Solving Bordered Almost Block Diagonal Linear Systems Stably and Efficiently

Abstract

Almost block diagonal (ABD) linear systems arise in a variety of contexts, specifically in the numerical solution of systems of ordinary differential equation two-point boundary value problems with separated boundary conditions. The stable, efficient solution of ABDs has received much attention recently. Bordered almost block diagonal (BABD) linear systems arise when the boundary conditions are non-separated. After an introduction to the problem, we describe a variety of direct approaches to solving BABD systems, some of them extensions of ABD techniques. We pay particular attention to questions of efficiency and stability.