Project B6:
Theory and Numerics of Linear Algebra Problems and Discrete Dynamical Systems

Head: Prof. Dr. L. Elsner

Previous and current investigations in this project were concerned with the following fields: Problems for structured matrices, parallel algorithms, variation theorems for eigenvalue problems and discrete dynamical systems. Here we list the current topics in more detail.

Problems for structured matrices

We focus on normal matrices, and consider here the following:

• Numerical methods for calculating the nearest normal matrix, or the nearest one with given spectrum.
• Normal completion problems.
• Comparisons between certain measures of nonnormality.
• Constructions of simple measures of nonnormality.

Nonnegative Toeplitz matrices and the asymptotic behaviour of their spectral radii are studied. Having determined this behaviour the generalization to the block-Toeplitz case is considered.

For stochastic matrices we try to give matrix theoretic proofs of bounds for the second eigenvalue, which are usually proved by probabilistic methods.

Parallel Algorithms

Currently the focus is on iterative methods for linear systems. In particular:

• Error bounds and convergence rates for Krylow space methods, such as GMRES.
• Asynchronous methods for consistent nonnegative linear systems and characterization of their convergence by graph theoretic means.
• The study of the decay rates of the entries of the inverses of certain block band matrices is continued.

Variation theorems for eigenvalue problems

We continue to try to improve bounds for the spectral variation or matching distance for pairs of normal and general matrices.

Discrete dynamical systems

The focus is on hyperbolic structures in discrete dynamical systems and on the loss of hyperbolicity which occurs through the variation of parameters. As a model case we analyze the transition from a transversal to a tangential homoclinic orbit. A numerical method was developed which approximates the various types of orbits by finite segments and an error analysis for these approximations was given. Based on these results the purpose of the project is to contribute to the following problems in the numerical analysis of hyperbolic and nonhyperbolic structures:

• Computation of projectors and exponents of linear systems with exponential dichotomies (exploring the relation to Liapunov exponents, rotation numbers and Oseledec spaces).
• Numerical analysis of the bifurcations occuring at a homoclinic tangency.
• Numerical detection and analysis of symbolic dynamics in a given system.
• Creation of these phenomena by time discretization of differential equations.