@Article{cmes.2011.076.083,
AUTHOR = {Cheng-Yu  Ku,Weichung Yeih, Chein-Shan  Liu},
TITLE = {Dynamical Newton-Like Methods for Solving Ill-Conditioned Systems of Nonlinear Equations with Applications to Boundary Value Problems},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {76},
YEAR = {2011},
NUMBER = {2},
PAGES = {83--108},
URL = {http://www.techscience.com/CMES/v76n2/26816},
ISSN = {1526-1506},
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.},
DOI = {10.3970/cmes.2011.076.083}
}