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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_sum_fft_complex_1d (c06pc)

## Purpose

nag_sum_fft_complex_1d (c06pc) calculates the discrete Fourier transform of a sequence of n$n$ complex data values (using complex data type).

## Syntax

[x, ifail] = c06pc(direct, x, 'n', n)
[x, ifail] = nag_sum_fft_complex_1d(direct, x, 'n', n)

## Description

Given a sequence of n$n$ complex data values zj ${z}_{\mathit{j}}$, for j = 0,1,,n1$\mathit{j}=0,1,\dots ,n-1$, nag_sum_fft_complex_1d (c06pc) calculates their (forward or backward) discrete Fourier transform (DFT) defined by
 n − 1 ẑk = 1/(sqrt(n)) ∑ zj × exp( ± i(2πjk)/n),  k = 0,1, … ,n − 1. j = 0
$z^k = 1n ∑ j=0 n-1 zj × exp( ±i 2πjk n ) , k= 0, 1, …, n-1 .$
(Note the scale factor of 1/(sqrt(n)) $\frac{1}{\sqrt{n}}$ 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 (c06pc) with direct = 'F'${\mathbf{direct}}=\text{'F'}$ followed by a call with direct = 'B'${\mathbf{direct}}=\text{'B'}$ will restore the original data.
nag_sum_fft_complex_1d (c06pc) 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$n$ is a large prime number or if n$n$ 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.

## 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

## Parameters

### Compulsory Input Parameters

1:     direct – string (length ≥ 1)
If the forward transform as defined in Section [Description] is to be computed, then direct must be set equal to 'F'.
If the backward transform is to be computed then direct must be set equal to 'B'.
Constraint: direct = 'F'${\mathbf{direct}}=\text{'F'}$ or 'B'$\text{'B'}$.
2:     x(n) – complex array
n, the dimension of the array, must satisfy the constraint n1${\mathbf{n}}\ge 1$.
If x is declared with bounds (0 : n1)$\left(0:{\mathbf{n}}-1\right)$ in the function from which nag_sum_fft_complex_1d (c06pc) is called, then x(j)${\mathbf{x}}\left(\mathit{j}\right)$ must contain zj${z}_{\mathit{j}}$, for j = 0,1,,n1$\mathit{j}=0,1,\dots ,n-1$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array x.
n$n$, the number of data values. The total number of prime factors of n, counting repetitions, must not exceed 30$30$.
Constraint: n1${\mathbf{n}}\ge 1$.

work

### Output Parameters

1:     x(n) – complex array
The components of the discrete Fourier transform. If x is declared with bounds (0 : n1)$\left(0:{\mathbf{n}}-1\right)$ in the function from which nag_sum_fft_complex_1d (c06pc) is called, then k${\stackrel{^}{z}}_{k}$ is contained in x(k)${\mathbf{x}}\left(k\right)$, for 0kn1$0\le k\le n-1$.
2:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, n < 1${\mathbf{n}}<1$.
ifail = 2${\mathbf{ifail}}=2$
 On entry, direct ≠ 'F'${\mathbf{direct}}\ne \text{'F'}$ or 'B'$\text{'B'}$.
ifail = 3${\mathbf{ifail}}=3$
 On entry, n has more than 30$30$ prime factors.
ifail = 4${\mathbf{ifail}}=4$
An unexpected error has occurred in an internal call. Check all function calls and array dimensions. Seek expert help.

## 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 × log(n)$n×\mathrm{log}\left(n\right)$, but also depends on the factorization of n$n$. nag_sum_fft_complex_1d (c06pc) is faster if the only prime factors of n$n$ are 2$2$, 3$3$ or 5$5$; and fastest of all if n$n$ is a power of 2$2$. This function requires a workspace of size 2n + 15$2n+15$ which is internally allocated. When the Bluestein’s FFT algorithm is in use, an additional workspace of size approximately 8n$8n$ is also allocated.

## Example

```function nag_sum_fft_complex_1d_example
direct = 'F';
x = [ 0.34907 - 0.37168i;
0.5489 - 0.35669i;
0.74776 - 0.31175i;
0.94459 - 0.23702i;
1.1385 - 0.13274i;
1.3285 + 0.00074i;
1.5137 + 0.16298i];
[xOut, ifail] = nag_sum_fft_complex_1d(direct, x)
```
```

xOut =

2.4836 - 0.4710i
-0.5518 + 0.4968i
-0.3671 + 0.0976i
-0.2877 - 0.0586i
-0.2251 - 0.1748i
-0.1483 - 0.3084i
0.0198 - 0.5650i

ifail =

0

```
```function c06pc_example
direct = 'F';
x = [ 0.34907 - 0.37168i;
0.5489 - 0.35669i;
0.74776 - 0.31175i;
0.94459 - 0.23702i;
1.1385 - 0.13274i;
1.3285 + 0.00074i;
1.5137 + 0.16298i];
[xOut, ifail] = c06pc(direct, x)
```
```

xOut =

2.4836 - 0.4710i
-0.5518 + 0.4968i
-0.3671 + 0.0976i
-0.2877 - 0.0586i
-0.2251 - 0.1748i
-0.1483 - 0.3084i
0.0198 - 0.5650i

ifail =

0

```