NAG CL Interfacec06rhc (fft_​qtrcosine)

1Purpose

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.

2Specification

 #include
 void c06rhc (Nag_TransformDirection direct, Integer m, Integer n, double x[], NagError *fail)
The function may be called by the names: c06rhc or nag_sum_fft_qtrcosine.

3Description

Given $m$ sequences of $n$ real data values ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, c06rhc simultaneously calculates the quarter-wave Fourier cosine transforms of all the sequences defined by
 $x^ k p = 1n 12 x0p + ∑ j=1 n-1 xjp × cos j 2k+1 π2n , if ​ direct=Nag_ForwardTransform ,$
or its inverse
 $xkp = 2n ∑ j=0 n-1 x^ j p × cos 2j+1 k π2n , if ​ direct=Nag_BackwardTransform ,$
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 c06rhc with ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$ followed by a call with ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$ 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 $5$.

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: $\mathbf{direct}$Nag_TransformDirection Input
On entry: indicates the transform, as defined in Section 3, to be computed.
${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$
Forward transform.
${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$
Inverse transform.
Constraint: ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$ or $\mathrm{Nag_BackwardTransform}$.
2: $\mathbf{m}$Integer Input
On entry: $m$, the number of sequences to be transformed.
Constraint: ${\mathbf{m}}\ge 1$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the number of real values in each sequence.
Constraint: ${\mathbf{n}}\ge 1$.
4: $\mathbf{x}\left[{\mathbf{n}}×{\mathbf{m}}\right]$double Input/Output
On entry: the $m$ data sequences to be transformed. The data values of the $\mathit{p}$th sequence to be transformed, denoted by ${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, must be stored in ${\mathbf{x}}\left[\left(p-1\right)×{\mathbf{n}}+j\right]$.
On exit: the $m$ quarter-wave cosine transforms, overwriting the corresponding original sequences. The $n$ components of the $\mathit{p}$th quarter-wave cosine transform, denoted by ${\stackrel{^}{x}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=0,1,\dots ,n-1$ and $\mathit{p}=1,2,\dots ,m$, are stored in ${\mathbf{x}}\left[\left(p-1\right)×{\mathbf{n}}+k\right]$.
5: $\mathbf{fail}$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

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.

The time taken by c06rhc is approximately proportional to $nm\mathrm{log}\left(n\right)$, 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 $\mathit{O}\left(n\right)$ double values.

10Example

This example reads in sequences of real data values and prints their quarter-wave cosine transforms as computed by c06rhc with ${\mathbf{direct}}=\mathrm{Nag_ForwardTransform}$. It then calls the function again with ${\mathbf{direct}}=\mathrm{Nag_BackwardTransform}$ and prints the results which may be compared with the original data.

10.1Program Text

Program Text (c06rhce.c)

10.2Program Data

Program Data (c06rhce.d)

10.3Program Results

Program Results (c06rhce.r)