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Please use this identifier to cite or link to this item: http://ntour.ntou.edu.tw:8080/ir/handle/987654321/51870

Title: Dynamical Newton-Like Methods for Solving Ill-Conditioned Systems of Nonlinear Equations with Applications to Boundary Value Problems
Authors: Cheng-Yu Ku
Weichung Yeih
Chein-Shan Liu
Contributors: 國立臺灣海洋大學:河海工程學系
Date: 2011-06
Issue Date: 2019-01-02T08:54:30Z
Publisher: CMES: Computer Modeling in Engineering & Science
Abstract: Abstract: In this paper, a general dynamical method based on the construction of a scalar homotopy function to transform a vector function of Non-Linear Algebraic Equations (NAEs) into a time-dependent scalar function by introducing a fictitious time-like variable is proposed. With the introduction of a transformation matrix, the proposed general dynamical method can be transformed into several dynamical Newton-like methods including the Dynamical Newton Method (DNM), the Dy-namical Jacobian-Inverse Free Method (DJIFM), and the Manifold-Based Expo-nentially Convergent Algorithm (MBECA). From the general dynamical method, we can also derive the conventional Newton method using a certain fictitious time-like function. The formulation presented in this paper demonstrates a variety of flexibility with the use of different transformation matrices to create other possible dynamical methods for solving NAEs. These three dynamical Newton-like meth-ods are then adopted for the solution of ill-conditioned systems of nonlinear equa-tions and applied to boundary value problems. Results reveal that taking advantages of the general dynamical method the proposed three dynamical Newton-like meth-ods can improve the convergence and increase the numerical stability for solving NAEs, especially for the system of nonlinear problems involving ill-conditioned Jacobian or poor initial values which cause convergence problems. Keywords: dynamical method, scalar homotopy function, fictitious time-like func-tion, Newton's method, dynamical Jacobian-inverse free method, manifold-based exponentially convergent algorithm.
Relation: 76(2) pp.83-108
URI: http://ntour.ntou.edu.tw:8080/ir/handle/987654321/51870
Appears in Collections:[河海工程學系] 期刊論文

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