NAG CL Interfacec06rec (fft_​sine)

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1Purpose

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.

2Specification

 #include
 void c06rec (Integer m, Integer n, double x[], NagError *fail)
The function may be called by the names: c06rec or nag_sum_fft_sine.

3Description

Given $m$ sequences of $n-1$ real data values ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=1,2,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, c06rec simultaneously calculates the Fourier sine transforms of all the sequences defined by
 $x^ k p = 2n ∑ j=1 n-1 xjp × sin(jkπn) , k= 1, 2, …, n-1 ​ and ​ p= 1, 2, …, m .$
(Note the scale factor $\sqrt{\frac{2}{n}}$ in this definition.)
This transform is also known as type-I 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 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 $5$.

4References

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

5Arguments

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 $n-1$.
Constraint: ${\mathbf{n}}\ge 1$.
3: $\mathbf{x}\left[\left({\mathbf{n}}-1\right)×{\mathbf{m}}\right]$double Input/Output
On entry: the $\mathit{p}$th sequence to be transformed, denoted by ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=1,2,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, must be stored in ${\mathbf{x}}\left[\left(p-1\right)×\left({\mathbf{n}}-1\right)+j-1\right]$.
On exit: the $m$ Fourier sine transforms, overwriting the corresponding original sequences. The $\left(n-1\right)$ components of the $p$th Fourier sine transform, denoted by ${\stackrel{^}{x}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=1,2,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, are stored in ${\mathbf{x}}\left[\left(p-1\right)×\left({\mathbf{n}}-1\right)+k-1\right]$.
4: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error 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.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
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.

7Accuracy

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).

8Parallelism 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 implementation-specific 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.

10Example

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.

10.1Program Text

Program Text (c06rece.c)

10.2Program Data

Program Data (c06rece.d)

10.3Program Results

Program Results (c06rece.r)