C06ECF (PDF version)
C06 Chapter Contents
C06 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

C06ECF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

C06ECF calculates the discrete Fourier transform of a sequence of n complex data values. (No extra workspace required.)

2  Specification

SUBROUTINE C06ECF ( X, Y, N, IFAIL)
INTEGER  N, IFAIL
REAL (KIND=nag_wp)  X(N), Y(N)

3  Description

Given a sequence of n complex data values zj, for j=0,1,,n-1, C06ECF 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 1n  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 calls of C06GCF to form the complex conjugates of the zj  and the z^k .
C06ECF 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: if X is declared with bounds 0:N-1 in the subroutine from which C06ECF is called, then Xj must contain xj, the real part of zj, for j=0,1,,n-1.
On exit: the real parts ak of the components of the discrete Fourier transform. If X is declared with bounds 0:N-1 in the subroutine from which C06ECF is called, then for 0 k n-1, ak is contained in Xk.
2:     Y(N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: if Y is declared with bounds 0:N-1 in the subroutine from which C06ECF is called, then Yj must contain yj, the imaginary part of zj, for j=0,1,,n-1.
On exit: the imaginary parts bk of the components of the discrete Fourier transform. If Y is declared with bounds 0:N-1 in the subroutine from which C06ECF is called, then for 0kn-1, bk is contained in Yk.
3:     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: N>1.
4:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ 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​ 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 -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry 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:
IFAIL=1
At least one of the prime factors of N is greater than 19.
IFAIL=2
N has more than 20 prime factors.
IFAIL=3
On entry,N1.
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).

8  Further Comments

The time taken is approximately proportional to n × logn, but also depends on the factorization of n. C06ECF 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, C06ECF is particularly slow if n has several unpaired prime factors, i.e., if the ‘square-free’ part of n has several factors. For such values of n, C06FCF (which requires an additional n real elements of workspace) is considerably faster.

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 C06ECF and C06GCF, and prints the sequence so obtained alongside the original data values.

9.1  Program Text

Program Text (c06ecfe.f90)

9.2  Program Data

Program Data (c06ecfe.d)

9.3  Program Results

Program Results (c06ecfe.r)


C06ECF (PDF version)
C06 Chapter Contents
C06 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012