Structured preconditioners for nonsingular matrices of block two-by-two structures |
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Authors: | Zhong-Zhi Bai |
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Institution: | State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100080, People's Republic of China |
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Abstract: | For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to efficient and high-quality preconditioning matrices for some typical matrices from the real-world applications. |
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Keywords: | Block two-by-two matrix preconditioner modified block relaxation iteration eigenvalue distribution positive definiteness |
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