Using Generalized Cayley Transformations within an Inexact Rational Krylov Sequence Method

The rational Krylov sequence (RKS) method is a generalization of Arnoldi's method. It constructs an orthogonal reduction of a matrix pencil into an upper Hessenberg pencil. The RKS method is useful when the matrix pencil may be eeciently factored. This article considers approximately solving the resulting linear systems with iterative methods. We show that a Cayley transformation leads to a more eecient and robust eigensolver than the usual shift-invert transformation when the linear systems are solved inexactly within the RKS method. A relationship with the recently introduced Jacobi{Davidson method is also established.