1998 Fiscal Year Final Research Report Summary
Mathematical analysis of linear/nonlinear iterative algorithms including GMRES, SOR, etc.
| Project/Area Number |
09640277
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Ehime University |
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Principal Investigator |
YAMAMOTO Tetsuro Ehime University, Faculty of Science, Professor, 理学部, 教授 (80034560)
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| Co-Investigator(Kenkyū-buntansha) |
方 青 愛媛大学, 理学部, 助手 (10243544)
TSUCHIYA Takuya Ehime University, Faculty of Science, Associate Prof., 理学部, 助教授 (00163832)
陳 小君 島根大学, 総合理工学部, 助教授 (70304251)
OYANAGI Yoshio University of Tokyo, Prof., 大学院・理学研究科, 教授 (60011673)
QING Fang Ehime University, Faculty of Science, Assistant
CHEN Xiaojun Shimane University, Associate Prof.
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| Project Period (FY) |
1997 – 1998
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| Keywords | Iterative methods for salring equations / GMRES method / SOR method / SSOR method / Global convergence theorem / Uzawa法 / Jacobi-Davidson法 |
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| Research Abstract |
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.
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