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_TransformDirection Input

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: $m$ – Integer Input

On entry:

m

, the number of sequences to be transformed.

Constraint:

m \geq 1

3: $n$ – Integer Input

On entry:

n

, the number of real values in each sequence.

Constraint:

n \geq 1

4: $x [n \times m]$ – double Input/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 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, $m = ⟨ value ⟩$ .
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.
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

c06rhc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

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 by c06rhc is approximately proportional to

n m \log (n)

, but also depends on the factors of

n

. 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 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.

c06rh: FL CL CPP AD

NAG CL Interfacec06rhc (fft_​qtrcosine)

▸▿ 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
c06rhc (fft_qtrcosine)