Symmetric Indefinite Lanczos Method
 Z. Bai, T. Ericsson, and T. Kowalski

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

$$A x = \lambda B x,$$ (234)

where matrices $A$ and $B$ are real or complex symmetric but neither $A$ nor $B$ nor a combination $\alpha A + \beta B$ for scalars $\alpha$ and $\beta$ is positive definite. $A - \lambda B$ is called a symmetric indefinite matrix pencil.

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 $B^{-1}A$ is symmetric with respect to the $B$ inner product.The three-term recurrence still holds with respect to the $B$ 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.