Francisco Arrieta Zuccalli

arXiv2026

Lambda admissible subspaces of self adjoint matrices

Given a self-adjoint matrix $A$ and an index $h$ such that $λ_h(A)$ lies in a cluster of eigenvalues of $A$, we introduce the novel class of $Λ$-admissible subspaces of $A$ of dimension $h$. First, we show that the low-rank approximation of the form $P_{\mathcal{T}} A P_{\mathcal{T}}$, for a subspace $\mathcal{T}$ that is close to any $Λ$-admissible subspace of $A$, has nice properties. Then, we prove that some well-known iterative algorithms (such as the Subspace Iteration Method, or the Krylov subspace method) produce subspaces that become arbitrarily close to $Λ$-admissible subspaces. We obtain upper bounds for the distance between subspaces obtained by the Rayleigh-Ritz method applied to $A$ and the class of $Λ$-admissible subspaces. We also find upper bounds for the condition number of the (set-valued) map computing the class of $Λ$-admissible subspaces of $A$. Finally, we include numerical examples that show the advantage of considering this new class of subspaces in the clustered eigenvalue setting.

math.NA↗

arXiv2024

Block subspace expansions for eigenvalues and eigenvectors approximation

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 $θ_i(\mathcal X,\mathcal V+A(\mathcal W_0))\leq θ_i(\mathcal V+A(\mathcal W))$ for every $\mathcal W\subset \mathcal V$ with $\dim \mathcal W\leq d$, where $θ_i(\mathcal X,\mathcal T)$ 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.

math.NA↗