c06 Chapter Contents
c06 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_sum_fft_complex_2d (c06puc)

## 1  Purpose

nag_sum_fft_complex_2d (c06puc) computes the two-dimensional discrete Fourier transform of a bivariate sequence of complex data values (using complex data type).

## 2  Specification

 #include #include
 void nag_sum_fft_complex_2d (Nag_TransformDirection direct, Integer m, Integer n, Complex x[], NagError *fail)

## 3  Description

nag_sum_fft_complex_2d (c06puc) computes the two-dimensional discrete Fourier transform of a bivariate sequence of complex data values ${z}_{{\mathit{j}}_{1}{\mathit{j}}_{2}}$, for ${j}_{1}=0,1,\dots ,m-1$ and ${j}_{2}=0,1,\dots ,n-1$.
The discrete Fourier transform is here defined by
 $z^ k1 k2 = 1mn ∑ j1=0 m-1 ∑ j2=0 n-1 z j1 j2 × exp ±2πi j1 k1 m + j2 k2 n ,$
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 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 nag_sum_fft_complex_2d (c06puc) with ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$ followed by a call with ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$ 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).

## 4  References

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

## 5  Arguments

1:    $\mathbf{direct}$Nag_TransformDirectionInput
On entry: if the forward transform as defined in Section 3 is to be computed, then direct must be set equal to $\mathrm{Nag_ForwardTransform}$.
If the backward transform is to be computed then direct must be set equal to $\mathrm{Nag_BackwardTransform}$.
Constraint: ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$ or $\mathrm{Nag_BackwardTransform}$.
2:    $\mathbf{m}$IntegerInput
On entry: $m$, the first dimension of the transform.
Constraint: ${\mathbf{m}}\ge 1$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the second dimension of the transform.
Constraint: ${\mathbf{n}}\ge 1$.
4:    $\mathbf{x}\left[{\mathbf{m}}×{\mathbf{n}}\right]$ComplexInput/Output
On entry: the complex data values. ${\mathbf{x}}\left[{\mathbf{m}}×{\mathit{j}}_{2}+{\mathit{j}}_{1}\right]$ must contain ${z}_{{\mathit{j}}_{1}{\mathit{j}}_{2}}$, for ${j}_{1}=0,1,\dots ,{\mathbf{m}}-1$ and ${j}_{2}=0,1,\dots ,{\mathbf{n}}-1$.
On exit: the corresponding elements of the computed transform.
5:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
$〈\mathit{\text{value}}〉$ is an invalid value of direct.
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.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

## 7  Accuracy

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).

## 8  Parallelism and Performance

nag_sum_fft_complex_2d (c06puc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_sum_fft_complex_2d (c06puc) 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 is approximately proportional to $mn×\mathrm{log}\left(mn\right)$, but also depends on the factorization of the individual dimensions $m$ and $n$. nag_sum_fft_complex_2d (c06puc) is faster if the only prime factors are $2$, $3$ or $5$; and fastest of all if they are powers of $2$. This function internally allocates a workspace of $mn+n+m+30$ Complex values.

## 10  Example

This example reads in a bivariate sequence of complex data values and prints the two-dimensional Fourier transform. It then performs an inverse transform and prints the sequence so obtained, which may be compared to the original data values.

### 10.1  Program Text

Program Text (c06puce.c)

### 10.2  Program Data

Program Data (c06puce.d)

### 10.3  Program Results

Program Results (c06puce.r)