Classical and vector sturm—liouville problems: recent advances in singular-point analysis and shooting-type algorithms

Significant advances have been made in the last year or two in algorithms and theory for Sturm—Liouville problems (SLPs). For the classical regular or singular SLP −(p(x)u′)′ + q(x)u = λw(x)u, a < x < b, we outline the algorithmic approaches of the recent library codes and what they can now routinely achieve.

For a library code, automatic treatment of singular problems is a must. New results are presented which clarify the effect of various numerical methods of handling a singular endpoint.

For the vector generalization −(P(x)u′)′+Q(x)u = λW(x)u where now u is a vector function of x, and P, Q, W are matrices, and for the corresponding higher-order vector self-adjoint problem, we outline the equally impressive advances in algorithms and theory.