{"title":"用块上三角分裂法求解由大不定最小二乘问题引起的块三乘三线性系统","authors":"Jun Li , Kailiang Xin , Lingsheng Meng","doi":"10.1016/j.amc.2025.129546","DOIUrl":null,"url":null,"abstract":"<div><div>In this research, we mainly utilize the stationary iteration method in conjunction with Krylov subspace techniques, such as GMRES, to tackle the large indefinite least squares problem. To accomplish this, the normal equation of the large indefinite least squares problem is firstly transformed into the sparse block three-by-three linear systems with non-singular diagonal blocks, then a block upper triangular matrix splitting of the coefficient matrix of the block three-by-three linear systems is given, the splitting not only produces the stationary iteration method, but also naturally derives a preconditioner, which can be used within GMRES method to solve the block linear systems. Thereafter, it is proved theoretically that the iteration method has unconditional convergence. Furthermore, the theory also shows that all the eigenvalues of the preconditioned matrix are real number and located in a positive interval. In the end, numerical results reflect that the theoretical results are correct and the studied methods are also effective.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"505 ","pages":"Article 129546"},"PeriodicalIF":3.4000,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A block upper triangular splitting method for solving block three-by-three linear systems arising from the large indefinite least squares problem\",\"authors\":\"Jun Li , Kailiang Xin , Lingsheng Meng\",\"doi\":\"10.1016/j.amc.2025.129546\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>In this research, we mainly utilize the stationary iteration method in conjunction with Krylov subspace techniques, such as GMRES, to tackle the large indefinite least squares problem. To accomplish this, the normal equation of the large indefinite least squares problem is firstly transformed into the sparse block three-by-three linear systems with non-singular diagonal blocks, then a block upper triangular matrix splitting of the coefficient matrix of the block three-by-three linear systems is given, the splitting not only produces the stationary iteration method, but also naturally derives a preconditioner, which can be used within GMRES method to solve the block linear systems. Thereafter, it is proved theoretically that the iteration method has unconditional convergence. Furthermore, the theory also shows that all the eigenvalues of the preconditioned matrix are real number and located in a positive interval. In the end, numerical results reflect that the theoretical results are correct and the studied methods are also effective.</div></div>\",\"PeriodicalId\":55496,\"journal\":{\"name\":\"Applied Mathematics and Computation\",\"volume\":\"505 \",\"pages\":\"Article 129546\"},\"PeriodicalIF\":3.4000,\"publicationDate\":\"2025-05-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Applied Mathematics and Computation\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0096300325002723\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics and Computation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0096300325002723","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
A block upper triangular splitting method for solving block three-by-three linear systems arising from the large indefinite least squares problem
In this research, we mainly utilize the stationary iteration method in conjunction with Krylov subspace techniques, such as GMRES, to tackle the large indefinite least squares problem. To accomplish this, the normal equation of the large indefinite least squares problem is firstly transformed into the sparse block three-by-three linear systems with non-singular diagonal blocks, then a block upper triangular matrix splitting of the coefficient matrix of the block three-by-three linear systems is given, the splitting not only produces the stationary iteration method, but also naturally derives a preconditioner, which can be used within GMRES method to solve the block linear systems. Thereafter, it is proved theoretically that the iteration method has unconditional convergence. Furthermore, the theory also shows that all the eigenvalues of the preconditioned matrix are real number and located in a positive interval. In the end, numerical results reflect that the theoretical results are correct and the studied methods are also effective.
期刊介绍:
Applied Mathematics and Computation addresses work at the interface between applied mathematics, numerical computation, and applications of systems – oriented ideas to the physical, biological, social, and behavioral sciences, and emphasizes papers of a computational nature focusing on new algorithms, their analysis and numerical results.
In addition to presenting research papers, Applied Mathematics and Computation publishes review articles and single–topics issues.