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<p>The Kaczmarz method is one of the most popular iterative methods for solving a consistent system of linear equation. Based on row and column selection criterions for sets of block controlled indices, we propose a deterministic block Kaczmarz method to solve matrix equation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(AXB=C\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>X</mi> <mi>B</mi> <mo>=</mo> <mi>C</mi> </mrow> </math></EquationSource> </InlineEquation>. Each iteration adaptively selects the sets of block control indices and is pseudoinverse-free. Moreover, the Polyak’s heavy ball momentum technique is integrated into the deterministic block Kaczmarz method to improve the performance. The theoretical analysis shows that the method will converge linearly to the unique minimal Frobenius norm solution. Numerical experiments are given to illustrate the feasibility and efficiency of the proposed methods.</p>

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An improved deterministic block Kaczmarz method for solving linear matrix equation \(AXB=C\)

  • 期刊论文

  • 发表日期：

    2025年9月8日

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  • Yihong Wang,
  • Yongzhong Song

摘要

The Kaczmarz method is one of the most popular iterative methods for solving a consistent system of linear equation. Based on row and column selection criterions for sets of block controlled indices, we propose a deterministic block Kaczmarz method to solve matrix equation \(AXB=C\) A X B = C . Each iteration adaptively selects the sets of block control indices and is pseudoinverse-free. Moreover, the Polyak’s heavy ball momentum technique is integrated into the deterministic block Kaczmarz method to improve the performance. The theoretical analysis shows that the method will converge linearly to the unique minimal Frobenius norm solution. Numerical experiments are given to illustrate the feasibility and efficiency of the proposed methods.