Recent Advances in the Numerical Solution of Quadratic Eigenvalue Problems
A wide variety of applications require the solution of a quadratic eigenvalue problem (QEP). The most common way of solving the QEP is to convert it into a linear problem of twice the dimension and solve the linear problem by the QZ algorithm or a Krylov method. Much work has been done to understand the linearization process and the effects of scaling. Structure preserving linearizations have been studied, along with algorithms that preserve the spectral properties in finite precision arithmetic. We present a new class of transformations that preserves the block structure of widely used linearizations and hence does not require computations with matrices of twice the original size. We show how to use these structure preserving transformations to decouple symmetric and nonsymmetric quadratic matrix polynomials.
Francoise Tisseur, The University of Manchester, United Kingdom