Symmetric Indefinite Lanczos Method

In this section, we present a Lanczos method for solving
the generalized eigenvalue problem

Such eigenvalue problems come from various applications, such as the linearization of a certain quadratic eigenvalue problem, which often arises in the modeling of damped structural systems; see §9.2.

Formally, the symmetric Lanczos algorithm may be used to compute some
eigenpairs, since the matrix
is symmetric with respect to the inner product.^{}The three-term recurrence still holds
with respect to the inner product
in this more general situation.
The algorithm is referred to as *a symmetric indefinite Lanczos method*.
The main trouble with this method is that the basis vectors are orthogonal
with respect to an indefinite inner product, so there is no assurance that
they will be linearly independent. The algorithm could occasionally fail
due to a breakdown. Nevertheless, this is an attractive
way to solve the problem because of potentially significant
savings in memory requirement and floating point operations.

- Some Properties of Symmetric Indefinite Matrix Pairs
- Algorithm
- Stopping Criteria and Accuracy Assessment
- Singular
- Software Availability
- Numerical Examples
- Notes and References