Rayleigh-Ritz and Lanczos methods for symmetric matrix pencils

Abstract

We are concerned with eigenvalue problems for definite and indefinite symmetric matrix pencils. First, Rayleigh-Ritz methods are formulated and, using Krylov subspaces, a convergence analysis is presented for definite pencils. Second, generalized symmetric Lanczos algorithms are introduced as a special Rayleigh-Ritz method. In particular, an a posteriori convergence criterion is demonstrated by using residuals. Local convergence to real and nonreal eigenvalues is also discussed. Numerical examples concerning vibrations of damped cantilever beams are included.