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

IF 2 3区化学 Q3 CHEMISTRY, MULTIDISCIPLINARY

Journal of Mathematical Chemistry Pub Date : 2025-12-16 DOI:10.1007/s10910-025-01759-0

Zheng Tang, Ji-Teng Jia

引用次数: 0

Abstract

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.

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由卷积方程出发的块循环三对角线性系统的数值解

本文研究了周期边界条件下一般由卷积方程产生的块循环三对角线性系统的数值解。利用块toeplitz结构，提出了一种新的保结构系数矩阵分解方法。基于保结构矩阵分解和Sherman-Morrison-Woodbury公式，提出了一种求解块循环三对角线性系统的有效的线性时间复杂度数值算法。此外，为了保证数值的稳定性，还进行了理论误差分析，并给出了块循环三对角矩阵行列式的数值公式。仿真结果表明，本文提出的算法具有较高的精度和效率，与块\(LU\)分解方法具有较强的竞争力。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Mathematical Chemistry 化学-化学综合

CiteScore

3.70

自引率

17.60%

发文量

105

审稿时长

6 months

期刊介绍： The Journal of Mathematical Chemistry (JOMC) publishes original, chemically important mathematical results which use non-routine mathematical methodologies often unfamiliar to the usual audience of mainstream experimental and theoretical chemistry journals. Furthermore JOMC publishes papers on novel applications of more familiar mathematical techniques and analyses of chemical problems which indicate the need for new mathematical approaches. Mathematical chemistry is a truly interdisciplinary subject, a field of rapidly growing importance. As chemistry becomes more and more amenable to mathematically rigorous study, it is likely that chemistry will also become an alert and demanding consumer of new mathematical results. The level of complexity of chemical problems is often very high, and modeling molecular behaviour and chemical reactions does require new mathematical approaches. Chemistry is witnessing an important shift in emphasis: simplistic models are no longer satisfactory, and more detailed mathematical understanding of complex chemical properties and phenomena are required. From theoretical chemistry and quantum chemistry to applied fields such as molecular modeling, drug design, molecular engineering, and the development of supramolecular structures, mathematical chemistry is an important discipline providing both explanations and predictions. JOMC has an important role in advancing chemistry to an era of detailed understanding of molecules and reactions.