Computational Variants of the Lanczos Method for the Eigenproblem

Abstract

The Lanczos process is theoretically ideal for finding several extreme eigenvalues of large sparse symmetric matrices, but because of loss of orthogonality in practice it has largely been ignored, or used with re-orthogonalization. Here it is shown that it can still be an extremely efficient and accurate algorithm if used in an iterative manner. An error analysis is given showing the most accurate algorithms, and the conclusions are supported by computational results.