NAG Library Routine Document

c06pqf (fft_realherm_1d_multi_col)

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NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

NAG C Library Manual, Mark 26

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

c06pqf computes the discrete Fourier transforms of

m

sequences, each containing

n

real data values or a Hermitian complex sequence stored column-wise in a complex storage format.

2

Specification

Fortran Interface

Subroutine c06pqf (

direct, n, m, x, work, ifail)

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

C Header Interface

#include <nagmk26.h>

void	c06pqf_ (const char direct, const Integer n, const Integer m, double x[], double work[], Integer ifail, const Charlen length_direct)

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

, c06pqf 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 .

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 (since

{\hat{z}}_{0}^{p}

is real, as is

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

for

n

even).

Alternatively, given

m

Hermitian sequences of

n

complex data values

z_{j}^{p}

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

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

The transformed values

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

are real.

(Note the scale factor

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

in the above definition.)

A call of c06pqf with

direct ='F'

followed by a call with

direct ='B'

will restore the original data.

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

and

5

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: $direct$ – 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: $n$ – IntegerInput

On entry:

n

, the number of real or complex values in each sequence.

Constraint:

n \geq 1

3: $m$ – IntegerInput

On entry:

m

, the number of sequences to be transformed.

Constraint:

m \geq 1

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

On entry: the

m

real or Hermitian data sequences to be transformed.

if $direct ='F'$ , the $m$ real data sequences, $x^{p} = (x_{0}^{p}, x_{1}^{p}, \dots, x_{n - 1}^{p})$ , for $p = 1, 2, \dots, m$ , should be stored sequentially in x, with a stride of $n + 2$ between sequences.
if $direct ='B'$ , the $m$ Hermitian data sequences, ${\hat{z}}^{p} = ({\hat{z}}_{0}^{p}, {\hat{z}}_{1}^{p}, \dots, {\hat{z}}_{n / 2}^{p}) = (Re ({\hat{z}}_{0}^{p}), Im ({\hat{z}}_{0}^{p}), Re ({\hat{z}}_{1}^{p}), Im ({\hat{z}}_{1}^{p}), \dots, Re ({\hat{z}}_{n / 2}^{p}), Im ({\hat{z}}_{n / 2}^{p}))$ , for $p = 1, 2, \dots, m$ , should be stored sequentially in x, with a stride of $n + 2$ between sequences.

In other words:

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

On exit:

if $direct ='F'$ then the $m$ sequences, ${\hat{z}}^{p}$ , for $p = 1, 2, \dots, m$ stored as described on entry for $direct ='B'$
if $direct ='B'$ then the $m$ sequences, $x^{p}$ , for $p = 1, 2, \dots, m$ stored as described on entry for $direct ='F'$

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

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

(m + 2) \times n + 15

The workspace requirements as documented for c06pqf may be an overestimate in some implementations.

On exit:

work (1)

contains the minimum workspace required for the current values of m and n with this implementation.

6: $ifail$ – 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, $direct = 〈value〉$ .
Constraint: $direct ='F'$ or $'B'$ .

$ifail = 4$: 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

c06pqf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

c06pqf 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 c06pqf is approximately proportional to

n m \log (n)

, but also depends on the factors of

n

. c06pqf 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 c06pqf with

direct ='F'

), after expanding them from complex Hermitian form into a full complex sequences.

Inverse transforms are then calculated by calling c06pqf with

direct ='B'

showing that the original sequences are restored.

NAG Library Routine Document

c06pqf (fft_realherm_1d_multi_col)

▸▿ 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