NAG Library Routine Document

C06RCF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

C06RCF computes the discrete quarter-wave Fourier sine transforms of

m

sequences of real data values.

2 Specification

SUBROUTINE C06RCF (

DIRECT, M, N, X, WORK, IFAIL)

INTEGER	M, N, IFAIL
REAL (KIND=nag_wp)	X(M(N+2)), WORK()
CHARACTER(1)	DIRECT

3 Description

Given

m

sequences of

n

real data values

x_{j}^{p}

, for

j = 1, 2, \dots, n

and

p = 1, 2, \dots, m

, C06RCF simultaneously calculates the quarter-wave Fourier sine transforms of all the sequences defined by

{\hat{x}}_{k}^{p} = \frac{1}{\sqrt{n}} (\sum_{j = 1}^{n - 1} x_{j}^{p} \times \sin (j (2 k - 1) \frac{π}{2 n}) + \frac{1}{2} {(- 1)}^{k - 1} x_{n}^{p}),   if ​ DIRECT ='F',

or its inverse

x_{k}^{p} = \frac{2}{\sqrt{n}} \sum_{j = 1}^{n} {\hat{x}}_{j}^{p} \times \sin ((2 j - 1) k \frac{π}{2 n}),   if ​ DIRECT ='B',

where

k = 1, 2, \dots, n

and

p = 1, 2, \dots, m

(Note the scale factor

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

in this definition.)

A call of C06RCF with

DIRECT ='F'

followed by a call with

DIRECT ='B'

will restore the original data.

The transform calculated by this routine can be used to solve Poisson's equation when the solution is specified at the left boundary, and the derivative of the solution is specified at the right boundary (see Swarztrauber (1977)).

The routine uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, described in Temperton (1983), together with pre- and post-processing stages described in Swarztrauber (1982). Special coding is provided for the factors

2

3

4

and

5

4 References

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

Swarztrauber P N (1977) The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle SIAM Rev. 19(3) 490–501

Swarztrauber P N (1982) Vectorizing the FFT's Parallel Computation (ed G Rodrique) 51–83 Academic Press

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

5 Parameters

1: $DIRECT$ – CHARACTER(1)Input

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

If the backward transform is to be computed then DIRECT must be set equal to 'B'.

Constraint:

DIRECT ='F'

'B'

2: $M$ – INTEGERInput

On entry:

m

, the number of sequences to be transformed.

Constraint:

M \geq 1

3: $N$ – INTEGERInput

On entry:

n

, the number of real values in each sequence.

Constraint:

N \geq 1

4: $X (M \times (N + 2))$ – 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, 1 : N + 2)

; 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 = 1, 2, \dots, n

and

p = 1, 2, \dots, m

, then the first

m n

elements of the array X must contain the values

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

The

(n + 1)

th and

(n + 2)

th elements of each row

x_{n + 1}^{p}, x_{n + 2}^{p}

, for

p = 1, 2, \dots, m

, are required as workspace. These

2 m

elements may contain arbitrary values as they are set to zero by the routine.

On exit: the

m

quarter-wave sine transforms stored as if in a two-dimensional array of dimension

(1 : M, 1 : N + 2)

. Each of the

m

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

n

components of the

p

th quarter-wave sine transform are denoted by

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

, for

k = 1, 2, \dots, n

and

p = 1, 2, \dots, m

, then the

m (n + 2)

elements of the array X contain the values

{\hat{x}}_{1}^{1}, {\hat{x}}_{1}^{2}, \dots, {\hat{x}}_{1}^{m}, {\hat{x}}_{2}^{1}, {\hat{x}}_{2}^{2}, \dots, {\hat{x}}_{2}^{m}, \dots, {\hat{x}}_{n}^{1}, {\hat{x}}_{n}^{2}, \dots, {\hat{x}}_{n}^{m}, 0, 0, \dots, 0 (2 m times) .

5: $WORK (*)$ – REAL (KIND=nag_wp) arrayWorkspace

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

M \times N + 2 \times N + 2 \times M + 15

The workspace requirements as documented for C06RCF 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 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:

$IFAIL = 1$

On entry,

M < 1

$IFAIL = 2$

On entry,

N < 1

$IFAIL = 3$

On entry,

DIRECT \neq'F'

'B'

$IFAIL = 4$: An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction 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

C06RCF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

C06RCF 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 C06RCF is approximately proportional to

n m \log (n)

, but also depends on the factors of

n

. C06RCF 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 quarter-wave sine transforms as computed by C06RCF with

DIRECT ='F'

. It then calls the routine again with

DIRECT ='B'

and prints the results which may be compared with the original data.

NAG Library Routine DocumentC06RCF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

NAG Library Routine Document

C06RCF