c06fq computes the discrete Fourier transforms of

m

Hermitian sequences, each containing

n

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

Syntax

C#
public static void c06fq( int m, int n, double[] x, string init, double[] trig, out int ifail )

Visual Basic
Public Shared Sub c06fq ( _ m As Integer, _ n As Integer, _ x As Double(), _ init As String, _ trig As Double(), _ <OutAttribute> ByRef ifail As Integer _ )

Visual C++
public: static void c06fq( int m, int n, array<double>^ x, String^ init, array<double>^ trig, [OutAttribute] int% ifail )

F#
static member c06fq : m : int * n : int * x : float[] * init : string * trig : float[] * ifail : int byref -> unit

Parameters

m: Type: System..::..Int32
On entry: $m$ , the number of sequences to be transformed.

Constraint: $m \geq 1$ .

n: Type: System..::..Int32
On entry: $n$ , the number of data values in each sequence.

Constraint: $n \geq 1$ .

x

Type: array<System..::..Double>[]()[][]

An array of size [

m \times n

]

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 - 1, j]

, and for

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

y_{j}^{p}

is contained in

x [p, n - j]

. (See also [] in the C06 class 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} .

init

Type: System..::..String

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 c06fp, c06fq or c06fr. The method performs a simple check that the current value of $n$ is consistent with the values stored in trig.

Constraint:

init = "I"

"S"

"R"

trig: Type: array<System..::..Double>[]()[][]
An array of size [ $2 \times n$ ]
On entry: if $init = "S"$ or $"R"$ , 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 method if $init = "I"$ ).

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

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

, c06fq 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 class).

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 method 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 method 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 method 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.

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

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$

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 method calls and array dimensions. Seek expert help.

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

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).

Parallelism and Performance

None.

Further Comments

The time taken by c06fq is approximately proportional to

n m \log (n)

, but also depends on the factors of

n

. c06fq 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.

Example

Example program (C#): c06fqe.cs

Example program data: c06fqe.d

Example program results: c06fqe.r

Syntax

Parameters

Description

References

Error Indicators and Warnings

Accuracy

Parallelism and Performance

Further Comments

Example

See Also