NAG Library Routine Document

Integer, Intent (In)	::	m, n
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (Inout)	::	x(mn), trig(2n)
Real (Kind=nag_wp), Intent (Out)	::	work(m*n)
Character (1), Intent (In)	::	init

C Header Interface

#include <nagmk26.h>

void	c06fpf_ (const Integer m, const Integer n, double x[], const char init, double trig[], double work[], Integer ifail, const Charlen length_init)

3

Description

Given

m

sequences of

n

real data values

x_{j}^{p}

, for

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

and

p = 1, 2, \dots, m

, c06fpf simultaneously calculates the Fourier transforms of all the sequences defined by

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

(Note the scale factor

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

in this definition.)

The transformed values

{\hat{z}}_{k}^{p}

are complex, but for each value of

p

the

{\hat{z}}_{k}^{p}

form a Hermitian sequence (i.e.,

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

is the complex conjugate of

{\hat{z}}_{k}^{p}

), so they are completely determined by

m n

real numbers (see also the C06 Chapter Introduction).

The discrete Fourier transform is sometimes defined using a positive sign in the exponential term:

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

To compute this form, this routine should be followed by forming the complex conjugates of the

{\hat{z}}_{k}^{p}

; that is

x (k) = - x (k)

, for

k = (n / 2 + 1) \times m + 1, \dots, m \times n

The routine 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). Special coding is provided for the factors

2

3

4

5

and

6

. This routine is designed to be particularly efficient on vector processors, and it becomes especially fast as

m

, the number of transforms to be computed in parallel, increases.

4

References

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

Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys. 52 340–350

5

Arguments

1: $m$ – IntegerInput

On entry:

m

, the number of sequences to be transformed.

Constraint:

m \geq 1

2: $n$ – IntegerInput

On entry:

n

, the number of real values in each sequence.

Constraint:

n \geq 1

3: $x (m \times n)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: the data must be stored in x as if in a two-dimensional array of dimension

(1 : m, 0 : n - 1)

; each of the

m

sequences is stored in a row of the array. In other words, if the data values of the

p

th sequence to be transformed are denoted by

x_{j}^{p}

, for

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

, the

m n

elements of the array x must contain the values

x_{0}^{1}, x_{0}^{2}, \dots, x_{0}^{m}, x_{1}^{1}, x_{1}^{2}, \dots, x_{1}^{m}, \dots, x_{n - 1}^{1}, x_{n - 1}^{2}, \dots, x_{n - 1}^{m} .

On exit: the

m

discrete Fourier transforms stored as if in a two-dimensional array of dimension

(1 : m, 0 : n - 1)

. Each of the

m

transforms is stored in a row of the array in Hermitian form, overwriting the corresponding original sequence. If the

n

components of the discrete Fourier transform

{\hat{z}}_{k}^{p}

are written as

a_{k}^{p} + i b_{k}^{p}

, then for

0 \leq k \leq n / 2

a_{k}^{p}

is contained in

x (p, k)

, and for

1 \leq k \leq (n - 1) / 2

b_{k}^{p}

is contained in

x (p, n - k)

. (See also Section 2.1.2 in the C06 Chapter Introduction.)

4: $init$ – Character(1)Input

On entry: indicates whether trigonometric coefficients are to be calculated.

$init ='I'$: Calculate the required trigonometric coefficients for the given value of $n$ , and store in the array trig.
$init ='S'$ or $'R'$: The required trigonometric coefficients are assumed to have been calculated and stored in the array trig in a prior call to one of c06fpf or c06fqf. The routine performs a simple check that the current value of $n$ is consistent with the values stored in trig.

Constraint:

init ='I'

'S'

'R'

5: $trig (2 \times n)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: if

init ='S'

'R'

, trig must contain the required trigonometric coefficients that have been previously calculated. Otherwise trig need not be set.

On exit: contains the required coefficients (computed by the routine if

init ='I'

6: $work (m \times n)$ – Real (Kind=nag_wp) arrayWorkspace

7: $ifail$ – 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, $m = 〈value〉$ .
Constraint: $m \geq 1$ .

$ifail = 2$: On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .

$ifail = 3$: On entry, $init = 〈value〉$ .
Constraint: $init ='I'$ , $'S'$ or $'R'$ .

$ifail = 5$: On entry, $init = 〈value〉$ but n and trig array incompatible.

$ifail = - 99$: 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

c06fpf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

c06fpf 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 by c06fpf is approximately proportional to

n m \log (n)

, but also depends on the factors of

n

. c06fpf is fastest if the only prime factors of

n

are

2

3

and

5

, and is particularly slow if

n

is a large prime, or has large prime factors.

10

Example

This example reads in sequences of real data values and prints their discrete Fourier transforms (as computed by c06fpf). The Fourier transforms are expanded into full complex form using and printed. Inverse transforms are then calculated by conjugating and calling c06fqf showing that the original sequences are restored.

NAG Library Routine Document

c06fpf (withdraw_fft_real_1d_multi_rfmt)

▸▿ 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