2001 Fiscal Year Final Research Report Summary
Development of practical methods for rigorous calculation with guaranteed accuracy
| Project/Area Number |
09640278
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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 | The University of Electro-Communications (1999-2001)
Kyushu University (1997-1998) |
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Principal Investigator |
YAMAMOTO Nobito Department of Electro-Communications, The University of Electro-Communications, Associate Professor, 電気通信学部, 助教授 (30210545)
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| Project Period (FY) |
1997 – 2000
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| Keywords | computation with guaranteed accuracy / numerical verification / numerical analysis / Newton method / eigenvalue problem |
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| Research Abstract |
Our objective in this study which is fonded by Grant-in-Aid for Scientific Research is development of practical methods for rigorous calculation with, guaranteed accuracy. Through the period of this study over 4 years, we have obtained some results on the following.
1. Verified computation of the maximum eigenvalue of Newton operators in infinite dimensional spaces
2. Verified computation methods for eigenvalues of symmetric band matrices together with their indices
3. Extension of the above methods to general eigenvalue problems
4. Methods for verification of uniqueness of solutions to fixed point equations
5. Research on a bifurcation diagram of Perturbed Gelfand Equation with guaranteed accuracy
6. Rigorous calculation of constants appearing in error estimations of FEM
7. Research on methods for transaction of rounding errors using Fortran 90 and quadruple-precision floating point numbers
8. Numerical verification of solutions to the Navier-Stockes equation using spectral methods
9. Estimation methods for influence of rounding error by interval arithmetic
10. Estimation of ability of approximation of FEM.
Consequently we can conclude that practical methods for verified computation of eigenvalue problems. are developed. On the methods for PDEs, they are also developed but there are some difficulties concerning mathematical matters in practical use for non-professional users.
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