This project will develop efficient, robust and reliable numerical methods for computing the eigen-structure of matrices and pencils with special algebraic structure. Its goals include building a foundation for algorithm development from the mathematical, numerical, perturbation, and bifurcation properties of structured matrices and pencils. This will provide effective and reliable computational tools for solving computational problems in science, engineering and industry. As an initial application, numerical algorithms will be developed for applications in robust control and stability analysis of dynamic systems. The investigator will develop structured eigenvalue methods that take advantage of symmetries among the eigenvalues and the invariant or deflating subspaces of algebraically structured matrices and pencils. He will use matrix splitting, embedding techniques, matrix transformations, and other novel techniques to decouple, extend or factor structured matrices to get matrices with simpler eigen-structures that can be computed in a relatively easy way.
Structured eigenvalue problems arise in almost every discipline of science and engineering. Applications include the study of planet motion, energy distribution in electron systems, stabilization of satellites, and vibration in nuclear power plants. All these applications require efficient, accurate information about both the eigen-structure and its geometry. Properties of the physical system under study often lead to eigenvalue problems with special structures and symmetries among eigenvalues and the invariant or deflating subspaces. Unfortunately, rounding errors in conventional numerical methods sometimes destroy these symmetries and lead to physically unrealistic computed results. Rounding errors in numerical methods that preserve special structure do not lead to unrealistic results. Such methods are typically more efficient, robust and accurate than conventional numerical methods.
