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.

翻译：Krylov子空间方法是高效求解高维线性代数问题的有力工具。本文研究了Krylov子空间在估计矩阵数值域时提供的近似质量。与以往通常依赖于特征值间隙的研究结果不同，我们的估计仅取决于矩阵和Krylov子空间的维度以及矩阵特征基的条件数。此外，我们给出了与估计值近乎匹配的下界，从而证明了我们论证的紧致性。