NAG Library Function Document

nag_sum_fft_realherm_1d (c06pac)

+− 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

nag_sum_fft_realherm_1d (c06pac) calculates the discrete Fourier transform of a sequence of

n

real data values or of a Hermitian sequence of

n

complex data values stored in compact form in a double array.

2 Specification

#include <nag.h>

#include <nagc06.h>

void	nag_sum_fft_realherm_1d (Nag_TransformDirection direct, double x[], Integer n, NagError *fail)

3 Description

Given a sequence of

n

real data values

x_{j}

, for

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

, nag_sum_fft_realherm_1d (c06pac) calculates their discrete Fourier transform (in the forward direction) defined by

{\hat{z}}_{k} = \frac{1}{\sqrt{n}} \sum_{j = 0}^{n - 1} x_{j} \times \exp (- i \frac{2 π j k}{n}), k = 0, 1, \dots, n - 1 .

The transformed values

{\hat{z}}_{k}

are complex, but they form a Hermitian sequence (i.e.,

{\hat{z}}_{n - k}

is the complex conjugate of

{\hat{z}}_{k}

), so they are completely determined by

n

real numbers (since

{\hat{z}}_{0}

is real, as is

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

for

n

even).

Alternatively, given a Hermitian sequence of

n

complex data values

z_{j}

, this function calculates their inverse (backward) discrete Fourier transform defined by

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

The transformed values

{\hat{x}}_{k}

are real.

(Note the scale factor of

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

in the above definitions.)

A call of nag_sum_fft_realherm_1d (c06pac) with

direct = Nag_ForwardTransform

followed by a call with

direct = Nag_BackwardTransform

will restore the original data.

nag_sum_fft_realherm_1d (c06pac) 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).

The same functionality is available using the forward and backward transform function pair: nag_sum_fft_real_2d (c06pvc) and nag_sum_fft_hermitian_2d (c06pwc) on setting

n = 1

. This pair use a different storage solution; real data is stored in a double array, while Hermitian data (the first unconjugated half) is stored in a Complex array.

4 References

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

Temperton C (1983) Self-sorting mixed-radix fast Fourier transforms J. Comput. Phys. 52 1–23

5 Arguments

1: direct – Nag_TransformDirectionInput

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

Nag_ForwardTransform

If the backward transform is to be computed then direct must be set equal to

Nag_BackwardTransform

Constraint:

direct = Nag_ForwardTransform

Nag_BackwardTransform

2: x[ $n + 2$ ] – doubleInput/Output

On entry:

if $direct = Nag_ForwardTransform$ , $x [j]$ must contain $x_{j}$ , for $j = 0, 1, \dots, n - 1$ ;
if $direct = Nag_BackwardTransform$ , $x [2 \times k]$ and $x [2 \times k + 1]$ must contain the real and imaginary parts respectively of $z_{k}$ , for $k = 0, 1, \dots, n / 2$ . (Note that for the sequence $z_{k}$ to be Hermitian, the imaginary part of $z_{0}$ , and of $z_{n / 2}$ for $n$ even, must be zero.)

On exit:

if $direct = Nag_ForwardTransform$ , $x [2 \times k]$ and $x [2 \times k + 1]$ will contain the real and imaginary parts respectively of ${\hat{z}}_{k}$ , for $k = 0, 1, \dots, n / 2$ ;
if $direct = Nag_BackwardTransform$ , $x [j]$ will contain ${\hat{x}}_{j}$ , for $j = 0, 1, \dots, n - 1$ .

3: n – IntegerInput

On entry:

n

, the number of data values.

Constraint:

n \geq 1

4: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
$"⟨value⟩"$ is an invalid value of direct.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 1$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

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

nag_sum_fft_realherm_1d (c06pac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_sum_fft_realherm_1d (c06pac) 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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken is approximately proportional to

n \times \log (n)

, but also depends on the factorization of

n

. nag_sum_fft_realherm_1d (c06pac) is faster if the only prime factors of

n

are

2

3

5

; and fastest of all if

n

is a power of

2

. This function internally allocates a workspace of

3 n + 100

double values.

10 Example

This example reads in a sequence of real data values and prints their discrete Fourier transform (as computed by nag_sum_fft_realherm_1d (c06pac) with

direct = Nag_ForwardTransform

), after expanding it from complex Hermitian form into a full complex sequence. It then performs an inverse transform using nag_sum_fft_realherm_1d (c06pac) with

direct = Nag_BackwardTransform

, and prints the sequence so obtained alongside the original data values.

NAG Library Function Documentnag_sum_fft_realherm_1d (c06pac)