Eine Variante des Lanczos-Verfahrens

Abstract

The computation of eigenfrequencies and eigenmodes of technical systems by the method of finite elements leads to generalized eigenvalue problems Ax = λBx with symmetric and sparse matrices A and B of high order n. If the classical Lanczos algorithm is applied to a sequence of inverse and shifted eigenvalue problems with appropriate chosen shifts, the resulting method is indeed very efficient. Each Lanczos run should yield only a certain group of eigenvalues and eigenvectors. In case of small group sizes reorthogonalization of the iterated basis vectors is unnecessary due to a small number of Lanczos steps required. The proposed version combines some techniques used in the bisection method in order to increase the reliability and efficiency of the process. Some examples show the superiority of the algorithm compared with other methods.