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<p>This paper presents the perturbation analysis of the sampling semi-randomized Kaczmarz (SSRK) method, proposed in <i>Advances in Computational Mathematics, 2023, 49(2):20</i>, for solving partially- and doubly-noisy linear systems. SSRK uses a computationally efficient greedy row selection strategy and generalizes the sampling Kaczmarz–Motzkin method. While the convergence of the standard randomized Kaczmarz (RK) method under perturbations of the coefficient matrix and the right-hand side vector has been studied recently in <i>SIAM Journal on Matrix Analysis and Applications, 45(2):992–1006, 2024</i>, our work focuses on SSRK, which is a fundamentally different and more efficient iterative solver. We rigorously derive the convergence rate of SSRK and characterize its behavior using perturbation analysis. A key contribution is the establishment of error bounds under various noise conditions, including additive and multiplicative noise. Numerical experiments validate our theoretical results, confirming that SSRK not only provides computational advantages but also maintains reliable performance in the presence of solving perturbed linear systems.</p>

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On the perturbation analysis of the sampling semi-randomized Kaczmarz method

  • 期刊论文

  • 发表日期：

    2026年6月26日

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  • Nian-Ci Wu,
  • Ya-Nan Zhao,
  • An-Yan Hu,
  • Chengzhi Liu

摘要

This paper presents the perturbation analysis of the sampling semi-randomized Kaczmarz (SSRK) method, proposed in Advances in Computational Mathematics, 2023, 49(2):20, for solving partially- and doubly-noisy linear systems. SSRK uses a computationally efficient greedy row selection strategy and generalizes the sampling Kaczmarz–Motzkin method. While the convergence of the standard randomized Kaczmarz (RK) method under perturbations of the coefficient matrix and the right-hand side vector has been studied recently in SIAM Journal on Matrix Analysis and Applications, 45(2):992–1006, 2024, our work focuses on SSRK, which is a fundamentally different and more efficient iterative solver. We rigorously derive the convergence rate of SSRK and characterize its behavior using perturbation analysis. A key contribution is the establishment of error bounds under various noise conditions, including additive and multiplicative noise. Numerical experiments validate our theoretical results, confirming that SSRK not only provides computational advantages but also maintains reliable performance in the presence of solving perturbed linear systems.