Authors:
Vasile Sima
1
and
Peter Benner
2
Affiliations:
1
National Institute for Research & Development in Informatics, Romania
;
2
Max Planck Institute for Dynamics of Complex Technical Systems, Germany
Keyword(s):
Deflating Subspaces, Eigenvalue Reordering, Generalized Eigenvalues, Generalized Schur Form, Numerical Methods, Skew-Hamiltonian/Hamiltonian Matrix Pencil, Software, Structure-preservation.
Related
Ontology
Subjects/Areas/Topics:
Engineering Applications
;
Industrial Engineering
;
Informatics in Control, Automation and Robotics
;
Intelligent Control Systems and Optimization
;
Optimization Algorithms
;
Performance Evaluation and Optimization
;
Robotics and Automation
;
Signal Processing, Sensors, Systems Modeling and Control
Abstract:
There is a continuous research effort worldwide to improve the reliability, efficiency, and accuracy of numerical
computations in various domains. One of the most promising research avenues is to exploit the structural
properties of the mathematical problems to be solved. This paper investigates some numerical algorithms for
the solution of common and structured eigenproblems, which have many applications in automatic control
(e.g., linear-quadratic optimization and H¥-optimization), but also in various areas of applied mathematics,
physics, and computational chemistry. Of much interest is to find the eigenvalues and certain deflating subspaces,
mainly those associated to the stable eigenvalues. Several simple examples are used to highlight the
pitfalls which may appear in such numerical computations, using state-of-the-art solvers. Balancing the matrices
and the use of condition numbers for eigenvalues are shown to be essential options in investigating the
behavior of the solvers an
d problem sensitivity.
(More)