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# A Scalar Homotopy Method with Optimal Hybrid Search Directions for Solving Nonlinear Algebraic Equations

Weichung Yeih1,2, Cheng-Yu Ku1,2,3, Chein-Shan Liu4, I-Yao Chan1,2

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
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan

Computer Modeling in Engineering & Sciences 2013, 90(4), 255-282. https://doi.org/10.3970/cmes.2013.090.255

## Abstract

In this paper, a scalar homotopy method with optimal hybrid search directions for solving nonlinear algebraic equations is proposed. To conduct the proposed method, we first convert the vector residual function to a scalar function by taking the square norm of the vector function and then, introduce a fictitious time variable to form a scalar homotopy function. To improve the convergence and the accuracy of the proposed method, a vector with multiple search directions and an iterative algorithm are introduced into the evolution dynamics of the solutions. Further, for obtaining the optimal search direction, linear and nonlinear optimization algorithms are develploped. Taking the advantages of finding the optimal search direction, the proposed novel method is able to consider hybrid search directions for solving the nonlinear algebraic equations. The formulation presented in this paper demonstrates a variety of flexibility with the use of the algorithm for finding optimal hybrid search directions. In addition, our proposed method does not necessarily need to calculate the inverse of the Jacobian matrix and has great numerical stability for solving nonlinear well-posed algebraic equations as well as the ill-posed problems which may have an ill-conditioned or singular Jacobian matrix. Results reveal that the proposed method can improve the convergence and increase the numerical stability for solving nonlinear algebraic equations.

## Keywords

APA Style
Yeih, W., Ku, C., Liu, C., Chan, I. (2013). A scalar homotopy method with optimal hybrid search directions for solving nonlinear algebraic equations. Computer Modeling in Engineering & Sciences, 90(4), 255-282. https://doi.org/10.3970/cmes.2013.090.255
Vancouver Style
Yeih W, Ku C, Liu C, Chan I. A scalar homotopy method with optimal hybrid search directions for solving nonlinear algebraic equations. Comput Model Eng Sci. 2013;90(4):255-282 https://doi.org/10.3970/cmes.2013.090.255
IEEE Style
W. Yeih, C. Ku, C. Liu, and I. Chan "A Scalar Homotopy Method with Optimal Hybrid Search Directions for Solving Nonlinear Algebraic Equations," Comput. Model. Eng. Sci., vol. 90, no. 4, pp. 255-282. 2013. https://doi.org/10.3970/cmes.2013.090.255

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