\documentstyle[twoside]{article}
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\markboth{\hfil Appendix to: Computing Eigenvalues \hfil EJDE--1994/06}%
{EJDE--1994/06\hfil H. I. Dwyer \& A. Zettl\hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.\ {\bf 1994}(1994), No.\ 06, pp. 1--10. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.swt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
 \vspace{\bigskipamount} \\
Appendix to: \\
 Computing Eigenvalues of Regular Sturm-
Liouville Problems
\thanks{Attached on January 9, 1996 to the article published on August 23,
 1994.} 
\date{}
\author{H. I. Dwyer \& A. Zettl}
}
\maketitle

\begin{abstract} 
This appendix contains a remark about the scalar problem.  
It also shows a FORTRAN implementation 
of the algorithm discussed in the main article. 
\end{abstract}
\section{Scalar Problems}

If the original problem to be solved is 
scalar, that is, the coefficients
are `matrices' of dimension~1, then 
the transformed, separated problem will
be of dimension~2.  In this case, it is 
possible to simplify the algorithm 
for solving separated matrix problems
considerably,
by taking advantage of the fact that there
are simple formulas for the inverse
and eigenvalues of 2x2 matrices.
This allows the references to routines
in a linear algebra library (such as NAG
or IMSL) to be replaced by simple
computational routines.                                      
Several other simplifications are also
possible:
\begin{enumerate}
\item  It becomes relatively simple to 
verify that the boundary conditions
specified by the user are correct; the 
requirements for a self-adjoint problem
as specified in Theorem~1 reduce to four simple
tests.
\item  The initial value for the Theta matrix
is easily computed `by-hand'.
\item  One of the difficulties
encountered in the early attempts to implement the
algorithm for separated matrix problems
was caused by the use of a
library routine to compute eigenvalues.
The algorithm tracks changes in the eigenvalues,
and this was made somewhat more difficult because the
library routine did not return the list of eigenvalues
in any predictable order.  Thus, it was necessary
to inspect the list, and `unshuffle' them if the
order had been changed.
Since the eigenvalues are now computed by
formulas, rather by a library routine, they will
always be listed in the same order, and the
`unshuffle' step becomes unnecessary.
\end{enumerate}   
%
%
\section{Sample FORTRAN Implementation}

A simple FORTRAN implementation of the algorithm, as
applied to scalar problems, is provided below.
The user is required to edit the program
and define the coefficient functions for the original,
coupled problem.  The code must then be compiled
before it can be run.  The coefficient functions in
this sample listing correspond to Problem~b.  The results 
using this implementation agree very well with the results
obtained using the more general version of the code.

Note: This particular version uses a basic Runge-Kutta method
for the numerical integration.  More advanced methods 
could certainly be used, but this simple approach
has worked well enough during the initial testing.
\bigskip                                               

The FORTRAN program can be found under the name \lq\lq Dwyer.fort"
in the directory EJDE/Volumes/1994/06-Dwyer-Zettl/

\end{document}