TY  - EJOU
AU  - Ku, Cheng-Yu  
AU  - Yeih, Weichung 
AU  - Liu, Chein-Shan  

TI  - Dynamical Newton-Like Methods for Solving Ill-Conditioned Systems of Nonlinear Equations with Applications to Boundary Value Problems
T2  - Computer Modeling in Engineering \& Sciences

PY  - 2011
VL  - 76
IS  - 2
SN  - 1526-1506

AB  - 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.
KW  - dynamical method
KW  -  scalar homotopy function
KW  -  fictitious time-like function
KW  -  Newton's method
KW  -  dynamical Jacobian-inverse free method
KW  -  manifold-based exponentially convergent algorithm

DO  - 10.3970/cmes.2011.076.083