@Article{cmes.2014.099.123,
AUTHOR = {Weichung  Yeih, I-Yao  Chan, Cheng-Yu  Ku, Chia-Ming Fan, Pai-Chen  Guan},
TITLE = {A Double Iteration Process for Solving the Nonlinear Algebraic Equations, Especially for Ill-posed Nonlinear Algebraic Equations},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {99},
YEAR = {2014},
NUMBER = {2},
PAGES = {123--149},
URL = {http://www.techscience.com/CMES/v99n2/27173},
ISSN = {1526-1506},
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 <b>x</b> in the selected direction <b>u</b> which is determined from the inner iteration process. For the inner iteration, the direction of evolution <b>u</b> is determined by solving a linear algebraic equation: <b>B<sup>T</sup>Bu </b> = <b>B<sup>T</sup>F</b> where <b>B</b> is the Jacobian matrix, <b>F</b> is the residual vector and the superscript ''<b>T</b>'' 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 <b>u</b> 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 <b>u</b> makes the value of <i>a<sub>0</sub></i> being smaller than the selected margin <i>a<sub>c</sub></i> or when the number of inner iteration steps exceeds the maximum tolerance I<sub>max</sub>. For the outer iteration process, it terminates once the root mean square error for the residual is less than the convergence criterion <i>ε</i> or when the number of inner iteration steps exceeds the maximum tolerance I<sub>max</sub>. Six numerical examples are given and it is found that the proposed method is very efficient especially for the nonlinear ill-posed systems.},
DOI = {10.3970/cmes.2014.099.123}
}