NAG Library Routine Document

C06FQF

+− 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

C06FQF computes the discrete Fourier transforms of

m

Hermitian sequences, each containing

n

complex data values. This routine is designed to be particularly efficient on vector processors.

2 Specification

SUBROUTINE C06FQF (

M, N, X, INIT, TRIG, WORK, IFAIL)

INTEGER	M, N, IFAIL
REAL (KIND=nag_wp)	X(MN), TRIG(2N), WORK(M*N)
CHARACTER(1)	INIT

3 Description

Given

m

Hermitian sequences of

n

complex data values

z_{j}^{p}

, for

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

and

p = 1, 2, \dots, m

, C06FQF simultaneously calculates the Fourier transforms of all the sequences 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 .

(Note the scale factor

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

in this definition.)

The transformed values are purely real (see also the C06 Chapter Introduction).

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

{\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}) .

To compute this form, this routine should be preceded 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 Parameters

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 data 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 Hermitian form. If the

n

data values

z_{j}^{p}

are written as

x_{j}^{p} + i y_{j}^{p}

, then for

0 \leq j \leq n / 2

x_{j}^{p}

is contained in

X (p, j)

, and for

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

y_{j}^{p}

is contained in

X (p, n - j)

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

On exit: the components of 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 as a row of the array, overwriting the corresponding original sequence. If the

n

components of the discrete Fourier transform are denoted by

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

, for

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

, then the

m n

elements of the array X contain the values

{\hat{x}}_{0}^{1}, {\hat{x}}_{0}^{2}, \dots, {\hat{x}}_{0}^{m}, {\hat{x}}_{1}^{1}, {\hat{x}}_{1}^{2}, \dots, {\hat{x}}_{1}^{m}, \dots, {\hat{x}}_{n - 1}^{1}, {\hat{x}}_{n - 1}^{2}, \dots, {\hat{x}}_{n - 1}^{m} .

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, C06FQF or C06FRF. 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 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,

INIT \neq'I'

'S'

'R'

$IFAIL = 4$: Not used at this Mark.

$IFAIL = 5$

On entry,

INIT ='S'

'R'

, but the array TRIG and the current value of N are inconsistent.

$IFAIL = 6$: 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 by C06FQF is approximately proportional to

n m \log n

, but also depends on the factors of

n

. C06FQF 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.

9 Example

This example reads in sequences of real data values which are assumed to be Hermitian sequences of complex data stored in Hermitian form. The sequences are expanded into full complex form and printed. The discrete Fourier transforms are then computed (using C06FQF) and printed out. Inverse transforms are then calculated by conjugating and calling C06FPF showing that the original sequences are restored.

NAG Library Routine DocumentC06FQF

+− 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

NAG Library Routine Document

C06FQF