# NAG Library Routine Document

## 1Purpose

c06pvf computes the two-dimensional discrete Fourier transform of a bivariate sequence of real data values.

## 2Specification

Fortran Interface
 Subroutine c06pvf ( m, n, x, y,
 Integer, Intent (In) :: m, n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x(m*n) Complex (Kind=nag_wp), Intent (Out) :: y((m/2+1)*n)
#include nagmk26.h
 void c06pvf_ (const Integer *m, const Integer *n, const double x[], Complex y[], Integer *ifail)

## 3Description

c06pvf computes the two-dimensional discrete Fourier transform of a bivariate sequence of real data values ${x}_{{j}_{1}{j}_{2}}$, for ${j}_{1}=0,1,\dots ,m-1$ and ${j}_{2}=0,1,\dots ,n-1$.
The discrete Fourier transform is here defined by
 $z^ k1 k2 = 1mn ∑ j1=0 m-1 ∑ j2=0 n-1 x j1 j2 × exp -2πi j1 k1 m + j2 k2 n ,$
where ${k}_{1}=0,1,\dots ,m-1$ and ${k}_{2}=0,1,\dots ,n-1$. (Note the scale factor of $\frac{1}{\sqrt{mn}}$ in this definition.)
The transformed values ${\stackrel{^}{z}}_{{k}_{1}{k}_{2}}$ are complex. Because of conjugate symmetry (i.e., ${\stackrel{^}{z}}_{{k}_{1}{k}_{2}}$ is the complex conjugate of ${\stackrel{^}{z}}_{\left(m-{k}_{1}\right)\left(n-{k}_{2}\right)}$), only slightly more than half of the Fourier coefficients need to be stored in the output.
A call of c06pvf followed by a call of c06pwf 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).

## 4References

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

## 5Arguments

1:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the first dimension of the transform.
Constraint: ${\mathbf{m}}\ge 1$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the second dimension of the transform.
Constraint: ${\mathbf{n}}\ge 1$.
3:     $\mathbf{x}\left({\mathbf{m}}×{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the real input dataset $x$, where ${x}_{{j}_{1}{j}_{2}}$ is stored in ${\mathbf{x}}\left({j}_{2}×m+{j}_{1}\right)$, for ${j}_{1}=0,1,\dots ,m-1$ and ${j}_{2}=0,1,\dots ,n-1$. That is, if x is regarded as a two-dimensional array of dimension $\left(0:{\mathbf{m}}-1,0:{\mathbf{n}}-1\right)$, ${\mathbf{x}}\left({j}_{1},{j}_{2}\right)$ must contain ${x}_{{j}_{1}{j}_{2}}$.
4:     $\mathbf{y}\left(\left({\mathbf{m}}/2+1\right)×{\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayOutput
On exit: the complex output dataset $\stackrel{^}{z}$, where ${\stackrel{^}{z}}_{{k}_{1}{k}_{2}}$ is stored in ${\mathbf{y}}\left({k}_{2}×\left(m/2+1\right)+{k}_{1}\right)$, for ${k}_{1}=0,1,\dots ,m/2$ and ${k}_{2}=0,1,\dots ,n-1$. That is, if y is regarded as a two-dimensional array of dimension $\left(0:{\mathbf{m}}/2,0:{\mathbf{n}}-1\right)$, ${\mathbf{y}}\left({k}_{1},{k}_{2}\right)$ contains ${\stackrel{^}{z}}_{{k}_{1}{k}_{2}}$. Note the first dimension is cut roughly by half to remove the redundant information due to conjugate symmetry.
5:     $\mathbf{ifail}$ – 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, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=3$
An internal error has occurred in this routine. Check the routine call and any array sizes. If the call is correct then please contact NAG for assistance.
${\mathbf{ifail}}=-99$
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 forward transform using c06pvf and a backward transform using c06pwf, and comparing the results with the original sequence (in exact arithmetic they would be identical).

## 8Parallelism and Performance

c06pvf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06pvf 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 by c06pvf is approximately proportional to $mn\mathrm{log}\left(mn\right)$, but also depends on the factors of $m$ and $n$. c06pvf is fastest if the only prime factors of $m$ and $n$ are $2$, $3$ and $5$, and is particularly slow if $m$ or $n$ is a large prime, or has large prime factors.
Workspace is internally allocated by c06pvf. The total size of these arrays is approximately proportional to $mn$.

## 10Example

This example reads in a bivariate sequence of real data values and prints their discrete Fourier transforms as computed by c06pvf. Inverse transforms are then calculated by calling c06pwf showing that the original sequences are restored.

### 10.1Program Text

Program Text (c06pvfe.f90)

### 10.2Program Data

Program Data (c06pvfe.d)

### 10.3Program Results

Program Results (c06pvfe.r)

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