\documentstyle[twoside]{article} \input amssym.def \pagestyle{myheadings} \markboth{\hfil Eigenvalue Computations \hfil EJDE--1995/05}% {EJDE--1995/05\hfil H.I. Dwyer \& A. Zettl \hfil} \begin{document} \ifx\Box\undefined \newcommand{\Box}{\diamondsuit}\fi \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}\newline Vol. {\bf 1995}(1995), No. 05, pp. 1-13. Published May 2, 1995.\newline ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113 } \vspace{\bigskipamount} \\ Eigenvalue Computations for Regular Matrix Sturm-Liouville Problems \thanks{ {\em 1991 Mathematics Subject Classifications:} 34B24, 34L15, 34L05.\newline\indent {\em Key words and phrases:} Sturm-Liouville problem, matrix coefficients, eigenvalues, \newline\indent numerical computation of eigenvalues \newline\indent \copyright 1995 Southwest Texas State University and University of North Texas.\newline\indent Submitted: August 25, 1994.} } \date{} \author{H.I. Dwyer \& A. Zettl } \maketitle % \begin{abstract} An algorithm is presented for computing eigenvalues of regular self-adjoint Sturm-Liouville~(SL) problems with matrix coefficients and separated boundary conditions. \end{abstract} % \newcommand{\ddx}{\frac{d}{dx}} % ordinary deriv wrt x \newcommand{\ddt}{\frac{d}{dt}} % ordinary deriv wrt t \newcommand{\pdx}{\frac{\partial}{\partial x}} % partial deriv wrt x \newtheorem{thm}{Theorem} \newtheorem{lemma}[thm]{Lemma} \newtheorem{defn}{Definition} \newtheorem{ex}{Example} % Main body of the paper % \section*{Introduction} In his book {\em Discrete and Continuous Boundary Problems}~\cite{atkinson}, F.V.~Atkinson characterizes eigenvalues of matrix Sturm-Liouville (SL) problems in terms of eigenvalues of certain unitary matrices. Based on this characterization the team of Atkinson, Krall, Leaf, and Zettl~(AKLZ) developed an algorithm, and constructed a prototype FORTRAN code for the numerical computation of eigenvalues of SL problems. % A description of the algorithm, with some test results, was published in an Argonne Laboratory Report~\cite{aklz87}. % While clearly demonstrating the feasibility of this algorithm, the tests showed that there were some difficulties remaining. % Dwyer, in his dissertation as a student of Zettl, refined and significantly extended the algorithm, and corrected the difficulties in the original code. In this paper, the completed algorithm for computing the eigenvalues for a regular Sturm-Liouville problem with separated boundary conditions is presented. % % % PROBLEM DEFINITIONS % % \section*{Problem Definitions} A SL problem consists of the second order linear ordinary differential equation $$\label{ode} -(py')'+qy = \lambda wy \,\,\,\mbox{on} \,\,\,(a,b)$$ together with boundary conditions. For the case when both endpoints $a$, $b$ are regular, these have the form $$\label{bc1} C \left(\begin{array}{c} y(a) \\ (py')(a) \end{array}\right) + D \left(\begin{array}{c} y(b) \\ (py')(b) \end{array}\right) = \left(\begin{array}{c} 0 \\ 0 \end{array}\right) .$$ % % Theorem % \begin{thm} \label{thm1} Let $p$, $q$, $w$ be complex $m\,\times\, m$ matrix functions defined on the interval $[a,b]$, \$-\infty < a