OBK-RCM: Accelerated Orthogonal Block Kaczmarz Algorithm via RCM Reordering and Dynamic Grouping for Sparse Linear Systems

Abstract: Existing block Kaczmarz methods face challenges in balancing computational efficiency and convergence for large sparse linear systems with scattered nonzero patterns, due to costly partitioning strategies and non-orthogonal projections. In this paper, we propose the orthogonal block Kaczmarz (OBK-RCM) algorithm with the Reverse Cuthill-McKee (RCM), which integrates the RCM reordering with a novel orthogonal block partitioning strategy. RCM transforms sparse matrices into banded structures to enhance inter-block orthogonality, while dynamic grouping of mutually orthogonal blocks based on angle cosine thresholds reduces iterative complexity. In addition, two extended versions (SOBK-RCM and UOBK-RCM) are proposed to deal with non-square systems by constructing extended matrices without sacrificing sparsity. This work offers a practical framework for efficient sparse linear algebra solvers. Experiments on 33 real-world and synthetic matrices show that OBK-RCM achieves 10-50 times faster CPU time (up to several hundred) and 50-90% fewer iterations than state-of-the-art methods (RBK,RBK(k),GREBK(k),aRBK), especially for scattered sparse structures in most cases. Theoretical analysis confirms linear convergence, driven by hyperplane orthogonality.