Applications of the Lanczos method

Abstract

In this paper we are concerned with the application of the unsymmetric two sided Lanczos method to systems of first order differential equations and its associated problems; namely generalized eigenproblems and linear systems of equations. The unsymmetric Lanczos method is first used to generate two sets of vectors; the left and right Lanczos vectors. These Lanczos vectors are used directly in a method of weighted residual to reduce the differential equations to a small unsymmetric tridiagonal system. The advantage of this method is that the reduced system can be solved directly thus eliminating the usual computation of the complex eigenpairs. An algorithm is then derived by simplifying the two sided Lanczos method for systems of equations with a symmetric matrix pair. By appropriate choice of the starting vectors we obtain an implementation of the Lanczos method that is remarkably close to that in ref. [9], but generalized to the case with indefinite matrix coefficients. This modification results in a simple relation between the left and right Lanczos vectors, a symmetric tridiagonal, and a diagonal matrix. The system of differential equations can then be reduced to one with a symmetric tridiagonal and diagonal coefficient matrices. The modified algorithm is also suitable for eigenproblems with symmetric indefinite matrices and for solution of systems of equations using indefinite preconditioning matrices. This approach is used to evaluate the vibration response of a damped beam problem and a space mast structure with a symmetric damping matrix arising from velocity feedback control forces. In both problems, accurate solutions were obtained with as few as 20 Lanczos vectors.