A note on harmonic Ritz values and their reciprocals

作者：Voemel Christof

来源：Numerical Linear Algebra with Applications, 2010, 17(1): 97-108.

DOI：10.1002/nla.653

• Rayleigh quotient
• Ritz value
• harmonic Ritz value
• reciprocal
• matrix-free computation
• interior eigenpair
• sparse iterative eigensolver
• symmetric eigenproblem
• generalized symmetric positive definite eigenproblem
• preconditioned conjugate gradients

This note summarizes an investigation of harmonic Ritz values to approximate the interior eigenvalues of a real symmetric matrix A while avoiding the explicit use of the inverse A(-1).
We consider a bounded functional psi that yields the reciprocals of the harmonic Ritz values of a symmetric matrix A. The crucial observation is that with an appropriate residual s, many results from Rayleigh quotient and Rayleigh-Ritz theory naturally extend. The same is true for the generalization to matrix pencils (A, B) when B is symmetric positive definite.
These observations have an application in the computation of eigenvalues in the interior of the spectrum of a large sparse matrix. The minimum and maximum of psi correspond to the eigenpairs just to the left and right of zero (or a chosen shift). As a spectral transformation, this distinguishes psi from the original harmonic approach where an interior eigenvalue remains at the interior of the transformed spectrum. As a consequence, psi is a very attractive vehicle for a matrix-free, optimization-based eigensolver. Instead of computing the smallest/largest eigenvalues by minimizing/maximizing the Rayleigh quotient, one can compute interior eigenvalues as the minimum/maximum of psi.

• 出版日期2010-1