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

Abstract

In this paper, a novel double iteration process for solving the nonlin-ear algebraic equations is developed. In this process, the outer iteration controls theevolution path of the unknown vector xin the selected direction uwhich is deter-mined from the inner iteration process. For the inner iteration, the direction of evo-lution uis determined by solving a linear algebraic equation: BTBu =BTFwhereBis the Jacobian matrix, Fis the residual vector and the superscript “T” denotesthe matrix transpose. For an ill-posed system, this linear algebraic equation is verydifficult to solve since the resulting leading coefficient matrix is ill-posed in nature.We adopted the modified Tikhonov’s regularization method (MTRM) developed byLiu (Liu, 2012) to solve the ill-posed linear algebraic equation. However, to exact-ly find the solution of the evolution direction umay consume too many iterationsteps for the inner iteration process, which is definitely not economic. Therefore,the inner iteration process stops while the direction umakes the value of a0beingsmaller than the selected margin acor when the number of inner iteration stepsexceeds the maximum tolerance Imax. For the outer iteration process, it terminatesonce the root mean square error for the residual is less than the convergence crite-rion εor when the number of inner iteration steps exceeds the maximum toleranceImax. Six numerical examples are given and it is found that the proposed method isvery efficient especially for the nonlinear ill-posed systems.