TY - GEN
T1 - The self-validating numerics - A new tool for computer assisted proofs of nonlinear problems
AU - Oishi, Shin'Ichi
N1 - Publisher Copyright:
© 1992 IEEE.
PY - 1992
Y1 - 1992
N2 - 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.
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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U2 - 10.1109/ISCAS.1992.230623
DO - 10.1109/ISCAS.1992.230623
M3 - Conference contribution
AN - SCOPUS:85067264558
T3 - Proceedings - IEEE International Symposium on Circuits and Systems
SP - 2773
EP - 2776
BT - 1992 IEEE International Symposium on Circuits and Systems, ISCAS 1992
PB - Institute of Electrical and Electronics Engineers Inc.
T2 - 1992 IEEE International Symposium on Circuits and Systems, ISCAS 1992
Y2 - 10 May 1992 through 13 May 1992
ER -