Partial Eigenvalue Assignment Problem for the Quadratic Matrix Pencil: comparisons of two types of methods for solving Quadratic Partial Eigenvalue Assignment Problem

Numerical experiments are performed to know which method, the direct second-order method or the first-order method is superior, both in eigenvalue assignment and norm reduction of the feedback matrices. Two standard approaches to compute eigenvalues and eigenvectors of a Quadratic Matrix Pencil are defined, one with finding the relation between standard eigenvalue problems and quadratic eigenvalue problems and the other with finding the relation between generalized and quadratic eigenvalue problems. The existence and uniqueness results for both the problems, the matrix second order case and for the partial eigenvalue assignment problem for the matrix pencil, and Orthogonality relations between the eigenvectors of the linear and quadratic matrix pencil are defined. The solutions are proposed for the partial eigenvalue assignment problems for the quadratic pencil where only the partial knowledge of eigenvalues and eigenvectors are required.