 
Summary: Twosided GrassmannRayleigh quotient iteration
P.A. Absil
P. Van Dooren
Submitted to Numer. Math. on 19 Apr 2007
Abstract
The twosided Rayleigh quotient iteration proposed by Ostrowski computes a pair of
corresponding leftright eigenvectors of a matrix C. We propose a Grassmannian ver
sion of this iteration, i.e., its iterates are pairs of pdimensional subspaces instead of
onedimensional subspaces in the classical case. The new iteration generically converges
locally cubically to the pairs of leftright pdimensional invariant subspaces of C. More
over, Grassmannian versions of the Rayleigh quotient iteration are given for the general
ized Hermitian eigenproblem, the Hamiltonian eigenproblem and the skewHamiltonian
eigenproblem.
Keywords. Block Rayleigh quotient iteration, twosided iteration, Grassmann manifold,
generalized eigenproblem, Hamiltonian eigenproblem.
AMS subject classification. 65F15
1 Introduction
The Rayleigh quotient iteration (RQI) is a classical method for computing eigenvectors of a
Hermitian matrix A = AH [Par74, Par98]. The RQI is a particular inverse iteration [Ips97]
where the shift is the Rayleigh quotient evaluated at the current iterate. The Rayleigh
