On the relaxed greedy deterministic row and column iterative methods

TL;DR

This paper introduces two new relaxed greedy deterministic iterative methods for large-scale linear systems, demonstrating their linear convergence and superior effectiveness over randomized counterparts through numerical experiments.

Contribution

The paper proposes two novel relaxed greedy deterministic row and column iterative methods with proven linear convergence, extending existing deterministic approaches and improving performance.

Findings

Algorithms have linear convergence rates with explicit bounds.

Numerical results show higher effectiveness than randomized methods.

Special case reduces to existing fast deterministic block Kaczmarz method.

Abstract

For solving the large-scale linear system by iteration methods, we utilize the Petrov-Galerkin conditions and relaxed greedy index selection technique and provide two relaxed greedy deterministic row (RGDR) and column (RGDC) iterative methods, in which one special case of RGDR reduces to the fast deterministic block Kaczmarz method proposed in Chen and Huang (Numer. Algor., 89: 1007-1029, 2021). Our convergence analyses reveal that the resulting algorithms all have the linear convergence rates, which are bounded by the explicit expressions. Numerical examples show that the proposed algorithms are more effective than the relaxed greedy randomized row and column iterative methods.