The self-validating numerics - A new tool for computer assisted proofs of nonlinear problems

Shin'Ichi Oishi

doi:10.1109/ISCAS.1992.230623

The self-validating numerics - A new tool for computer assisted proofs of nonlinear problems

Shin'Ichi Oishi^*

^*Corresponding author for this work

School of Fundamental Science and Engineering

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Abstract

The self-validating numerical method is sueveyed for nonlinear problems. By taking into account of the effect of rounding error rigorously, this method provides a method of computer assisted proofs. In the first place, Kantrovich's approach to this problem is surveyed. His method is based on his convergence theorem of Newton's method and can be seen as an a posteriori error estimation method. Then, Urabe's approach to this problem is discussed. He treated practical nonlinear differential equations such as the van der Pol equation and the Duffing equation and proved the existence of their periodic and quasi-periodic solutions by the self-validating numerics. Generalizations and abstraction of Urabe's method to more general functional equations is also discussed. Then methods for rigorous estimation of rounding errors are surveyed.

Original language	English
Title of host publication	1992 IEEE International Symposium on Circuits and Systems, ISCAS 1992
Publisher	Institute of Electrical and Electronics Engineers Inc.
Pages	2773-2776
Number of pages	4
ISBN (Electronic)	0780305930
DOIs	https://doi.org/10.1109/ISCAS.1992.230623
Publication status	Published - 1992
Event	1992 IEEE International Symposium on Circuits and Systems, ISCAS 1992 - San Diego, United States Duration: 1992 May 10 → 1992 May 13

Publication series

Name	Proceedings - IEEE International Symposium on Circuits and Systems
Volume	6
ISSN (Print)	0271-4310

Conference

Conference	1992 IEEE International Symposium on Circuits and Systems, ISCAS 1992
Country/Territory	United States
City	San Diego
Period	92/5/10 → 92/5/13

ASJC Scopus subject areas

Electrical and Electronic Engineering

Access to Document

10.1109/ISCAS.1992.230623

Cite this

Oishi, SI. (1992). The self-validating numerics - A new tool for computer assisted proofs of nonlinear problems. In 1992 IEEE International Symposium on Circuits and Systems, ISCAS 1992 (pp. 2773-2776). Article 230623 (Proceedings - IEEE International Symposium on Circuits and Systems; Vol. 6). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ISCAS.1992.230623

The self-validating numerics - A new tool for computer assisted proofs of nonlinear problems. / Oishi, Shin'Ichi.
1992 IEEE International Symposium on Circuits and Systems, ISCAS 1992. Institute of Electrical and Electronics Engineers Inc., 1992. p. 2773-2776 230623 (Proceedings - IEEE International Symposium on Circuits and Systems; Vol. 6).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Oishi, SI 1992, The self-validating numerics - A new tool for computer assisted proofs of nonlinear problems. in 1992 IEEE International Symposium on Circuits and Systems, ISCAS 1992., 230623, Proceedings - IEEE International Symposium on Circuits and Systems, vol. 6, Institute of Electrical and Electronics Engineers Inc., pp. 2773-2776, 1992 IEEE International Symposium on Circuits and Systems, ISCAS 1992, San Diego, United States, 92/5/10. https://doi.org/10.1109/ISCAS.1992.230623

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AB - The self-validating numerical method is sueveyed for nonlinear problems. By taking into account of the effect of rounding error rigorously, this method provides a method of computer assisted proofs. In the first place, Kantrovich's approach to this problem is surveyed. His method is based on his convergence theorem of Newton's method and can be seen as an a posteriori error estimation method. Then, Urabe's approach to this problem is discussed. He treated practical nonlinear differential equations such as the van der Pol equation and the Duffing equation and proved the existence of their periodic and quasi-periodic solutions by the self-validating numerics. Generalizations and abstraction of Urabe's method to more general functional equations is also discussed. Then methods for rigorous estimation of rounding errors are surveyed.

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The self-validating numerics - A new tool for computer assisted proofs of nonlinear problems

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