Iterative Refinement for a Subset of Eigenvectors of Symmetric Matrices via Matrix Multiplications

TL;DR

This paper introduces an efficient iterative refinement method for improving a selected subset of eigenvectors of symmetric matrices, leveraging matrix multiplications and eigenvalue separation conditions for convergence.

Contribution

It presents a novel iterative refinement technique specifically for a subset of eigenvectors, using compact WY form and efficient approximations from Rayleigh quotients.

Findings

Method converges linearly under eigenvalue separation conditions.

Practical variants enable targeting different spectrum parts.

Preprocessing can restore convergence for clustered eigenvalues.

Abstract

We develop an iterative refinement method that improves the accuracy of a user-chosen subset of $k$ eigenvectors ($k\ll n$) of an $n\times n$ real symmetric matrix. Using an orthogonal matrix represented in compact WY form, the method expresses the eigenvector error through a correction matrix that can be approximated efficiently from Rayleigh quotients and residuals. Unlike refinement methods for a single eigenpair or for a full eigenbasis, the proposed method refines only the selected $k$ eigenvectors using $\mathcal{O}(nk)$ additional storage, and its dominant work can be organized as matrix--matrix multiplications. Under an eigenvalue separation condition, the refinement converges linearly; we also provide a conservative sufficient condition. Practical variants of the separation condition (e.g., via shifting) enable targeting other extremal parts of the spectrum. For tightly clustered eigenvalues, we discuss limitations and show that preprocessing can restore convergence in a representative sparse example. Numerical experiments on dense test matrices and sparse matrices from the SuiteSparse Matrix Collection illustrate attainable accuracy and problem-dependent convergence.