The first call of c06fqc must be preceded by a call to c06gzc to initialize the array trig with trigonometric coefficients according to the value of n.

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} \exp (+ 2 π i j k / n) .

For that form, this function should be preceded by a call to c06gqc to form the complex conjugates of the

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

The function uses a variant of the fast Fourier transform algorithm (Brigham (1974)) known as the Stockham self-sorting algorithm, which is described in Temperton (1983). Special code is included for the factors

2

3

4

, 5 and 6.

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: $m$ – Integer Input

On entry: the number of sequences to be transformed,

m

Constraint:

m \geq 1

2: $n$ – Integer Input

On entry: the number of data values in each sequence,

n

Constraint:

n \geq 1

3: $x [m \times n]$ – double Input/Output

On entry: the

m

Hermitian sequences must be stored consecutively in x in Hermitian form. Sequence 1 should occupy the first

n

elements of x, sequence 2 the elements

n

2 n - 1

, so that in general sequence

p

occupies the array elements

(p - 1) n

p n - 1

. 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}

should be in array element

x [(p - 1) \times n + j]

and for

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

y_{j}^{p}

should be in array element

x [(p - 1) \times n + n - j]

On exit: the components of the

m

discrete Fourier transforms, stored consecutively. Transform

p

occupies the elements

(p - 1) n

p n - 1

of x overwriting the corresponding original sequence; thus 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}}_{1}^{1}, \dots, {\hat{x}}_{n - 1}^{1}, {\hat{x}}_{0}^{2}, {\hat{x}}_{1}^{2}, \dots, {\hat{x}}_{n - 1}^{2}, \dots, {\hat{x}}_{0}^{m}, {\hat{x}}_{1}^{m}, \dots, {\hat{x}}_{n - 1}^{m} .

4: $trig [2 \times n]$ – const double Input

On entry: trigonometric coefficients as returned by a call of c06gzc. c06fqc makes a simple check to ensure that trig has been initialized and that the initialization is compatible with the value of n

5: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_C06_NOT_TRIG: Value of n and trig array are incompatible or trig array not initialized.
NE_INT_ARG_LT: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 1$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

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

c06fqc is not threaded in any implementation.

9 Further Comments

The time taken is approximately proportional to

n m \log (n)

, but also depends on the factors of

n

. The function 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 program 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 using c06gsc and printed. The discrete Fourier transforms are then computed (using c06fqc) and printed out. Inverse transforms are then calculated by calling c06fpc followed by c06gqc showing that the original sequences are restored.

c06fq: FL CL CPP AD

NAG CL Interfacec06fqc (withdraw_​fft_​hermitian_​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

NAG CL Interface
c06fqc (withdraw_fft_hermitian_1d_multi_rfmt)