Author(s) :
Volker Mehrmann
,
David Watkins
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 724-2002
MSC 2000
- 65F15 Eigenvalues, eigenvectors
-
15A18 Eigenvalues, singular values, and eigenvectors
Abstract :
We discuss the numerical solution of eigenvalue
problems for matrix polynomials, where the coefficient matrices
are alternating symmetric and skew symmetric or Hamiltonian
and skew Hamiltonian. We discuss several applications
that lead to such structures. Matrix polynomials of this type have a
symmetry in the spectrum that is the same as that of Hamiltonian
matrices or skew-Hamiltonian/Hamiltonian pencils. The numerical methods
that we derive are designed to preserve this eigenvalue symmetry.
We also discuss linearization techniques that transform the
polynomial into a skew-Hamiltonian/Hamiltonian linear eigenvalue problem
with a specific substructure. For this linear eigenvalue problem
we discuss special factorizations that are useful
in shift-and-invert Krylov subspace methods for the solution
of the eigenvalue problem. We present a numerical example
that demonstrates the effectiveness of our approach.
Keywords :
matrix polynomial, Hamiltonian
matrix, skew-Hamiltonian matrix, skew-Hamiltonian/Hamiltonian pencil,
matrix factorizations