Lanczos Methods

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.