c06pa: FL CL CPP AD

NAG CL Interface
c06pac (fft_realherm_1d)

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NAG Library Manual, Mark 27

Interfaces: FL CL CPP AD

NAG CL Interface Introduction

C06 (Sum) Chapter Contents

C06 (Sum) Chapter Introduction

c06pa: FL CL CPP AD

▸▿ 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

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>

void	c06pac (Nag_TransformDirection direct, double x[], Integer n, NagError *fail)

The function may be called by the names: c06pac or nag_sum_fft_realherm_1d.

3 Description

Given a sequence of

n

real data values

x_{j}

, for

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

, 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 c06pac with

direct = Nag_ForwardTransform

followed by a call with

direct = Nag_BackwardTransform

will restore the original data.

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: c06pvc and 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_TransformDirection Input

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

Nag_ForwardTransform

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

Nag_BackwardTransform

Constraint:

direct = Nag_ForwardTransform

Nag_BackwardTransform

2: $x [n + 2]$ – double Input/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: $n$ – Integer Input

On entry:

n

, the number of data values.

Constraint:

n \geq 1

4: $fail$ – 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.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface 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

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

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 X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also 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

. 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 c06pac with

direct = Nag_ForwardTransform

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

direct = Nag_BackwardTransform

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

c06pa: FL CL CPP AD

NAG CL Interfacec06pac (fft_​realherm_​1d)

▸▿ 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
c06pac (fft_realherm_1d)