2025/06/11 by Vance Faber, Faber, Vance, Jörg Liesen +3
Computer Science · Mathematics · #65F10 #Advanced Optimization Algorithms Research #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2506.09687
openalex publication_date 2025/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We derive, similar to Lau and Riha, a matrix formulation of a general best approximation theorem of Singer for the special case of spectral approximations of a given matrix from a given subspace. Using our matrix formulation we describe the relation of the spectral approximation problem to semidefinite programming, and we present a simple MATLAB code to solve the problem numerically. We then obtain geometric characterizations of spectral approximations that are based on the k-dimensional field of k matrices, which we illustrate with several numerical examples. The general spectral approximation problem is a min-max problem, whose value is bounded from below by the corresponding max-min problem. Using our geometric characterizations of spectral approximations, we derive several necessary and sufficient as well as sufficient conditions for equality of the max-min and min-max values. Finally, we prove that the max-min and min-max values are always equal when we ``double'' the problem. Several results in this paper generalize results that have been obtained in the convergence analysis of the GMRES method for solving linear algebraic systems.