A Model-Trust-Region Framework for Symmetric Generalized Eigenvalue Problems

A general inner-outer iteration for computing extreme eigenpairs of symmetric/positive-definite matrix pencils is proposed. The principle of the method is to produce a sequence of p-dimensional bases {Xk} that converge to a minimizer of a generalized Rayleigh quotient. The role of the inner iteration is to produce an “update” vector by (approximately) minimizing a quadratic model of the Rayleigh quotient within a neighbourhood of Xk where the model is trusted. The role of the outer iteration is to make the best out of the proposed update vector combined with previously-obtained information; it consists of a Rayleigh-Ritz process that minimizes the exact Rayleigh quotient in a low-dimensional subspace. This general scheme leaves a lot of leeway for choosing the algorithmic details of the inner and outer iterations. The global and local convergence of the scheme are analytically studied under weak assumptions on these algorithmic choices. Moreover, numerous experiments are carried out to explore the influence of the algorithmic choices on the performance of the scheme. In particular, the question of balancing the computational effort between the inner and outer iterations is investigated. ∗School of Computational Science, Florida State University, Tallahassee, FL 323064120, USA (http://www.csit.fsu.edu/{∼cbaker,∼absil,∼gallivan}). This work was supported by the USA National Science Foundation under Grants ACI0324944 and CCR9912415, and by the School of Computational Science of Florida State University through a postdoctoral fellowship.