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An adaptive Newton-method based on a dynamical systems approach |
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| Institution: | 1. Friedrich-Alexander Universität Erlangen-Nürnberg, Germany;2. TU Kaiserslautern, Germany;3. Technische Universiteit Eindhoven, Netherlands;1. Institute of Mathematics of the National Academy of Sciences of Ukraine, 3 Tereshchenkivska Str., Kyiv-4 01601, Ukraine;2. Department of Mathematics, University of Nicosia, P.O. Box 24005, 1700 Nicosia, Cyprus;3. Department of Mathematics and Statistics, University of Cyprus, Nicosia CY 1678, Cyprus;1. Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang 212013, Jiangsu, China;2. Key Laboratory of Measurement and Control for Complex System of Ministry of Education, Research Institute of Automation, Southeast University, Nanjing 210096, Jiangsu, China |
| Abstract: | The traditional Newton method for solving nonlinear operator equations in Banach spaces is discussed within the context of the continuous Newton method. This setting makes it possible to interpret the Newton method as a discrete dynamical system and thereby to cast it in the framework of an adaptive step size control procedure. In so doing, our goal is to reduce the chaotic behavior of the original method without losing its quadratic convergence property close to the roots. The performance of the modified scheme is illustrated with various examples from algebraic and differential equations. |
| Keywords: | Newton–Raphson methods Continuous Newton–Raphson method Adaptive step size control Nonlinear differential equations Chaotic behavior |
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