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On a Chebyshev accelerated splitting iteration method with application to two‐by‐two block linear systems
Author(s) -
Wang ZengQi
Publication year - 2018
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2172
Subject(s) - mathematics , chebyshev iteration , linear system , iterative method , preconditioner , rate of convergence , chebyshev filter , coefficient matrix , eigenvalues and eigenvectors , chebyshev polynomials , upper and lower bounds , arnoldi iteration , mathematical optimization , mathematical analysis , computer science , channel (broadcasting) , computer network , physics , quantum mechanics
Summary The Chebyshev accelerated preconditioned modified Hermitian and skew‐Hermitian splitting (CAPMHSS) iteration method is presented for solving the linear systems of equations, which have two‐by‐two block coefficient matrices. We derive an iteration error bound to show that the new method is convergent as long as the eigenvalue bounds are not underestimated. Even when the spectral information is lacking, the CAPMHSS iteration method could be considered as an exponentially converging iterative scheme for certain choices of the method parameters. In this case, the convergence rate is independent of the parameters. Besides, the linear subsystems in each iteration can be solved inexactly, which leads to the inexact CAPMHSS iteration method. The iteration error bound of the inexact method is derived also. We discuss in detail the implementation of CAPMHSS for solving two models arising from the Galerkin finite‐element discretizations of distributed control problems and complex symmetric linear systems. The numerical results show the robustness and the efficiency of the new methods.