The solution of eigenvalue problems is an integral part of many scientific
computations. For example, the numerical solution of problems in structural
dynamics, electrical networks, macro-economics, quantum chemistry, and con-
trol theory often requires solving eigenvalue problems. The coefficient matrix
of the eigenvalue problem may be small to medium sized and dense, or large
and sparse (containing many zero elements). In the past tremendous advances
have been achieved in the solution methods for symmetric eigenvalue prob-
lems. The state of the art for nonsymmetric problems is not so advanced;
nonsymmetric eigenvalue problems can be hopelessly difficult to solve in some
situations due, for example, to poor conditioning. Good numerical algorithms
for nonsymmetric eigenvalue problems also tend to be far more complex than
their symmetric counterparts.
This book deals with methods for solving a special nonsymmetric eigen-
value problem; the symplectic eigenvalue problem. The symplectic eigenvalue
problem is helpful, e.g., in analyzing a number of different questions that arise
in linear control theory for discrete-time systems. Certain quadratic eigenvalue
problems arising, e.g., in finite element discretization in structural analysis, in
acoustic simulation of poro-elastic materials, or in the elastic deformation of
anisotropic materials can also lead to symplectic eigenvalue problems. The
problem appears in other applications as well.