Necessary and sufficient conditions for GMRES complete and partial stagnation |
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| Institution: | 1. IRMA, Université de Strasbourg and CNRS, France;2. Inria Nancy Grand Est, France;1. Harbin Institute of Technology, Shenzhen Graduate School, Shenzhen, 518055, PR China;2. College of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao, 066004, PR China;3. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, PR China;1. Department of Computational and Applied Mathematics, China University of Petroleum, Qingdao 266580, PR China;2. Department of Mathematics, East China Normal University, Shanghai 200062, PR China;3. Laboratório Nacional de Computação Científica, MCTI Avenida Getúlio Vargas 333, 25651-075 Petrópolis, RJ, Brazil |
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| Abstract: | In this paper we give necessary and sufficient conditions for the complete or partial stagnation of the GMRES iterative method for solving real linear systems. Our results rely on a paper by Arioli, Pták and Strako? (1998), characterizing the matrices having a prescribed convergence curve for the residual norms. We show that we have complete stagnation if and only if the matrix A is orthonormally similar to an upper or lower Hessenberg matrix having a particular first row or column or a particular last row or column. Partial stagnation is characterized by a particular pattern of the matrix Q in the QR factorization of the upper Hessenberg matrix generated by the Arnoldi process. |
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| Keywords: | GMRES Complete stagnation Partial stagnation |
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