Geodesic
Using successive approximations for improving the convergence of GMRES method
Applications of Mathematics, Tome 43 (1998) no. 5, pp. 321-350.

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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}=Ty_l+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 $m$th 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.
DOI : 10.1023/A:1022291601664
Classification : 65F10, 65N22, 65N35
Keywords: GMRES; iterative method; numerical experiments; solution of discretized equations 
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     title = {Using successive approximations for improving the convergence of {GMRES} method},
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Zítko, Jan. Using successive approximations for improving the convergence of GMRES method. Applications of Mathematics, Tome 43 (1998) no. 5, pp. 321-350. doi : 10.1023/A:1022291601664. http://geodesic.mathdoc.fr/articles/10.1023/A:1022291601664/

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