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# Dynamical Newton-Like Methods with Adaptive Stepsize for Solving Nonlinear Algebraic Equations

Cheng-Yu Ku1,2,3, Weichung Yeih1,2
Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan.
Computation and Simulation Center, National Taiwan Ocean University, Keelung, Taiwan.
Corresponding Author, E-mail: chkst26@mail.ntou.edu.tw

Computers, Materials & Continua 2012, 31(3), 173-200. https://doi.org/10.3970/cmc.2012.031.173

### Abstract

In this paper, a dynamical Newton-like method with the adaptive stepsize 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 the fictitious time-like function, we derived the adaptive stepsize using the dynamics of the residual vector. Based on the proposed dynamical Newton-like method, we can also derive the dynamical Newton method (DNM) and the dynamical Jacobian-inverse free method (DJIFM) with the transformation matrix as the inverse of the Jacobian and the identity matrix, respectively. These two dynamical Newton-like methods are then adopted for the solution of NAEs. Numerical illustrations demonstrate that taking advantages of the dynamical Newton-like method with the adaptive stepsize the proposed two dynamical Newton-like methods can release limitations of the conventional Newton method such as root jumping, the divergence at inflection points, root oscillations, and the divergence of the root. Results reveal that with the use of the fictitious time-like function the proposed method presents exponential convergence. In addition, taking the advantages of the transformation matrix, the proposed method does not need to calculate the inverse of the Jacobian matrix and thus has great numerical stability.

### Keywords

the scalar homotopy method, adaptive stepsize, Jacobian, dynamical Newton-like method, Newton's method

C. . Ku and W. . Yeih, "Dynamical newton-like methods with adaptive stepsize for solving nonlinear algebraic equations," Computers, Materials & Continua, vol. 31, no.3, pp. 173–200, 2012. This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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