The transform calculated by this function can be used to solve Poisson's equation when the derivative of the solution is specified at both left and right boundaries (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: $m$ – Integer Input: On entry: $m$ , the number of sequences to be transformed.

Constraint: $m \geq 1$ .
2: $n$ – Integer Input: On entry: one less than the number of real values in each sequence, i.e., the number of values in each sequence is $n + 1$ .

Constraint: $n \geq 1$ .
3: $x [(n + 1) \times m]$ – double Input/Output: On entry: the $m$ data sequences to be transformed. The $(n + 1)$ data values of the $p$ th sequence to be transformed, denoted by $x_{j}^{p}$ , for $j = 0, 1, \dots, n$ and $p = 1, 2, \dots, m$ , must be stored in $x [(p - 1) \times (n + 1) + j]$ .

On exit: the $m$ Fourier cosine transforms, overwriting the corresponding original sequences. The $(n + 1)$ components of the $p$ th Fourier cosine transform, denoted by ${\hat{x}}_{k}^{p}$ , for $k = 0, 1, \dots, n$ and $p = 1, 2, \dots, m$ , are stored in $x [(p - 1) \times (n + 1) + k]$ .
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, $m = ⟨ value ⟩$ .
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

Background information to multithreading can be found in the Multithreading documentation.

c06rfc 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 c06rfc is approximately proportional to

n m \log (n)

, but also depends on the factors of

n

. c06rfc 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 Fourier cosine transforms (as computed by c06rfc). It then calls c06rfc again and prints the results which may be compared with the original sequence.

c06rf: FL CL CPP AD PY MB

NAG CL Interfacec06rfc (fft_​cosine)

▸▿ 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
c06rfc (fft_cosine)