Using successive approximations for improving the convergence of GMRES method |
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Authors: | Jan Zítko |
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Institution: | (1) Katedra numerické matematiky MFF UK, Malostranské, námsti 25, 11800 Praha 1, Czech Republic |
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Abstract: | In this paper, our attention is concentrated on the GMRES method for the solution of the system (I–T)x=b of linear algebraic equations with a nonsymmetric matrix. We perform m pre-iterations y
l+1
=T
yl
+b before starting GMRES and put y
m for the initial approximation in GMRES. We derive an upper estimate for the norm of the error vector in dependence on the mth powers of eigenvalues of the matrix T Further we study under what eigenvalues lay-out this upper estimate is the best one. The estimate shows and numerical experiments verify that it is advisable to perform pre-iterations before starting GMRES as they require fewer arithmetic operations than GMRES. Towards the end of the paper we present a numerical experiment for a system obtained by the finite difference approximation of convection-diffusion equations. |
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Keywords: | GMRES iterative method numerical experiments solution of discretized equations |
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