All Spectral Dominant Norms Are Stable.
Abstract
When solving partial differential equations numerically one often has to use iterations involving matrices restricted to a given set alpha of n x n complex valued matrices. It then follows that the iteration scheme is stable if and only if this set of matrices is stable. That is all powers of all matrices from the set alpha are uniformly bounded. Such sets were completely characterized by H. O. Kreiss. However, his criteria are hard to use. In this paper the authors characterize in a very simple way stable sets of matrices alpha, whenever the set alpha is closed, convex, balanced, and contains a neighborhood of the origin. Such a set alpha is a unit ball of some vector norm true value of. on matrices. They then show that alpha is stable if and only if the above norm dominates the spectral radius rho. That is rho < or = true value of A for all matrices A. The necessity of the above condition is obvious, and is sometime referred to as the Neumann condition. To prove the sufficiency the authors use the Kriess matrix theorem and other results.
Document Details
- Document Type
- Technical Report
- Publication Date
- Apr 01, 1983
- Accession Number
- ADA129076
Entities
People
- C. Zenger
- S. Friedland
Organizations
- University of Wisconsin–Madison