Transformation to Standard Problems

A common approach for the numerical solution of the large sparse generalized eigenvalue problem (8.1) is to transform the problem first to an equivalent standard eigenvalue problem and then to apply an appropriate iterative method as described in Chapter 7. In this section, we will discuss three approaches for the transformation to a standard eigenproblem. The first approach (invert $B$) is recommended only if the matrix $B$ has a very simple structure and when linear systems of equations with matrices $B$ and $B^{\ast}$ can be solved efficiently. The second approach (split-invert $B$) is suitable when the matrix $B$ is Hermitian positive definite and when the Cholesky decomposition of $B$ can be computed efficiently a priori. For numerical stability, these two approaches require that the matrix $B$ be well-conditioned. The third approach is the shift-and-invert spectral transformation (SI), and this is the most common approach. We recommend using it whenever possible.