Two-sided Grassmann-Rayleigh quotient iteration

TL;DR

This paper introduces a Grassmannian extension of the two-sided Rayleigh quotient iteration that computes pairs of invariant subspaces for matrices, achieving local cubic convergence and applicable to various eigenproblems.

Contribution

It develops a novel Grassmannian version of the iteration, generalizing the classical method to subspaces and extending its applicability to several structured eigenproblems.

Findings

Achieves local cubic convergence for invariant subspace pairs.

Extends to generalized Hermitian, Hamiltonian, and skew-Hamiltonian eigenproblems.

Provides a new iterative method for structured eigenproblems.

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 version 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 $C$. Moreover, Grassmannian versions of the Rayleigh quotient iteration are given for the generalized Hermitian eigenproblem, the Hamiltonian eigenproblem and the skew-Hamiltonian eigenproblem.