Mathematical analysis of linear/nonlinear iterative algorithms including GMRES, SOR, etc.

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The purpose of this research was to give mathematical foundations for linear/nonlinear iterative methods including GMRES, SOR, etc. Concerning SOR, we obtained the following results :1. Unified treatment of known convergence theorems for linear SOR methods : The Ostrowski-Reich theorem is the most famous result for convergence of the SOR method applied to the linear system Ax = b which asserts that if A is hermitian with positive diagonals, then the SOR method converges for 0 < omega < 2 if and only if A is positive definite. Some results by Householder-John, Ortega-Plemmons, etc. are also known, which are closely related to the Ostrowski-Reich theorem. We found out that. these results can uniformly be derived from the Stein theorem, which asserts that a matrix H is a convergent matrix if and only if there exists a positive definite matrix B such that BETA - H^*BH is positive definite. This simplifies and unifies the convergence proofs by Ostrowski and others.2. Global convergence of nonlinear SOR-like methods. Although Brewster-Kannan's result is known for convergence of SOR-Newton's method, it only asserts that for any initial vector there exists a sequence of parameter {omega_k}, 0< omega_k < 2 such that the process with wk at each step converges to the solution. however, the explicit choice of wk is not known. We obtained a global convergence theorem for SOR-Newton's method applied to a discretized equation for semilinear PDE, which guarantees the convergence for 0 < omega< omega^<**>_h = 2 - OMICRON(h^2) where Ii denotes the mesh size.