1996 Fiscal Year Final Research Report Summary
MATHEMATICAL ANALYSIS AND NUMERICAL ANALYSIS OF SEVERAL KINDS OF DIFFERENTIAL EQUATIONS.
| Project/Area Number |
06640335
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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 | WASEDA UNIVERSITY |
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Principal Investigator |
MUROYA Yoshiaki WASEDA UNIVERSITY,Sch.Sci.& Eng., PROFESSOR, 理工学部, 教授 (90063718)
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| Co-Investigator(Kenkyū-buntansha) |
TANAKA Kazunaga SCH.SCI.AND ENG., WASEDA UNIVERSITY ASSOCIATE PROFESSOR, 理工学部, 助教授 (20188288)
TSUTSUMI Masayoshi SCH.SCI.AND ENG., WASEDA UNIVERSITY PROFESSOR, 理工学部, 教授 (70063774)
KORI Toshiaki SCH.SCI.AND ENG., WASEDA UNIVERSITY PROFESSOR, 理工学部, 教授 (50063730)
OTANI Mitsuharu SCH.SCI.AND ENG., WASEDA UNIVERSITY PROFESSOR, 理工学部, 教授 (30119656)
YAMADA Yoshio SCH.SCI.AND ENG., WASEDA UNIVERSITY PROFESSOR, 理工学部, 教授 (20111825)
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| Project Period (FY) |
1994 – 1996
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| Keywords | IMPROVED SOR METHOD / MULTIPLE RELAXATION PARAMETERS / ADAPTIVE IMPROVED SOR METHOD / IMPROVED SSOR METHOD |
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| Research Abstract |
To solve non-symmetric linear systems derived from the discretization of singular pertur-bation problems, we propose a generalized SOR method with multiple relaxation parameters, that is the improved SOR method with orderings and study its theory and practical use.
In the case of tridiagonal matrices, optimal choices of the parameters are examined : It is shown that the spectral radius of the iterative matrix is reduced to zero for a pair of parameter values which are computed from the pivots of the Gaussian elimination applied to the system. A proper choice of orderings and starting vectors for the iteration is also proposed.
We apply the above method to two-dimensional cases, and propose the "adaptive improved block SOR method with orderings" for block tridiafonal matrices. The point of this method is to change the multiple relaxation parameters not only for each block but also for each iteration. If special multiple relaxation parameters are selected and used with this method for an n * n block tridiagonal matrix whose block matrices are all n * n matrices, then this iterative method converges at most n^2 iterations.
We also proposed the improved SSOR method with orderings, which converges at most only one iteration for a tridiagonal system, and n iterations for a block tridiagonal system.
The generalized convergence theorems to the improved SOR method with orderings are also considered, and we study necessary and sufficient conditions for a matrix to be a generalized diagonally dominant.
Using the notation 'basic LUL factorization' of matrices, we give some techniques to obtain special multiple relaxation parameters such that the spectral radius of the iterative matrix is zero for the Hessenberg matrices and a class of matrices.
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