# NAG Library Function Document

## 1Purpose

nag_sum_fft_real_2d (c06pvc) computes the two-dimensional discrete Fourier transform of a bivariate sequence of real data values.

## 2Specification

 #include #include
 void nag_sum_fft_real_2d (Integer m, Integer n, const double x[], Complex y[], NagError *fail)

## 3Description

nag_sum_fft_real_2d (c06pvc) 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 nag_sum_fft_real_2d (c06pvc) followed by a call of nag_sum_fft_hermitian_2d (c06pwc) will restore the original data.
This function performs 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:    $\mathbf{x}\left[{\mathbf{m}}×{\mathbf{n}}\right]$const doubleInput
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 ${j}_{2}=0,1,\dots ,n-1$.
4:    $\mathbf{y}\left[\left({\mathbf{m}}/2+1\right)×{\mathbf{n}}\right]$ComplexOutput
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 ${k}_{2}=0,1,\dots ,n-1$. Note the first dimension is cut roughly by half to remove the redundant information due to conjugate symmetry.
5:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_INTERNAL_ERROR
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.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 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 nag_sum_fft_real_2d (c06pvc) and a backward transform using nag_sum_fft_hermitian_2d (c06pwc), and comparing the results with the original sequence (in exact arithmetic they would be identical).

## 8Parallelism and Performance

nag_sum_fft_real_2d (c06pvc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_sum_fft_real_2d (c06pvc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The time taken by nag_sum_fft_real_2d (c06pvc) is approximately proportional to $mn\mathrm{log}\left(mn\right)$, but also depends on the factors of $m$ and $n$. nag_sum_fft_real_2d (c06pvc) 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 nag_sum_fft_real_2d (c06pvc). 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 nag_sum_fft_real_2d (c06pvc). Inverse transforms are then calculated by calling nag_sum_fft_hermitian_2d (c06pwc) showing that the original sequences are restored.

### 10.1Program Text

Program Text (c06pvce.c)

### 10.2Program Data

Program Data (c06pvce.d)

### 10.3Program Results

Program Results (c06pvce.r)

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