NAG Library Function Document

nag_sum_fft_qtrcosine (c06rhc)

+− 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_qtrcosine (c06rhc) computes the discrete quarter-wave Fourier cosine transforms of

m

sequences of real data values. The elements of each sequence and its transform are stored contiguously.

2 Specification

#include <nag.h>

#include <nagc06.h>

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

3 Description

Given

m

sequences of

n

real data values

x_{j}^{p}

, for

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

and

p = 1, 2, \dots, m

, nag_sum_fft_qtrcosine (c06rhc) simultaneously calculates the quarter-wave Fourier cosine transforms of all the sequences defined by

{\hat{x}}_{k}^{p} = \frac{1}{\sqrt{n}} (\frac{1}{2} x_{0}^{p} + \sum_{j = 1}^{n - 1} x_{j}^{p} \times \cos (j (2 k + 1) \frac{π}{2 n})),   if ​ direct = Nag_ForwardTransform,

or its inverse

x_{k}^{p} = \frac{2}{\sqrt{n}} \sum_{j = 0}^{n - 1} {\hat{x}}_{j}^{p} \times \cos ((2 j + 1) k \frac{π}{2 n}),   if ​ direct = Nag_BackwardTransform,

where

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

and

p = 1, 2, \dots, m

(Note the scale factor

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

in this definition.)

A call of nag_sum_fft_qtrcosine (c06rhc) with

direct = Nag_ForwardTransform

followed by a call with

direct = Nag_BackwardTransform

will restore the original data.

The two transforms are also known as type-III DCT and type-II DCT, respectively.

The transform calculated by this function can be used to solve Poisson's equation when the derivative of the solution is specified at the left boundary, and the solution is specified at the right boundary (see Swarztrauber (1977)).

The function uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, described in Temperton (1983), together with pre- and post-processing stages described in Swarztrauber (1982). Special coding is provided for the factors

2

3

4

and

5

4 References

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

Swarztrauber P N (1977) The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle SIAM Rev. 19(3) 490–501

Swarztrauber P N (1982) Vectorizing the FFT's Parallel Computation (ed G Rodrique) 51–83 Academic Press

Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys. 52 340–350

5 Arguments

1: direct – Nag_TransformDirectionInput

On entry: indicates the transform, as defined in Section 3, to be computed.

$direct = Nag_ForwardTransform$: Forward transform.
$direct = Nag_BackwardTransform$: Inverse transform.

Constraint:

direct = Nag_ForwardTransform

Nag_BackwardTransform

2: m – IntegerInput

On entry:

m

, the number of sequences to be transformed.

Constraint:

m \geq 1

3: n – IntegerInput

On entry:

n

, the number of real values in each sequence.

Constraint:

n \geq 1

4: x[ $n \times m$ ] – doubleInput/Output

On entry: the

m

data sequences to be transformed. The data values of the

p

th sequence to be transformed, denoted by

x_{j}^{p}

, for

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

and

p = 1, 2, \dots, m

, must be stored in

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

On exit: the

m

quarter-wave cosine transforms, overwriting the corresponding original sequences. The

n

components of the

p

th quarter-wave cosine transform, denoted by

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

, for

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

and

p = 1, 2, \dots, m

, are stored in

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

5: 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.
NE_INT: On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 1$ .
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_qtrcosine (c06rhc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken by nag_sum_fft_qtrcosine (c06rhc) is approximately proportional to

n m \log (n)

, but also depends on the factors of

n

. nag_sum_fft_qtrcosine (c06rhc) 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. This function internally allocates a workspace of order

O (n)

double values.

10 Example

This example reads in sequences of real data values and prints their quarter-wave cosine transforms as computed by nag_sum_fft_qtrcosine (c06rhc) with

direct = Nag_ForwardTransform

. It then calls the function again with

direct = Nag_BackwardTransform

and prints the results which may be compared with the original data.

NAG Library Function Documentnag_sum_fft_qtrcosine (c06rhc)

+− 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 Library Function Document

nag_sum_fft_qtrcosine (c06rhc)