TY - EJOU
AU - Yeih, Weichung
AU - Chan, I-Yao
AU - Ku, Cheng-Yu
AU - Fan, Chia-Ming
AU - Guan, Pai-Chen
TI - A Double Iteration Process for Solving the Nonlinear Algebraic Equations, Especially for Ill-posed Nonlinear Algebraic Equations
T2 - Computer Modeling in Engineering \& Sciences
PY - 2014
VL - 99
IS - 2
SN - 1526-1506
AB - 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: BTBu = BTF 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 ac or when the number of inner iteration steps exceeds the maximum tolerance Imax. 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 Imax. Six numerical examples are given and it is found that the proposed method is very efficient especially for the nonlinear ill-posed systems.
KW - double iteration process
KW - ill-posed
KW - the modified Tikhnov’s regularization method
DO - 10.3970/cmes.2014.099.123