IEICE TRANSACTIONS on Fundamentals
A Modified Newton Method with Guaranteed Accuracy Based on Rational Arithmetic
Summary :
In this paper, we are concerned with a problem of obtaining an approximate solution of a finite-dimensional nonlinear equation with guaranteed accuracy. Assuming that an approximate solution of a nonlinear equation is already calculated by a certain numerical method, we present computable conditions to validate whether there exists an exact solution in a neighborhood of this approximate solution or not. In order to check such conditions by computers, we present a method using rational arithmetic. In this method, both the effects of the truncation errors and the rounding errors of numerical computation are taken into consideration. Moreover, based on rational arithmetic we propose a new modified Newton interation to obtain an improved approximate solution with desired accuracy.
- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E76-A No.5 pp.795-807
- Publication Date
- 1993/05/25
- Publicized
- Online ISSN
- DOI
- Type of Manuscript
- Special Section PAPER (Special Section on Neural Nets,Chaos and Numerics)
- Category
- Numerical Homotopy Method and Self-Validating Numerics
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Akira INOUE, Masahide KASHIWAGI, Shin'ichi OISHI, Mitsunori MAKINO, "A Modified Newton Method with Guaranteed Accuracy Based on Rational Arithmetic" in IEICE TRANSACTIONS on Fundamentals,
vol. E76-A, no. 5, pp. 795-807, May 1993, doi: .
Abstract: In this paper, we are concerned with a problem of obtaining an approximate solution of a finite-dimensional nonlinear equation with guaranteed accuracy. Assuming that an approximate solution of a nonlinear equation is already calculated by a certain numerical method, we present computable conditions to validate whether there exists an exact solution in a neighborhood of this approximate solution or not. In order to check such conditions by computers, we present a method using rational arithmetic. In this method, both the effects of the truncation errors and the rounding errors of numerical computation are taken into consideration. Moreover, based on rational arithmetic we propose a new modified Newton interation to obtain an improved approximate solution with desired accuracy.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e76-a_5_795/_p
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@ARTICLE{e76-a_5_795,
author={Akira INOUE, Masahide KASHIWAGI, Shin'ichi OISHI, Mitsunori MAKINO, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={A Modified Newton Method with Guaranteed Accuracy Based on Rational Arithmetic},
year={1993},
volume={E76-A},
number={5},
pages={795-807},
abstract={In this paper, we are concerned with a problem of obtaining an approximate solution of a finite-dimensional nonlinear equation with guaranteed accuracy. Assuming that an approximate solution of a nonlinear equation is already calculated by a certain numerical method, we present computable conditions to validate whether there exists an exact solution in a neighborhood of this approximate solution or not. In order to check such conditions by computers, we present a method using rational arithmetic. In this method, both the effects of the truncation errors and the rounding errors of numerical computation are taken into consideration. Moreover, based on rational arithmetic we propose a new modified Newton interation to obtain an improved approximate solution with desired accuracy.},
keywords={},
doi={},
ISSN={},
month={May},}
Copy
TY - JOUR
TI - A Modified Newton Method with Guaranteed Accuracy Based on Rational Arithmetic
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 795
EP - 807
AU - Akira INOUE
AU - Masahide KASHIWAGI
AU - Shin'ichi OISHI
AU - Mitsunori MAKINO
PY - 1993
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E76-A
IS - 5
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - May 1993
AB - In this paper, we are concerned with a problem of obtaining an approximate solution of a finite-dimensional nonlinear equation with guaranteed accuracy. Assuming that an approximate solution of a nonlinear equation is already calculated by a certain numerical method, we present computable conditions to validate whether there exists an exact solution in a neighborhood of this approximate solution or not. In order to check such conditions by computers, we present a method using rational arithmetic. In this method, both the effects of the truncation errors and the rounding errors of numerical computation are taken into consideration. Moreover, based on rational arithmetic we propose a new modified Newton interation to obtain an improved approximate solution with desired accuracy.
ER -
English
