# NAG FL Interfacec06fcf (fft_​complex_​1d_​sep)

## 1Purpose

c06fcf calculates the discrete Fourier transform of a sequence of $n$ complex data values (using a work array for extra speed).

## 2Specification

Fortran Interface
 Subroutine c06fcf ( x, y, n, work,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Inout) :: x(n), y(n) Real (Kind=nag_wp), Intent (Out) :: work(n)
#include <nag.h>
 void c06fcf_ (double x[], double y[], const Integer *n, double work[], Integer *ifail)
The routine may be called by the names c06fcf or nagf_sum_fft_complex_1d_sep.

## 3Description

Given a sequence of $n$ complex data values ${z}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$, c06fcf calculates their discrete Fourier transform defined by
 $z^k = ak + i bk = 1n ∑ j=0 n-1 zj × exp -i 2πjk n , k= 0, 1, …, n-1 .$
(Note the scale factor of $\frac{1}{\sqrt{n}}$ in this definition.)
To compute the inverse discrete Fourier transform defined by
 $w^k = 1n ∑ j=0 n-1 zj × exp +i 2πjk n ,$
this routine should be preceded and followed by the complex conjugation of the data values and the transform (by negating the imaginary parts stored in $y$).
c06fcf uses the fast Fourier transform (FFT) algorithm (see Brigham (1974)).

## 4References

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

## 5Arguments

1: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input/Output
On entry: if x is declared with bounds $\left(0:{\mathbf{n}}-1\right)$ in the subroutine from which c06fcf is called, ${\mathbf{x}}\left(\mathit{j}\right)$ 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 ${a}_{k}$ of the components of the discrete Fourier transform. If x is declared with bounds $\left(0:{\mathbf{n}}-1\right)$ in the subroutine from which c06fcf is called, for $0\le k\le n-1$, ${a}_{k}$ is contained in ${\mathbf{x}}\left(k\right)$.
2: $\mathbf{y}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input/Output
On entry: if y is declared with bounds $\left(0:{\mathbf{n}}-1\right)$ in the subroutine from which c06fcf is called, ${\mathbf{y}}\left(\mathit{j}\right)$ 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 ${b}_{k}$ of the components of the discrete Fourier transform. If y is declared with bounds $\left(0:{\mathbf{n}}-1\right)$ in the subroutine from which c06fcf is called, then for $0\le k\le n-1$, ${b}_{k}$ is contained in ${\mathbf{y}}\left(k\right)$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the number of data values.
Constraint: ${\mathbf{n}}>1$.
4: $\mathbf{work}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Workspace
5: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=3$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>1$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

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).

## 8Parallelism and Performance

c06fcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c06fcf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The time taken is approximately proportional to $n×\mathrm{log}\left(n\right)$, but also depends on the factorization of $n$. c06fcf is faster if the only prime factors of $n$ are $2$, $3$ or $5$; and fastest of all if $n$ is a power of $2$.

## 10Example

This example reads in a sequence of complex data values and prints their discrete Fourier transform (as computed by c06fcf). It then performs an inverse transform using c06fcf, and prints the sequence so obtained alongside the original data values.

### 10.1Program Text

Program Text (c06fcfe.f90)

### 10.2Program Data

Program Data (c06fcfe.d)

### 10.3Program Results

Program Results (c06fcfe.r)