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On the convergence of the Gauss-Seidel method applied to Dirichlet difference problems over various types of regions. Teng, Koit

Abstract

The main problem considered is the effect due to changes in the shape of the region on the convergence rate of the Gauss-Seidel iterative method for solving the Dirichlet Difference Problem. Experimentally, it is found that as a rule the number of iterations required to attain convergence decreases as the perimeter of the region is increased. The ensuing theoretical investigation leads to the examination of the corresponding iteration matrices and a qualitative theory results which predicts that the number of iterations should increase with the number of nonzero off - diagonal elements in the matrix of the linear system. Further experiments indicate that the latter relationship is no more precise than the former; the lack of rigour in the theory is undoubtedly to blame. Better results, are obtained in the sub-problem of estimating the number of iterations necessary to satisfy a suitable convergence criterion, given a good estimate of the spectral radius of the iteration matrix corresponding to the region under study.

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