The Lanczos algorithm with selective orthogonalization

Abstract

A new stable and efficient implementation of the Lanczos algorithm is presented. The algorithm is a powerful method for finding a few eigenvalues and eigenvectors at one or both ends of the spectrum of a symmetric matrix A. The algorithm is particularly effective if A is large and sparse in that the only way in which A enters the calculation is through a subroutine which computes Av for any vector v. Thus the user is free to take advantage of any sparsity structure in A and A need not even be represented as a matrix et al.