Yesterday I gave a talk on the preliminary results of my master's thesis on Projection Methods for Generalized Eigenvalue Problems. Below is the announcement for the talk; it mentions only complex (Hermitian) matrices but all results are immediately applicable to real matrices. The slides are here (PDF).

Projection Methods for Generalized Eigenvalue Problems

In this talk, we will discuss backward errors, dense solvers, and a new projection method for generalized eigenvalue problems (GEPs) with Hermitian positive semidefinite (HPSD) matrices.

We will present properties and origins of such GEPs. We will also address quickly computable and structure preserving backward error bounds for these kinds of GEPs. There is an abundance of literature on backward error measures possessing one of these features but only recently, the author came across a backward error providing both.

We will elaborate on dense solvers for GEPs with HPSD matrices. The standard solver for GEPs with Hermitian matrices is fast but requires positive definite mass matrices and is only conditionally backward stable; the QZ algorithm for general GEPs is backward stable but it is also magnitudes slower and does not preserve any problem properties. In the talk, we will present two new backward stable and structure preserving solvers, one using deflation, the other one using the generalized singular value decomposition (GSVD). In comparison to the QZ algorithm, both solvers are competitive with the standard solver in our tests.

Finally, we will touch on a new solver for large, sparse GEPs.