AN AUGMENTED STABILITY RESULT FOR THE LANCZOS HERMITIAN MATRIX TRIDIAGONALIZATION PROCESS

摘要

It is shown that a good implementation of the Hermitian matrix tridiagonalization process of Lanczos [J. Research Nat. Bur. Standards, 45 (1950), pp. 255-282] produces a tridiagonal matrix that is, at each step, the exact result for the process applied to a strange augmented problem. Since the process is not stable in the standard sense, this augmented stability result cannot be transformed to prove standard stability. The intent is to obtain an increased understanding of the Lanczos tridiagonalization process, and this result could later be used to analyze the many applications of the process to large sparse matrix problems, such as the solution of the eigenproblem, compatible linear systems, least squares, and the singular value decomposition.