{"title":"An efficient numerical algorithm for solving linear systems with cyclic tridiagonal coefficient matrices","authors":"Ji-Teng Jia, Fu-Rong Wang, Rong Xie, Yi-Fan Wang","doi":"10.1007/s10910-024-01631-7","DOIUrl":null,"url":null,"abstract":"<div><p>In the present paper, we mainly consider the direct solution of cyclic tridiagonal linear systems. By using the specific low-rank and Toeplitz-like structure, we derive a structure-preserving factorization of the coefficient matrix. Based on the combination of such matrix factorization and Sherman–Morrison–Woodbury formula, we then propose a cost-efficient algorithm for numerically solving cyclic tridiagonal linear systems, which requires less memory storage and data transmission. Furthermore, we show that the structure-preserving matrix factorization can provide us with an explicit formula for <i>n</i>-th order cyclic tridiagonal determinants. Numerical examples are given to demonstrate the performance and efficiency of our algorithm. All of the experiments are performed on a computer with the aid of programs written in MATLAB.</p></div>","PeriodicalId":648,"journal":{"name":"Journal of Mathematical Chemistry","volume":"62 8","pages":"1808 - 1821"},"PeriodicalIF":1.7000,"publicationDate":"2024-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Chemistry","FirstCategoryId":"92","ListUrlMain":"https://link.springer.com/article/10.1007/s10910-024-01631-7","RegionNum":3,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
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Abstract
In the present paper, we mainly consider the direct solution of cyclic tridiagonal linear systems. By using the specific low-rank and Toeplitz-like structure, we derive a structure-preserving factorization of the coefficient matrix. Based on the combination of such matrix factorization and Sherman–Morrison–Woodbury formula, we then propose a cost-efficient algorithm for numerically solving cyclic tridiagonal linear systems, which requires less memory storage and data transmission. Furthermore, we show that the structure-preserving matrix factorization can provide us with an explicit formula for n-th order cyclic tridiagonal determinants. Numerical examples are given to demonstrate the performance and efficiency of our algorithm. All of the experiments are performed on a computer with the aid of programs written in MATLAB.
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.
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