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Two-sided Grassmann-Rayleigh quotient iteration P.-A. Absil
 

Summary: Two-sided Grassmann-Rayleigh quotient iteration
P.-A. Absil
P. Van Dooren
Submitted to Numer. Math. on 19 Apr 2007
Abstract
The two-sided Rayleigh quotient iteration proposed by Ostrowski computes a pair of
corresponding left-right eigenvectors of a matrix C. We propose a Grassmannian ver-
sion of this iteration, i.e., its iterates are pairs of p-dimensional subspaces instead of
one-dimensional subspaces in the classical case. The new iteration generically converges
locally cubically to the pairs of left-right p-dimensional 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 skew-Hamiltonian
eigenproblem.
Keywords. Block Rayleigh quotient iteration, two-sided 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

  

Source: Absil, Pierre-Antoine - Département d'ingénierie Mathématique, Université Catholique de Louvain

 

Collections: Mathematics