Block subspace expansions for eigenvalues and eigenvectors approximation

TL;DR

This paper introduces a method for expanding subspaces to better approximate eigenvalues and eigenvectors of a matrix, relating optimal expansions to block Krylov subspaces and demonstrating convergence for Hermitian matrices.

Contribution

It proposes a theoretical framework for optimal subspace expansions for eigenvalue approximation, connecting them to block Krylov subspaces and providing computable algorithms.

Findings

Optimal subspace expansions improve eigenvector approximation.

The method converges for Hermitian matrices with simple eigenvalues.

Numerical experiments validate the effectiveness of the algorithms.

Abstract

Let $A\in\mathbb C^{n\times n}$ and let $\mathcal X\subset \mathbb C^n$ be an $A$-invariant subspace with $\dim \mathcal X=d\geq 1$, corresponding to exterior eigenvalues of $A$. Given an initial subspace $\mathcal V\subset \mathbb C^n$ with $\dim \mathcal V=r\geq d$, we search for expansions of $\mathcal V$ of the form $\mathcal V+A(\mathcal W_0)$, where $\mathcal W_0\subset \mathcal V$ is such that $\dim \mathcal W_0\leq d$ and such that the expanded subspace is closer to $\mathcal X$ than the initial $\mathcal V$. We show that there exist (theoretical) optimal choices of such $\mathcal W_0$, in the sense that $\theta_i(\mathcal X,\mathcal V+A(\mathcal W_0))\leq \theta_i(\mathcal V+A(\mathcal W))$ for every $\mathcal W\subset \mathcal V$ with $\dim \mathcal W\leq d$, where $\theta_i(\mathcal X,\mathcal T)$ denotes the $i$-th principal angle between $\mathcal X$ and $\mathcal T$, for $1\leq i\leq d\leq \dim \mathcal T$. We relate these optimal expansions to block Krylov subspaces generated by $A$ and $\mathcal V$. We also show that the corresponding iterative sequence of subspaces constructed in this way approximate $\mathcal X$ arbitrarily well, when $A$ is Hermitian and $\mathcal X$ is simple. We further introduce computable versions of this construction and compute several numerical examples that show the performance of the computable algorithms and test our convergence analysis.