IEICE TRANSACTIONS on Fundamentals
The Self-Validating Numerical Method--A New Tool for Computer Assisted Proofs of Nonlinear Problems--
Summary :
The purpose of the present paper is to review a state of the art of nonlinear analysis with the self-validating numerical method. The self-validating numerics based method provides a tool for performing computer assisted proofs of nonlinear problems by taking the effect of rounding errors in numerical computations rigorously into account. First, Kantorovich's approach of a posteriori error estimation method is surveyed, which is based on his convergence theorem of Newton's method. Then, Urabe's approach for computer assisted existence proofs is likewise discussed. Based on his convergence theorem of the simplified Newton method, he treated practical nonlinear differential equations such as the Van der Pol equation ahd the Duffing equation, and proved the existence of their periodic and quasi-periodic solutions by the self-validating numerics. An approach of the author for generalization and abstraction of Urabe's method are also discribed to more general funcional equations. Furthermore, methods for rigorous estimation of rounding errors are surveyed. Interval analytic methods are discussed. Then an approach of the author which uses rational arithmetic is reviewed. Finally, approaches for computer assisted proofs of nonlinear problems are surveyed, which are based on the self-validating numerics.
- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E75-A No.5 pp.595-612
- Publication Date
- 1992/05/25
- Publicized
- Online ISSN
- DOI
- Type of Manuscript
- INVITED SURVEY PAPER
- Category
- Nonlinear Systems
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Shin'ichi OISHI, "The Self-Validating Numerical Method--A New Tool for Computer Assisted Proofs of Nonlinear Problems--" in IEICE TRANSACTIONS on Fundamentals,
vol. E75-A, no. 5, pp. 595-612, May 1992, doi: .
Abstract: The purpose of the present paper is to review a state of the art of nonlinear analysis with the self-validating numerical method. The self-validating numerics based method provides a tool for performing computer assisted proofs of nonlinear problems by taking the effect of rounding errors in numerical computations rigorously into account. First, Kantorovich's approach of a posteriori error estimation method is surveyed, which is based on his convergence theorem of Newton's method. Then, Urabe's approach for computer assisted existence proofs is likewise discussed. Based on his convergence theorem of the simplified Newton method, he treated practical nonlinear differential equations such as the Van der Pol equation ahd the Duffing equation, and proved the existence of their periodic and quasi-periodic solutions by the self-validating numerics. An approach of the author for generalization and abstraction of Urabe's method are also discribed to more general funcional equations. Furthermore, methods for rigorous estimation of rounding errors are surveyed. Interval analytic methods are discussed. Then an approach of the author which uses rational arithmetic is reviewed. Finally, approaches for computer assisted proofs of nonlinear problems are surveyed, which are based on the self-validating numerics.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e75-a_5_595/_p
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@ARTICLE{e75-a_5_595,
author={Shin'ichi OISHI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={The Self-Validating Numerical Method--A New Tool for Computer Assisted Proofs of Nonlinear Problems--},
year={1992},
volume={E75-A},
number={5},
pages={595-612},
abstract={The purpose of the present paper is to review a state of the art of nonlinear analysis with the self-validating numerical method. The self-validating numerics based method provides a tool for performing computer assisted proofs of nonlinear problems by taking the effect of rounding errors in numerical computations rigorously into account. First, Kantorovich's approach of a posteriori error estimation method is surveyed, which is based on his convergence theorem of Newton's method. Then, Urabe's approach for computer assisted existence proofs is likewise discussed. Based on his convergence theorem of the simplified Newton method, he treated practical nonlinear differential equations such as the Van der Pol equation ahd the Duffing equation, and proved the existence of their periodic and quasi-periodic solutions by the self-validating numerics. An approach of the author for generalization and abstraction of Urabe's method are also discribed to more general funcional equations. Furthermore, methods for rigorous estimation of rounding errors are surveyed. Interval analytic methods are discussed. Then an approach of the author which uses rational arithmetic is reviewed. Finally, approaches for computer assisted proofs of nonlinear problems are surveyed, which are based on the self-validating numerics.},
keywords={},
doi={},
ISSN={},
month={May},}
Copy
TY - JOUR
TI - The Self-Validating Numerical Method--A New Tool for Computer Assisted Proofs of Nonlinear Problems--
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 595
EP - 612
AU - Shin'ichi OISHI
PY - 1992
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E75-A
IS - 5
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - May 1992
AB - The purpose of the present paper is to review a state of the art of nonlinear analysis with the self-validating numerical method. The self-validating numerics based method provides a tool for performing computer assisted proofs of nonlinear problems by taking the effect of rounding errors in numerical computations rigorously into account. First, Kantorovich's approach of a posteriori error estimation method is surveyed, which is based on his convergence theorem of Newton's method. Then, Urabe's approach for computer assisted existence proofs is likewise discussed. Based on his convergence theorem of the simplified Newton method, he treated practical nonlinear differential equations such as the Van der Pol equation ahd the Duffing equation, and proved the existence of their periodic and quasi-periodic solutions by the self-validating numerics. An approach of the author for generalization and abstraction of Urabe's method are also discribed to more general funcional equations. Furthermore, methods for rigorous estimation of rounding errors are surveyed. Interval analytic methods are discussed. Then an approach of the author which uses rational arithmetic is reviewed. Finally, approaches for computer assisted proofs of nonlinear problems are surveyed, which are based on the self-validating numerics.
ER -
English
