On optimal improvements of classical iterative schemes for Z-matrices

Authors:
D. Noutsos;M. Tzoumas
Affiliations:
Department of Mathematics, University of Ioannina, GR-451 10 Ioannina, Greece;Department of Environment and Natural Resources Management, Agrinion University School, University of Ioannina, GR-301 00 Agrinion, Greece
Venue:
Journal of Computational and Applied Mathematics
Year:
2006

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Abstract

Many researchers have considered preconditioners, applied to linear systems, whose matrix coefficient is a Z- or an M-matrix, that make the associated Jacobi and Gauss-Seidel methods converge asymptotically faster than the unpreconditioned ones. Such preconditioners are chosen so that they eliminate the off-diagonal elements of the same column or the elements of the first upper diagonal [Milaszewicz, LAA 93 (1987) 161-170], Gunawardena et al. [LAA 154-156 (1991) 123-143]. In this work we generalize the previous preconditioners to obtain optimal methods. ''Good'' Jacobi and Gauss-Seidel algorithms are given and preconditioners, that eliminate more than one entry per row, are also proposed and analyzed. Moreover, the behavior of the above preconditioners to the Krylov subspace methods is studied.