NAG Library Routine Document

C06EAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

C06EAF calculates the discrete Fourier transform of a sequence of

n

real data values. (No extra workspace required.)

2 Specification

SUBROUTINE C06EAF (

X, N, IFAIL)

INTEGER	N, IFAIL
REAL (KIND=nag_wp)	X(N)

3 Description

Given a sequence of

n

real data values

x_{j}

, for

j = 0, 1, \dots, n - 1

, C06EAF calculates their discrete Fourier transform defined by

{\hat{z}}_{k} = \frac{1}{\sqrt{n}} \sum_{j = 0}^{n - 1} x_{j} \times \exp (- i \frac{2 π j k}{n}), k = 0, 1, \dots, n - 1 .

(Note the scale factor of

\frac{1}{\sqrt{n}}

in this definition.) The transformed values

{\hat{z}}_{k}

are complex, but they form a Hermitian sequence (i.e.,

{\hat{z}}_{n - k}

is the complex conjugate of

{\hat{z}}_{k}

), so they are completely determined by

n

real numbers (see also the C06 Chapter Introduction).

To compute the inverse discrete Fourier transform defined by

{\hat{w}}_{k} = \frac{1}{\sqrt{n}} \sum_{j = 0}^{n - 1} x_{j} \times \exp (+ i \frac{2 π j k}{n}),

this routine should be followed by a call of C06GBF to form the complex conjugates of the

{\hat{z}}_{k}

C06EAF 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 C06EAF is called, then $X (j)$ must contain $x_{j}$ , for $j = 0, 1, \dots, n - 1$ .

On exit: the discrete Fourier transform stored in Hermitian form. If the components of the transform ${\hat{z}}_{k}$ are written as $a_{k} + i b_{k}$ , and if X is declared with bounds $(0 : N - 1)$ in the subroutine from which C06EAF is called, then for $0 \leq k \leq n / 2$ , $a_{k}$ is contained in $X (k)$ , and for $1 \leq k \leq (n - 1) / 2$ , $b_{k}$ is contained in $X (n - k)$ . (See also Section 2.1.2 in the C06 Chapter Introduction and Section 9.)
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: $N > 1$ .
3: 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

- 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,

N \leq 1

$IFAIL = 4$: 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 \times \log n

, but also depends on the factorization of

n

. C06EAF is faster if the only prime factors of

n

are

2

3

5

; and fastest of all if

n

is a power of

2

On the other hand, C06EAF 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

, C06FAF (which requires additional real workspace) is considerably faster.

9 Example

This example reads in a sequence of real data values and prints their discrete Fourier transform (as computed by C06EAF), after expanding it from Hermitian form into a full complex sequence. It then performs an inverse transform using C06GBF followed by C06EBF, and prints the sequence so obtained alongside the original data values.

NAG Library Routine DocumentC06EAF