Open Access

ARTICLE

A Double Iteration Process for Solving the Nonlinear Algebraic Equations, Especially for Ill-posed Nonlinear Algebraic Equations

Weichung Yeih^1,2, I-Yao Chan¹, Cheng-Yu Ku¹, Chia-Ming Fan¹, Pai-Chen Guan³

1 Department of Harbor and River Engineering & Computation and Simulation Center, National Taiwan Ocean University, Keelung, 202 Taiwan.
2 Corresponding author, Tel: +886-2-24622192-ext6119, e-mail: wcyeih@mail.ntou.edu.tw
3 Department of Naval Architecture and System Engineering & Computation and Simulation Center, National Taiwan Ocean University, Keelung, 202 Taiwan.

Computer Modeling in Engineering & Sciences 2014, 99(2), 123-149. https://doi.org/10.3970/cmes.2014.099.123

Abstract

In this paper, a novel double iteration process for solving the nonlinear algebraic equations is developed. In this process, the outer iteration controls the evolution path of the unknown vector x in the selected direction u which is determined from the inner iteration process. For the inner iteration, the direction of evolution u is determined by solving a linear algebraic equation: B^TBu = B^TF where B is the Jacobian matrix, F is the residual vector and the superscript ''T'' denotes the matrix transpose. For an ill-posed system, this linear algebraic equation is very difficult to solve since the resulting leading coefficient matrix is ill-posed in nature. We adopted the modified Tikhonov's regularization method (MTRM) developed by Liu (Liu, 2012) to solve the ill-posed linear algebraic equation. However, to exactly find the solution of the evolution direction u may consume too many iteration steps for the inner iteration process, which is definitely not economic. Therefore, the inner iteration process stops while the direction u makes the value of a₀ being smaller than the selected margin a_c or when the number of inner iteration steps exceeds the maximum tolerance I_max. For the outer iteration process, it terminates once the root mean square error for the residual is less than the convergence criterion ε or when the number of inner iteration steps exceeds the maximum tolerance I_max. Six numerical examples are given and it is found that the proposed method is very efficient especially for the nonlinear ill-posed systems.

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Keywords

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