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<p>Row-action methods provide a powerful framework for large-scale linear problems by iteratively enforcing constraints through low-dimensional projections. While highly successful for over-determined linear systems, their potential for matrix equations remains underexplored. This paper extends the row-action paradigm to the matrix equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(AXB = F\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>X</mi> <mi>B</mi> <mo>=</mo> <mi>F</mi> </mrow> </math></EquationSource> </InlineEquation> by introducing the <b>A</b>lternating <b>R</b>andomized <b>B</b>lock <b>K</b>aczmarz (ARBK) method. ARBK operates by alternately applying randomized block Kaczmarz iterations to two interrelated linear systems with multiple right-hand sides, effectively decomposing the matrix problem into a sequence of manageable row-action steps. We establish a tight linear convergence rate for ARBK, with an explicit bound that quantifies its performance. Numerical experiments demonstrate that our method not only validates the theory but also achieves competitive performance on large-scale data fitting tasks, highlighting the efficacy of row-action methods in this extended setting.</p>

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On the Alternating randomized Row-action Methods with the Application to Data Fitting

  • 期刊论文

  • 发表日期：

    2026年7月21日

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  • Nian-Ci Wu,
  • Yang Zhou

摘要

Row-action methods provide a powerful framework for large-scale linear problems by iteratively enforcing constraints through low-dimensional projections. While highly successful for over-determined linear systems, their potential for matrix equations remains underexplored. This paper extends the row-action paradigm to the matrix equation \(AXB = F\) A X B = F by introducing the Alternating Randomized Block Kaczmarz (ARBK) method. ARBK operates by alternately applying randomized block Kaczmarz iterations to two interrelated linear systems with multiple right-hand sides, effectively decomposing the matrix problem into a sequence of manageable row-action steps. We establish a tight linear convergence rate for ARBK, with an explicit bound that quantifies its performance. Numerical experiments demonstrate that our method not only validates the theory but also achieves competitive performance on large-scale data fitting tasks, highlighting the efficacy of row-action methods in this extended setting.