A block upper triangular splitting method for solving block three-by-three linear systems arising from the large indefinite least squares problem

Jun Li, Kailiang Xin and Lingsheng Meng

Applied Mathematics and Computation, 2025, vol. 505, issue C

Abstract: 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.

Keywords: Indefinite least squares problem; Block three-by-three linear systems; Iteration method; Preconditioning; GMRES method (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:505:y:2025:i:c:s0096300325002723

DOI: 10.1016/j.amc.2025.129546

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