Among randomized numerical linear algebra strategies, so-called sketching procedures are emerging as effective reduction means to accelerate the computation of Krylov subspace methods for, e.g., the solution of linear systems, eigenvalue computations, and the approximation of matrix functions. While there is plenty of experimental evidence showing that sketched Krylov solvers may dramatically improve performance over standard Krylov methods, many features of these schemes are still unexplored. We derive a new sketched Arnoldi-type relation that allows us to obtain several different new theoretical results. These lead to an improvement of our understanding of sketched Krylov methods, and to identifying, among several possible equivalent formulations, the most suitable sketched approximations according to their numerical stability properties. These results are also employed to analyze the error of sketched Krylov methods in the approximation of the action of matrix functions, significantly contributing to the theory available in the current literature.

翻译：在随机数值线性代数策略中，所谓的草稿化方法正逐步成为有效的简化手段，用于加速Krylov子空间方法在求解线性系统、特征值计算以及矩阵函数逼近等问题中的应用。尽管大量实验证据表明，草稿化Krylov求解器相比标准Krylov方法能显著提升性能，但这些方案的诸多特性仍未得到充分探索。本文推导出一种新的草稿化Arnoldi型关系，由此可得出多项全新的理论结果。这些结果加深了我们对草稿化Krylov方法的理解，并能够从多种等效表述中，根据数值稳定性特性识别出最合适的草稿化近似。本文还利用这些结果分析了草稿化Krylov方法在逼近矩阵函数作用时的误差，显著丰富了现有文献中的相关理论。