NAG Library Function Document
nag_sum_fft_cosine (c06rfc)
1
Purpose
nag_sum_fft_cosine (c06rfc) computes the discrete 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_cosine (Integer m,
Integer n,
double x[],
NagError *fail) 

3
Description
Given
$m$ sequences of
$n+1$ real data values
${x}_{\mathit{j}}^{\mathit{p}}$, for
$\mathit{j}=0,1,\dots ,n$ and
$\mathit{p}=1,2,\dots ,m$,
nag_sum_fft_cosine (c06rfc) simultaneously calculates the Fourier cosine 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 DCT.
Since the Fourier cosine transform defined above is its own inverse, two consecutive calls of nag_sum_fft_cosine (c06rfc) will restore the original data.
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 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:
$\mathbf{m}$ – IntegerInput

On entry: $m$, the number of sequences to be transformed.
Constraint:
${\mathbf{m}}\ge 1$.
 2:
$\mathbf{n}$ – IntegerInput

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:
${\mathbf{n}}\ge 1$.
 3:
$\mathbf{x}\left[\left({\mathbf{n}}+1\right)\times {\mathbf{m}}\right]$ – doubleInput/Output

On entry: the $m$ data sequences to be transformed. The $\left(n+1\right)$ data values of the $\mathit{p}$th sequence to be transformed, denoted by
${x}_{\mathit{j}}^{\mathit{p}}$, for $\mathit{j}=0,1,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, must be stored in ${\mathbf{x}}\left[\left(p1\right)\times \left({\mathbf{n}}+1\right)+j\right]$.
On exit: the $m$ Fourier cosine transforms, overwriting the corresponding original sequences. The $\left(n+1\right)$ components of the $\mathit{p}$th Fourier cosine transform, denoted by
${\hat{x}}_{\mathit{k}}^{\mathit{p}}$, for $\mathit{k}=0,1,\dots ,n$ and $\mathit{p}=1,2,\dots ,m$, are stored in ${\mathbf{x}}\left[\left(p1\right)\times \left({\mathbf{n}}+1\right)+k\right]$.
 4:
$\mathbf{fail}$ – 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_cosine (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 implementationspecific information.
The time taken by nag_sum_fft_cosine (c06rfc) is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. nag_sum_fft_cosine (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 $\mathit{O}\left(n\right)$ double values.
10
Example
This example reads in sequences of real data values and prints their Fourier cosine transforms (as computed by nag_sum_fft_cosine (c06rfc)). It then calls nag_sum_fft_cosine (c06rfc) again and prints the results which may be compared with the original sequence.
10.1
Program Text
Program Text (c06rfce.c)
10.2
Program Data
Program Data (c06rfce.d)
10.3
Program Results
Program Results (c06rfce.r)