Eigenvalue perturbation

Abstract: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography Index.

Abstract: Preliminaries. Norms and Metrics. Linear Systems and Least Squares Problems. The Perturbation of Eigenvalues. Invariant Subspaces. Generalized Eigenvalue Problems.

Abstract: An interpretation of Dr. Cornelius Lanczos' iteration method, which he has named \"minimized iterations\", is discussed in this article, expounding the method as applied to the solution of the characteristic matrix equations both in homogeneous and nonhomogeneous form. This interpretation leads to a variation of the Lanczos procedure which may frequently be advantageous by virtue of reducing the volume of numerical work in practical applications. Both methods employ essentially the same algorithm, requiring the generation of a series of orthogonal functions through which a simple matrix equation of reduced order is established. The reduced matrix equation may be solved directly in terms of certain polynomial functions obtained in conjunction with the generated orthogonal functions, and the convergence of the solution may be observed as the order of the reduced matrix is successively increased with the order of the original matrix as a limit. The method of minimized iterations is recommended as a rapid means for determining a small number of the larger eigenvalues and modal columns of a large matrix and as a desirable alternative for various series expansions of the Fredholm problem. 1. The conventional iterative procedures. It is frequently required that real latent roots, or eigenvalues, and modal columns be determined for a real numerical matrix, u, of order, n, in the characteristic homogeneous equation,*

Abstract: List of symbols and acronyms List of iterative algorithm templates List of direct algorithms List of figures List of tables 1: Introduction 2: A brief tour of Eigenproblems 3: An introduction to iterative projection methods 4: Hermitian Eigenvalue problems 5: Generalized Hermitian Eigenvalue problems 6: Singular Value Decomposition 7: Non-Hermitian Eigenvalue problems 8: Generalized Non-Hermitian Eigenvalue problems 9: Nonlinear Eigenvalue problems 10: Common issues 11: Preconditioning techniques Appendix: of things not treated Bibliography Index .