Estimating the numerical range with a Krylov subspace

TL;DR

The paper develops gap-free, probabilistic bounds for how accurately the Krylov-subspace projection $H_m$ captures the numerical range $W(A)$ of a matrix $A$. By reframing eigenvalue approximation as a polynomial extremal problem and leveraging convex-geometry and probabilistic tools, it derives sharp upper and lower bounds for normal matrices, including enhancements when $\Lambda(A)$ lies on the unit circle. It then extends the analysis to non-normal matrices, showing degradation with eigenvector conditioning but identifying regimes (notably β-normal repulsion) where the convex hull of the spectrum remains well-approximated by $W(H_m)$. The results provide concrete, dimension-dependent rates (e.g., $O(\tfrac{\ln n}{m})$ for normal and $O(\tfrac{\ln^2 n}{m^2})$ on the unit circle) and support practical Krylov-based estimates for extremal eigenvalues and convergence analyses in GMRES-type contexts.

Abstract

Krylov subspace methods are a powerful tool for efficiently solving high-dimensional linear algebra problems. In this work, we study the approximation quality that a Krylov subspace provides for estimating the numerical range of a matrix. In contrast to prior results, which often depend on the gaps between eigenvalues, our estimates depend only on the dimensions of the matrix and Krylov subspace, and the conditioning of the eigenbasis of the matrix. In addition, we provide nearly matching lower bounds for our estimates, illustrating the tightness of our arguments.

Figures (5)

• Figure 1: Estimating the eigenvalues of $A = \mathrm{diag}(\{e^{2 \pi i j/n}\}_{j=1}^n)$ using $\mathcal{K}_m(A,\bm b)$ for $\bm{b} \sim \mathrm{Unif}(\mathbb{S}^{n-1})$, where $n = 10^4$ and $m = 30$. In Subfigure (a), note that, while there is a decent gap between $\Lambda(A)$ and $\Lambda(H_m)$, the difference between $\Lambda(A)$ and $\partial W(H_m)$ is imperceptible. This is further quantified in Subfigure (b), where we note that the Hausdorff distance between $\mathrm{conv}(\Lambda(A))$ and $\mathrm{conv}(\Lambda(H_m))$ scales roughly like $1/m$, while the Hausdorff distance between $\mathrm{conv}(\Lambda(A))$ and $W(H_m)$ scales like $1/m^2$.
• Figure 2: The mapping from eigenvalues to inputs for Remez-type polynomials. The desired extremal eigenvalue is shown in red, and the blue region represents all possible eigenvalues at least $\delta$ distance away. The illustrated transformation maps the blue region $D_\delta$ into the outlined region $R_\varepsilon$ from Proposition \ref{['prop:remez_disk']}
• Figure 3: Numerical range estimates of $A$ from Theorem \ref{['thm:norm_lower']} (top row) and Theorem \ref{['thm:circle_upper']} (bottom row). Whereas $n$ changes as a function of $m$ to simplify the proof, the essence of the statement holds for each matrix individually. Here we fix $A \in \mathbb{C}^{n \times n}$ with $n=10240$ with eigenvalue multiplicity $10$ and run Arnoldi for $m$ up to 50. In Subfigures (a) and (d), we plot part of the numerical range of $A$ along with the estimated numerical range for 10 random starting vectors $\bm{b}$ and $m=50$. In Subfigures (b) and (e), we plot the Hausdorff distance between the estimated and actual numerical ranges as $m$ increases. To better see the $1/m$ and $1/m^2$ behavior respectively, we multiply the error by $m$ in Subfigure (c) and by $m^2$ in Subfigure (f).
• Figure 4: The numerical range of $H_{m}$ for $A = \mathrm{diag}(\cos \tfrac{(j-1)\pi}{n}) + 2 \bm{e}_1 \bm{e}_n^T$ and ten different $\bm{b} \sim \mathrm{Unif}(\mathbb{S}^{n-1})$, where $n = 1000$ and $m = 100$. The numerical range quickly (approximately) contains $[-1,1]$ (the convex hull of eigenvalues of $A$) in all instances, but fails to reasonably approximate the numerical range of $A$. In over half the cases, $W(H_{100})$ is barely bigger than $[-1,1]$, and in all cases the Hausdorff distance between $W(A)$ and $W(H_{100})$ is still at least $1/2$.
• Figure 5: $W(H_{m})$ for $A$ in Example \ref{['ex:eigenangles']}. Here we let $n=2560$, $m=16$, and $l=10$.

Theorems & Definitions (55)

• Proposition 2.1: polytope,popov
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4: erdelyi1994remez
• Proposition 2.5
• Proposition 2.6
• Proposition 2.7: wainwright2019high
• Proposition 2.8