The recently proposed Riemannian Trust-Region method can be applied to the problem of computing extreme eigenpairs of a matrix
pencil, with strong global convergence and local convergence properties. This paper addresses inherent inefficiencies of an
explicit trust-region mechanism. We propose a new algorithm, the Implicit Riemannian Trust-Region method for extreme eigenpair
computation, which seeks to overcome these inefficiencies while still retaining the favorable convergence properties.
This work was supported by NSF Grant ACI0324944. The first author was in part supported by the CSRI, Sandia National Laboratories.
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department
of Energy; contract/grant number: DE-AC04-94AL85000. The second author was partially supported by Microsoft Research through
a Research Fellowship at Peterhouse, Cambridge.