NAG Library Routine Document

c06paf (fft_realherm_1d)

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NAG Library Manual, Mark 26

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C06 (sum) Chapter Contents

C06 (sum) Chapter Introduction

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

c06paf calculates the discrete Fourier transform of a sequence of

n

real data values or of a Hermitian sequence of

n

complex data values stored in compact form in a real array.

2

Specification

Fortran Interface

Subroutine c06paf (

direct, x, n, work, ifail)

Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (Inout)	::	x(n+2), work(*)
Character (1), Intent (In)	::	direct

C Header Interface

#include <nagmk26.h>

void	c06paf_ (const char direct, double x[], const Integer n, double work[], Integer *ifail, const Charlen length_direct)

3

Description

Given a sequence of

n

real data values

x_{j}

, for

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

, c06paf calculates their discrete Fourier transform (in the forward direction) 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 .

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 (since

{\hat{z}}_{0}

is real, as is

{\hat{z}}_{n / 2}

for

n

even).

Alternatively, given a Hermitian sequence of

n

complex data values

z_{j}

, this routine calculates their inverse (backward) discrete Fourier transform defined by

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

The transformed values

{\hat{x}}_{k}

are real.

(Note the scale factor of

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

in the above definitions.)

A call of c06paf with

direct ='F'

followed by a call with

direct ='B'

will restore the original data.

c06paf 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).

The same functionality is available using the forward and backward transform routine pair: c06pvf and c06pwf on setting

n = 1

. This pair use a different storage solution; real data is stored in a real array, while Hermitian data (the first unconjugated half) is stored in a complex array.

4

References

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

5

Arguments

1: $direct$ – Character(1)Input

On entry: if the forward transform as defined in Section 3 is to be computed, direct must be set equal to 'F'.

If the backward transform is to be computed, direct must be set equal to 'B'.

Constraint:

direct ='F'

'B'

2: $x (n + 2)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: if x is declared with bounds

(0 : n + 1)

in the subroutine from which c06paf is called:

if $direct ='F'$ , $x (j)$ must contain $x_{j}$ , for $j = 0, 1, \dots, n - 1$ ;
if $direct ='B'$ , $x (2 \times k)$ and $x (2 \times k + 1)$ must contain the real and imaginary parts respectively of $z_{k}$ , for $k = 0, 1, \dots, n / 2$ . (Note that for the sequence $z_{k}$ to be Hermitian, the imaginary part of $z_{0}$ , and of $z_{n / 2}$ for $n$ even, must be zero.)

On exit:

if $direct ='F'$ and x is declared with bounds $(0 : n + 1)$ , $x (2 \times k)$ and $x (2 \times k + 1)$ will contain the real and imaginary parts respectively of ${\hat{z}}_{k}$ , for $k = 0, 1, \dots, n / 2$ ;
if $direct ='B'$ and x is declared with bounds $(0 : n + 1)$ , $x (j)$ will contain ${\hat{x}}_{j}$ , for $j = 0, 1, \dots, n - 1$ .

3: $n$ – IntegerInput

On entry:

n

, the number of data values.

Constraint:

n \geq 1

4: $work (*)$ – Real (Kind=nag_wp) arrayWorkspace

Note: the dimension of the array work must be at least

3 \times n + 100

The workspace requirements as documented for c06paf may be an overestimate in some implementations.

On exit:

work (1)

contains the minimum workspace required for the current value of n with this implementation.

5: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 or 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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:

$ifail = 1$: On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .

$ifail = 2$: On entry, $direct = 〈value〉$ .
Constraint: $direct ='F'$ or $'B'$ .

$ifail = 3$: An internal error has occurred in this routine. Check the routine call and any array sizes. If the call is correct then please contact NAG for assistance.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

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

Parallelism and Performance

c06paf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

c06paf 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.

9

Further Comments

The time taken is approximately proportional to

n \times \log (n)

, but also depends on the factorization of

n

. c06paf is faster if the only prime factors of

n

are

2

3

5

; and fastest of all if

n

is a power of

2

10

Example

This example reads in a sequence of real data values and prints their discrete Fourier transform (as computed by c06paf with

direct ='F'

), after expanding it from complex Hermitian form into a full complex sequence. It then performs an inverse transform using c06paf with

direct ='B'

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

NAG Library Routine Document

c06paf (fft_realherm_1d)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results