c06 Chapter Contents
c06 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_fft_complex (c06ecc)

## 1  Purpose

nag_fft_complex (c06ecc) calculates the discrete Fourier transform of a sequence of $n$ complex data values.

## 2  Specification

 #include #include
 void nag_fft_complex (Integer n, double x[], double y[], NagError *fail)

## 3  Description

Given a sequence of $n$ complex data values ${z}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$, nag_fft_complex (c06ecc) calculates their discrete Fourier transform defined by
 $z^k = ak + ib k = 1n ∑ j=0 n-1 zj exp -i 2πjk n , k= 0, 1, …, n-1 .$
(Note the scale factor of $\frac{1}{\sqrt{n}}$ in this definition.)
To compute the inverse discrete Fourier transform defined by
 $w^k = 1n ∑ j=0 n-1 zj exp +i 2πjk n , ​ k= 0, 1, …, n-1 ,$
this function should be preceded and followed by calls of nag_conjugate_complex (c06gcc) to form the complex conjugates of the ${z}_{j}$ and the ${\stackrel{^}{z}}_{k}$.
nag_fft_complex (c06ecc) uses the fast Fourier transform (FFT) algorithm (see Brigham (1974)). There are some restrictions on the value of $n$ (see Section 5).

## 4  References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall

## 5  Arguments

1:     nIntegerInput
On entry: $n$, the number of data values.
Constraint: ${\mathbf{n}}>1$. The largest prime factor of n must not exceed 19, and the total number of prime factors of n, counting repetitions, must not exceed 20.
2:     x[n]doubleInput/Output
On entry: ${\mathbf{x}}\left[\mathit{j}\right]$ must contain ${x}_{\mathit{j}}$, the real part of ${z}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$.
On exit: the real parts ${a}_{\mathit{k}}$ of the components of the discrete Fourier transform. ${a}_{\mathit{k}}$ is contained in ${\mathbf{x}}\left[\mathit{k}\right]$, for $\mathit{k}=0,1,\dots ,n-1$.
3:     y[n]doubleInput/Output
On entry: ${\mathbf{y}}\left[\mathit{j}\right]$ must contain ${y}_{\mathit{j}}$, the imaginary part of ${z}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$.
On exit: the imaginary parts ${b}_{\mathit{k}}$ of the components of the discrete Fourier transform. ${b}_{\mathit{k}}$ is contained in ${\mathbf{y}}\left[\mathit{k}\right]$, for $\mathit{k}=0,1,\dots ,n-1$.
4:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_C06_FACTOR_GT
At least one of the prime factors of n is greater than 19.
NE_C06_TOO_MANY_FACTORS
n has more than 20 prime factors.
NE_INT_ARG_LE
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>1$.

## 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).

The time taken is approximately proportional to $n\mathrm{log}n$, but also depends on the factorization of $n$. nag_fft_complex (c06ecc) is somewhat faster than average if the only prime factors of $n$ are 2, 3 or 5; and fastest of all if $n$ is a power of 2.
On the other hand, nag_fft_complex (c06ecc) is particularly slow if $n$ has several unpaired prime factors, i.e., if the ‘square-free’ part of $n$ has several factors.

## 9  Example

This example reads in a sequence of complex data values and prints their discrete Fourier transform. It then performs an inverse transform using nag_fft_complex (c06ecc) and nag_conjugate_complex (c06gcc), and prints the sequence so obtained alongside the original data values.

### 9.1  Program Text

Program Text (c06ecce.c)

### 9.2  Program Data

Program Data (c06ecce.d)

### 9.3  Program Results

Program Results (c06ecce.r)