Generalized Gearhart-Koshy acceleration is a Krylov subspace method

Abstract

The Kaczmarz method is a row-action method for solving consistent non-square linear systems, and Gearhart-Koshy acceleration is a line-search that minimizes the Euclidean norm of the error along a ray in the direction of a Kaczmarz step. Recently one of the authors generalized this procedure to a search for the point with minimal Euclidean error norm within a sequence of nested affine subspaces. In this paper, we demonstrate that this generalization can be interpreted as a Krylov subspace method for a square linear system, which is equivalent with the original system to be solved. In exact arithmetic the method cannot break down prematurely, and it makes progress in every step. We also present a mathematically equivalent reformulation of the algorithm in terms of the Gram-Schmidt orthogonalization procedure, and we illustrate the convergence behavior of the new method with numerical experiments.