NAG Library Function Document
nag_sum_fft_qtrcosine (c06rhc)
1
Purpose
nag_sum_fft_qtrcosine (c06rhc) computes the discrete quarterwave Fourier cosine transforms of $m$ sequences of real data values. The elements of each sequence and its transform are stored contiguously.
2
Specification
#include <nag.h> 
#include <nagc06.h> 
void 
nag_sum_fft_qtrcosine (Nag_TransformDirection direct,
Integer m,
Integer n,
double x[],
NagError *fail) 

3
Description
Given
$m$ sequences of
$n$ real data values
${x}_{\mathit{j}}^{\mathit{p}}$, for
$\mathit{j}=0,1,\dots ,n1$ and
$\mathit{p}=1,2,\dots ,m$,
nag_sum_fft_qtrcosine (c06rhc) simultaneously calculates the quarterwave Fourier cosine transforms of all the sequences defined by
or its inverse
where
$k=0,1,\dots ,n1$ and
$p=1,2,\dots ,m$.
(Note the scale factor $\frac{1}{\sqrt{n}}$ in this definition.)
A call of nag_sum_fft_qtrcosine (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 typeIII DCT and typeII 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 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:
$\mathbf{direct}$ – Nag_TransformDirectionInput

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:
$\mathbf{m}$ – IntegerInput

On entry: $m$, the number of sequences to be transformed.
Constraint:
${\mathbf{m}}\ge 1$.
 3:
$\mathbf{n}$ – IntegerInput

On entry: $n$, the number of real values in each sequence.
Constraint:
${\mathbf{n}}\ge 1$.
 4:
$\mathbf{x}\left[{\mathbf{n}}\times {\mathbf{m}}\right]$ – doubleInput/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 $\mathit{j}=0,1,\dots ,n1$ and $\mathit{p}=1,2,\dots ,m$, must be stored in ${\mathbf{x}}\left[\left(p1\right)\times {\mathbf{n}}+j\right]$.
On exit: the $m$ quarterwave cosine transforms, overwriting the corresponding original sequences. The $n$ components of the $\mathit{p}$th quarterwave cosine transform, denoted by
${\hat{x}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=0,1,\dots ,n1$ and $\mathit{p}=1,2,\dots ,m$, are stored in ${\mathbf{x}}\left[\left(p1\right)\times {\mathbf{n}}+k\right]$.
 5:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation 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
nag_sum_fft_qtrcosine (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 implementationspecific information.
The time taken by nag_sum_fft_qtrcosine (c06rhc) is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. nag_sum_fft_qtrcosine (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.
10
Example
This example reads in sequences of real data values and prints their quarterwave cosine transforms as computed by nag_sum_fft_qtrcosine (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.
10.1
Program Text
Program Text (c06rhce.c)
10.2
Program Data
Program Data (c06rhce.d)
10.3
Program Results
Program Results (c06rhce.r)