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<p>In this paper, we first propose a novel block diagonalization approach for evaluating the determinant of the block circulant tridiagonal matrix. Based on this approach, we transform the determinant of the block circulant tridiagonal matrix into the determinants of block matrices with lower order. Then we develop a breakdown-free algorithm for the block circulant matrices with circulant tridiagonal blocks and derive an explicit formula for those with Toeplitz-structured blocks. Furthermore, the block diagonalization is further extended to applications such as computing matrix integer powers, inverses, and condition numbers. Numerical experiments demonstrate the superiority of the proposed algorithms over MATLAB built-in functions in terms of computational efficiency and stability.</p>

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Numerical algorithms for determinant evaluation of the block circulant tridiagonal matrices arising from convolution equations

  • 期刊论文

  • 发表日期：

    2025年7月29日

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  • Zheng Tang,
  • Ji-Teng Jia

摘要

In this paper, we first propose a novel block diagonalization approach for evaluating the determinant of the block circulant tridiagonal matrix. Based on this approach, we transform the determinant of the block circulant tridiagonal matrix into the determinants of block matrices with lower order. Then we develop a breakdown-free algorithm for the block circulant matrices with circulant tridiagonal blocks and derive an explicit formula for those with Toeplitz-structured blocks. Furthermore, the block diagonalization is further extended to applications such as computing matrix integer powers, inverses, and condition numbers. Numerical experiments demonstrate the superiority of the proposed algorithms over MATLAB built-in functions in terms of computational efficiency and stability.