     Next: Algorithm. Up: Generalized Hermitian Eigenvalue Problems Previous: Rayleigh Quotient Iteration.   Contents   Index

# Lanczos Methods A. Ruhe

There are several variants of the Lanczos algorithm for the GHEP (5.1). Theoretically they correspond to a reformulation of (5.1) as a standard problem , with a matrix chosen as either , which corresponds to a direct iteration, or , which corresponds to inverse iteration. We will actually study the slightly more general formulation , shift-and-invert iteration. This is the variant that is preferred in most practical cases because it gives fast convergence to eigenvalues close to the target value , provided that we can solve linear systems with the shifted matrix.

The cause for using shift-and-invert iterations is stronger in this generalized case (5.1) than in the standard (4.1), since also a direct iteration needs solution of linear systems in each step, now with as a matrix. Even if most often is better conditioned than , e.g., when stands for a mass matrix in a vibration problem, it is only when has a much simpler structure, like being diagonal, that direct iterations need substantially less work in each step than shift-and-invert iterations.

In all the variants, a basis of the Krylov subspace, is built up one column at a time, alternately solving systems and doing matrix-vector multiplications.

Subsections     Next: Algorithm. Up: Generalized Hermitian Eigenvalue Problems Previous: Rayleigh Quotient Iteration.   Contents   Index
Susan Blackford 2000-11-20