{N1}}/2+1\right)×{\mathbf{N2}}×{\mathbf{N3}}$) – COMPLEX (KIND=nag_wp) arrayOutput
On exit: the complex output dataset $\stackrel{^}{z}$, where ${\stackrel{^}{z}}_{{k}_{1}{k}_{2}{k}_{3}}$ is stored in ${\mathbf{Y}}\left({k}_{3}×\left({n}_{1}/2+1\right){n}_{2}+{k}_{2}×\left({n}_{1}/2+1\right)+{k}_{1}+1\right)$, for ${k}_{1}=0,1,\dots ,{n}_{1}/2$, ${k}_{2}=0,1,\dots ,{n}_{2}-1$ and ${k}_{3}=0,1,\dots ,{n}_{3}-1$. That is, if Y is regarded as a three-dimensional array of dimension $\left(0:{\mathbf{N1}}/2,0:{\mathbf{N2}}-1,0:{\mathbf{N3}}-1\right)$, then ${\mathbf{Y}}\left({k}_{1},{k}_{2},{k}_{3}\right)$ contains ${\stackrel{^}{z}}_{{k}_{1}{k}_{2}{k}_{3}}$. Note the first dimension is cut roughly by half to remove the redundant information due to conjugate symmetry.
6:     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, ${\mathbf{N1}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{N1}}\ge 1$.
${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{N2}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{N2}}\ge 1$.
${\mathbf{IFAIL}}=3$
On entry, ${\mathbf{N3}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{N3}}\ge 1$.
${\mathbf{IFAIL}}=4$
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.
${\mathbf{IFAIL}}=-999$
Dynamic memory allocation failed.

## 7  Accuracy

Some indication of accuracy can be obtained by performing a forward transform using C06PYF and a backward transform using C06PZF, and comparing the results with the original sequence (in exact arithmetic they would be identical).

The time taken by C06PYF 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 factors of ${n}_{1}$, ${n}_{2}$ and ${n}_{3}$. C06PYF is fastest if the only prime factors of ${n}_{1}$, ${n}_{2}$ and ${n}_{3}$ are $2$, $3$ and $5$, and is particularly slow if one of the dimensions is a large prime, or has large prime factors.
Workspace is internally allocated by C06PYF. The total size of these arrays is approximately proportional to ${n}_{1}{n}_{2}{n}_{3}$.

## 9  Example

This example reads in a trivariate sequence of real data values and prints their discrete Fourier transforms as computed by C06PYF. Inverse transforms are then calculated by calling C06PZF showing that the original sequences are restored.

### 9.1  Program Text

Program Text (c06pyfe.f90)

### 9.2  Program Data

Program Data (c06pyfe.d)

### 9.3  Program Results

Program Results (c06pyfe.r)