NAG Library Routine Document
c06rhf (fft_qtrcosine)
1
Purpose
c06rhf 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
Fortran Interface
Integer, Intent (In)  ::  idir, m, n  Integer, Intent (Inout)  ::  ifail  Real (Kind=nag_wp), Intent (Inout)  ::  x(0:n1,m) 

C Header Interface
#include nagmk26.h
void 
c06rhf_ (const Integer *idir, const Integer *m, const Integer *n, double x[], Integer *ifail) 

3
Description
Given
$m$ sequences of
$n$ real data values
${x}_{\mathit{j}}^{\mathit{p}}$, for
$\mathit{j}=0,1,\dots ,n1$ and
$\mathit{p}=1,2,\dots ,m$,
c06rhf 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 c06rhf with ${\mathbf{idir}}=1$ followed by a call with ${\mathbf{idir}}=1$ will restore the original data.
The two transforms are also known as typeIII DCT and typeII DCT, respectively.
The transform calculated by this routine 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 routine 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: $\mathbf{idir}$ – IntegerInput

On entry: indicates the transform, as defined in
Section 3, to be computed.
 ${\mathbf{idir}}=1$
 Forward transform.
 ${\mathbf{idir}}=1$
 Inverse transform.
Constraint:
${\mathbf{idir}}=1$ or $1$.
 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: $\mathbf{x}\left(0:{\mathbf{n}}1,{\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: 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 ${\mathbf{x}}\left(j,p\right)$.
On exit: 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 ${\mathbf{x}}\left(k,p\right)$, overwriting the corresponding original values.
 5: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
 ${\mathbf{ifail}}=1$

On entry, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{m}}\ge 1$.
 ${\mathbf{ifail}}=2$

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 1$.
 ${\mathbf{ifail}}=3$

On entry, ${\mathbf{idir}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{idir}}=1$ or $1$.
 ${\mathbf{ifail}}=4$

An internal error has occurred in this routine.
Check the routine call and any array sizes.
If the call is correct then please contact
NAG for assistance.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 3.7 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
c06rhf 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 routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The time taken by c06rhf is approximately proportional to $nm\mathrm{log}\left(n\right)$, but also depends on the factors of $n$. c06rhf 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 routine. The total amount of memory allocated is $\mathit{O}\left(n\right)$.
10
Example
This example reads in sequences of real data values and prints their quarterwave cosine transforms as computed by c06rhf with ${\mathbf{idir}}=1$. It then calls the routine again with ${\mathbf{idir}}=1$ and prints the results which may be compared with the original data.
10.1
Program Text
Program Text (c06rhfe.f90)
10.2
Program Data
Program Data (c06rhfe.d)
10.3
Program Results
Program Results (c06rhfe.r)