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ARTICLE

Dynamical Newton-Like Methods for Solving Ill-Conditioned Systems of Nonlinear Equations with Applications to Boundary Value Problems

by Cheng-Yu Ku¹, Chein-Shan Liu⁴

1 Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung.
2 Computation and Simulation Center, National Taiwan Ocean University, Keelung.
3 Corresponding Author, E-mail: chkst26@mail.ntou.edu.tw
4 Department of Civil Engineering, National Taiwan University, Taipei

Computer Modeling in Engineering & Sciences 2011, 76(2), 83-108. https://doi.org/10.3970/cmes.2011.076.083

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 Dynamical Jacobian-Inverse Free Method (DJIFM), and the Manifold-Based Exponentially 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 methods are then adopted for the solution of ill-conditioned systems of nonlinear equations and applied to boundary value problems. Results reveal that taking advantages of the general dynamical method the proposed three dynamical Newton-like methods 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.

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Keywords

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