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,