Strang-type preconditioners for systems of LMF-based ODE codes |
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Authors: | Chan, Raymond H. Ng, Michael K. Jin, Xiao-Qing |
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Affiliation: | 1 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong 2 Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong 3 Faculty of Science and Technology, The University of Macau, Macau |
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Abstract: | We consider the solution of ordinary differential equations(ODEs) using boundary value methods. These methods require thesolution of one or more unsymmetric, large and sparse linearsystems. The GMRES method with the Strang-type block-circulantpreconditioner is proposed for solving these linear systems.We show that if an Ak1,k2 -stable boundary value method is usedfor an m-by-m system of ODEs, then our preconditioners are invertibleand all the eigenvalues of the preconditioned systems are 1except for at most 2m(k1 + k2) outliers. It follows that whenthe GMRES method is applied to solving the preconditioned systems,the method will converge in at most 2m(k1 + k2) + 1 iterations.Numerical results are given to illustrate the effectivenessof our methods. Received 8 October 1999. Accepted 30 May 2000. |
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