In many cases we can transform the pencil (5.1) into a standard HEP discussed in Chapter 4, or in some cases into a standard NHEP discussed in Chapter 7, and use any of the algorithms described in these two chapters. We give a short review of such transformations in §5.2.

The main body of this chapter is devoted to algorithms that take a special form for generalized Hermitian pencils (5.1):

*Power method and inverse iteration,*- §5.4, is the basic iterative method. It starts
with an appropriate starting vector and makes one matrix-vector
multiplication and one linear system solve in each iteration.
Needing both these operations, it is
less appealing than in the standard case, but it is included here because
of its simplicity and to clarify its relations to more sophisticated
schemes described later.
*Lanczos method,*- §5.5,
builds up a orthogonal basis of a Krylov sequence of vectors in
which the matrix operator is represented by a tridiagonal matrix ,
whose eigenvalues yield Ritz approximations to several of the
eigenvalues of the original matrix pencil (5.1). We will
describe two variants in §5.5, one direct
iteration that needs multiplications with and the solution of
systems with , and one shift-and-invert iteration that needs
solution of systems with and multiplication with .
*Jacobi-Davidson method,*- §5.6, updates a sequence of subspaces, operating with a preconditioned shifted matrix. It uses orthogonality and does not need the solution of linear systems. This makes it applicable in several cases when the matrices are too large to allow for solving linear systems, as needed in the other algorithms described in this chapter.