NAG CL Interface
c06rec (fft_sine)
1
Purpose
c06rec computes the discrete Fourier sine transforms of $m$ sequences of real data values. The elements of each sequence and its transform are stored contiguously.
2
Specification
void 
c06rec (Integer m,
Integer n,
double x[],
NagError *fail) 

The function may be called by the names: c06rec or nag_sum_fft_sine.
3
Description
Given
$m$ sequences of
$n1$ real data values
${x}_{\mathit{j}}^{\mathit{p}}$, for
$\mathit{j}=1,2,\dots ,n1$ and
$\mathit{p}=1,2,\dots ,m$,
c06rec simultaneously calculates the Fourier sine transforms of all the sequences defined by
(Note the scale factor $\sqrt{\frac{2}{n}}$ in this definition.)
This transform is also known as typeI DST.
Since the Fourier sine transform defined above is its own inverse, two consecutive calls of c06rec will restore the original data.
The transform calculated by this function can be used to solve Poisson's equation when 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 selfsorting algorithm, described in
Temperton (1983), together with pre and postprocessing 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 mixedradix real Fourier transforms J. Comput. Phys. 52 340–350
5
Arguments

1:
$\mathbf{m}$ – Integer
Input

On entry: $m$, the number of sequences to be transformed.
Constraint:
${\mathbf{m}}\ge 1$.

2:
$\mathbf{n}$ – Integer
Input

On entry: one more than the number of real values in each sequence, i.e., the number of values in each sequence is $n1$.
Constraint:
${\mathbf{n}}\ge 1$.

3:
$\mathbf{x}\left[\left({\mathbf{n}}1\right)\times {\mathbf{m}}\right]$ – double
Input/Output

On entry: the $\mathit{p}$th sequence to be transformed, denoted by
${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=1,2,\dots ,n1$ and $\mathit{p}=1,2,\dots ,m$, must be stored in ${\mathbf{x}}\left[\left(p1\right)\times \left({\mathbf{n}}1\right)+j1\right]$.
On exit: the $m$ Fourier sine transforms, overwriting the corresponding original sequences. The $\left(n1\right)$ components of the $p$th Fourier sine transform, denoted by
${\hat{x}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=1,2,\dots ,n1$ and $\mathit{p}=1,2,\dots ,m$, are stored in ${\mathbf{x}}\left[\left(p1\right)\times \left({\mathbf{n}}1\right)+k1\right]$.

4:
$\mathbf{fail}$ – 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 $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 1$.
 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
c06rec 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 implementationspecific information.
The time taken by c06rec is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. c06rec 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.
Workspace is internally allocated by this function. The total amount of memory allocated is $\mathit{O}\left(n\right)$ double values.
10
Example
This example reads in sequences of real data values and prints their Fourier sine transforms (as computed by c06rec). It then calls c06rec again and prints the results which may be compared with the original sequence.
10.1
Program Text
10.2
Program Data
10.3
Program Results