Computational Methods for Structured and Singular Matrix Polynomials

Western Michigan University, Kalamazoo MI

Investigators

Abstract

Matrix polynomials frequently arise in the engineering and applied sciences,especially in structural dynamics, vibrational analysis, control systems, and differential-algebraic equations (DAEs), to give a few examples. Principal among the associated problems are the computation of the eigenstructure of regular matrix polynomials, and in the case of singular polynomials, the computation of minimal indices and minimal bases. In recent work, the investigators and their colleagues identified rich spaces of linearizations which led to the construction of new structured linearizations,condensed forms, and accurate structure-preserving algorithms. By using new techniques, they have also made progress on singular polynomials, showing that linearizations provide a pathway to the reliable computation of minimal indices and bases. This proposal singles out several important tasks for investigation concerning linearizations, quadratifications and minimal indices and bases. The goal is to develop new algorithms for these computations, and increase theoretical understanding so as to aid in the formulation of effective algorithms. The problems studied in this proposal are ubiquitous in a wide range of important problems in engineering and applied sciences. Numerical methods for their solution are critical in structural mechanics, molecular dynamics, vibrational analysis, the simulation of electrical circuits, elastic deformation of anisotropic materials, and optical waveguide design, to give a few examples. The trend towards extreme designs, such as high speed trains, optoelectronic devices, micro-electromechanical systems, and ``superjumbo'' jets such as the Airbus 380, presents a challenge for the computation of the resonant frequencies of these structures. These extreme designs often lead to computationally sensitive problems, while the physics of the underlying problem leads to structure that numerical methods should exploit in order to obtain physically meaningful results. The aim of this project is to increase our theoretical understanding of mathematical transformations that preserve these structures and thereby advance the development of computationally effective algorithms. Consequently, this work will have direct benefit to scientists and engineers across a wide range of disciplines.