On Convergence of dqds Algorithm that Uses Ostrowski and Brauer Type Lower Bounds as Shifts(Theory,Algorithms for Matrix/Eigenvalue Problems and their Applications,<Special Issue>Joint Symposium of JSIAM Activity Groups 2007)

Abstract

Recently, Aishima et al. proved that the dqds algorithm that uses the Johnson bound for the smallest singular value as shifts is globally convergent and its asymptotic convergence rate is 1.5/ In this paper, we study the convergence of the dqds algorithm when Ostrowski and Brauer type bounds, which are stronger lower bounds than the Johnson bound, are used as shifts. Both shifting strategies satisfy the conditions for global convergence. The asymptotic convergence rate is shown to be 1.5 for the Ostrowski bound and super-1.5 for the Brauer bound. Numerical experiments support our theoretical analysis.