nag_sum_fft_complex_1d (c06pcc) (PDF version)
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NAG Library Manual

# NAG Library Function Documentnag_sum_fft_complex_1d (c06pcc)

## 1  Purpose

nag_sum_fft_complex_1d (c06pcc) calculates the discrete Fourier transform of a sequence of $n$ complex data values (using complex data type).

## 2  Specification

 #include #include
 void nag_sum_fft_complex_1d (Nag_TransformDirection direct, Complex x[], Integer n, NagError *fail)

## 3  Description

Given a sequence of $n$ complex data values ${z}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$, nag_sum_fft_complex_1d (c06pcc) calculates their (forward or backward) discrete Fourier transform (DFT) defined by
 $z^k = 1n ∑ j=0 n-1 zj × exp ±i 2πjk n , k= 0, 1, …, n-1 .$
(Note the scale factor of $\frac{1}{\sqrt{n}}$ in this definition.) 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_1d (c06pcc) with ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$ followed by a call with ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$ will restore the original data.
nag_sum_fft_complex_1d (c06pcc) uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, which is described in Temperton (1983). If $n$ is a large prime number or if $n$ contains large prime factors, then the Fourier transform is performed using Bluestein's algorithm (see Bluestein (1968)), which expresses the DFT as a convolution that in turn can be efficiently computed using FFTs of highly composite sizes.

## 4  References

Bluestein L I (1968) A linear filtering approach to the computation of the discrete Fourier transform Northeast Electronics Research and Engineering Meeting Record 10 218–219
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:    $\mathbf{x}\left[{\mathbf{n}}\right]$ComplexInput/Output
On entry: ${\mathbf{x}}\left[\mathit{j}\right]$ must contain ${z}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$.
On exit: the components of the discrete Fourier transform. ${\stackrel{^}{z}}_{k}$ is contained in ${\mathbf{x}}\left[k\right]$, for $0\le k\le n-1$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of data values.
Constraint: ${\mathbf{n}}\ge 1$.
4:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error 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.
NE_BAD_PARAM
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
$〈\mathit{\text{value}}〉$ is an invalid value of direct.
NE_INT
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 unexpected error has been triggered by this function. Please contact NAG.
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.

## 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_1d (c06pcc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_sum_fft_complex_1d (c06pcc) 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.

## 9  Further Comments

The time taken is approximately proportional to $n×\mathrm{log}\left(n\right)$, but also depends on the factorization of $n$. nag_sum_fft_complex_1d (c06pcc) is faster if the only prime factors of $n$ are $2$, $3$ or $5$; and fastest of all if $n$ is a power of $2$. This function internally allocates a workspace of $2n+15$ Complex values. When the Bluestein's FFT algorithm is in use, an additional Complex workspace of size approximately $8n$ is allocated.

## 10  Example

This example reads in a sequence of complex data values and prints their discrete Fourier transform (as computed by nag_sum_fft_complex_1d (c06pcc) with ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$). It then performs an inverse transform using nag_sum_fft_complex_1d (c06pcc) with ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$, and prints the sequence so obtained alongside the original data values.

### 10.1  Program Text

Program Text (c06pcce.c)

### 10.2  Program Data

Program Data (c06pcce.d)

### 10.3  Program Results

Program Results (c06pcce.r)

nag_sum_fft_complex_1d (c06pcc) (PDF version)
c06 Chapter Contents
c06 Chapter Introduction
NAG Library Manual