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<p>In the current paper, we consider the numerical solution of a block circulant tridiagonal linear system which commonly originates from convolution equations under periodic boundary conditions. By leveraging the block-Toeplitz structure, we propose a novel structure-preserving factorization of the coefficient matrix. Based on the structure-preserving matrix factorization and Sherman-Morrison-Woodbury formula, we then develop an efficient numerical algorithm with linear time complexity for solving block circulant tridiagonal linear systems. Additionally, a theoretical error analysis is provided to ensure numerical stability, and a numerical formula for the determinant of the block circulant tridiagonal matrix is also presented. Numerical results with simulations in MATLAB implementation are provided to demonstrate the accuracy and efficiency of our proposed algorithm, and its competitiveness with the block <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(LU\)</EquationSource> </InlineEquation> decomposition method.</p>

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Numerical solution of block circulant tridiagonal linear systems originating from convolution equations

  • 期刊论文

  • 发表日期：

    2025年12月16日

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  • Zheng Tang,
  • Ji-Teng Jia

摘要

In the current paper, we consider the numerical solution of a block circulant tridiagonal linear system which commonly originates from convolution equations under periodic boundary conditions. By leveraging the block-Toeplitz structure, we propose a novel structure-preserving factorization of the coefficient matrix. Based on the structure-preserving matrix factorization and Sherman-Morrison-Woodbury formula, we then develop an efficient numerical algorithm with linear time complexity for solving block circulant tridiagonal linear systems. Additionally, a theoretical error analysis is provided to ensure numerical stability, and a numerical formula for the determinant of the block circulant tridiagonal matrix is also presented. Numerical results with simulations in MATLAB implementation are provided to demonstrate the accuracy and efficiency of our proposed algorithm, and its competitiveness with the block \(LU\) decomposition method.