NAG Library Routine Document
C06EBF
1 Purpose
C06EBF calculates the discrete Fourier transform of a Hermitian sequence of $n$ complex data values. (No extra workspace required.)
2 Specification
INTEGER 
N, IFAIL 
REAL (KIND=nag_wp) 
X(N) 

3 Description
Given a Hermitian sequence of
$n$ complex data values
${z}_{\mathit{j}}$ (i.e., a sequence such that
${z}_{0}$ is real and
${z}_{n\mathit{j}}$ is the complex conjugate of
${z}_{\mathit{j}}$, for
$\mathit{j}=1,2,\dots ,n1$), C06EBF calculates their discrete Fourier transform defined by
(Note the scale factor of
$\frac{1}{\sqrt{n}}$ in this definition.) The transformed values
${\hat{x}}_{k}$ are purely real (see also the
C06 Chapter Introduction).
To compute the inverse discrete Fourier transform defined by
this routine should be preceded by a call of
C06GBF to form the complex conjugates of the
${z}_{j}$.
C06EBF 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 Parameters
 1: X(N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: the sequence to be transformed stored in Hermitian form. If the data values
${z}_{j}$ are written as
${x}_{j}+i{y}_{j}$, and if
X is declared with bounds
$\left(0:{\mathbf{N}}1\right)$ in the subroutine from which C06EBF is called, then for
$0\le j\le n/2$,
${x}_{j}$ is contained in
${\mathbf{X}}\left(j\right)$, and for
$1\le j\le \left(n1\right)/2$,
${y}_{j}$ is contained in
${\mathbf{X}}\left(nj\right)$. (See also
Section 2.1.2 in the C06 Chapter Introduction and
Section 9.)
On exit: the components of the discrete Fourier transform
${\hat{x}}_{k}$. If
X is declared with bounds
$\left(0:{\mathbf{N}}1\right)$ in the subroutine from which C06EBF is called, then
${\hat{x}}_{\mathit{k}}$ is stored in
${\mathbf{X}}\left(\mathit{k}\right)$, for
$\mathit{k}=0,1,\dots ,n1$.
 2: N – INTEGERInput
On entry:
$n$, the number of data values. The largest prime factor of
N must not exceed
$19$, and the total number of prime factors of
N, counting repetitions, must not exceed
$20$.
Constraint:
${\mathbf{N}}>1$.
 3: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ 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).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$

At least one of the prime factors of
N is greater than
$19$.
 ${\mathbf{IFAIL}}=2$

N has more than
$20$ prime factors.
 ${\mathbf{IFAIL}}=3$

On entry,  ${\mathbf{N}}\le 1$. 
 ${\mathbf{IFAIL}}=4$

An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.
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 $n\times \mathrm{log}n$, but also depends on the factorization of $n$. C06EBF is faster if the only prime factors of $n$ are $2$, $3$ or $5$; and fastest of all if $n$ is a power of $2$.
On the other hand, C06EBF is particularly slow if
$n$ has several unpaired prime factors, i.e., if the ‘squarefree’ part of
$n$ has several factors.
For such values of
$n$,
C06FBF (which requires an additional
$n$ elements of workspace) is considerably faster.
9 Example
This example reads in a sequence of real data values which is assumed to be a Hermitian sequence of complex data values stored in Hermitian form. The input sequence is expanded into a full complex sequence and printed alongside the original sequence. The discrete Fourier transform (as computed by C06EBF) is printed out. It then performs an inverse transform using
C06EAF and
C06GBF, and prints the sequence so obtained alongside the original data values.
9.1 Program Text
Program Text (c06ebfe.f90)
9.2 Program Data
Program Data (c06ebfe.d)
9.3 Program Results
Program Results (c06ebfe.r)