Several Methods Of The Large Scale Nonlinear Equations

Along with the widely usage of computer, more and more fields in science and en-gineering calculation involve the large nonlinear equations. Such as the weather fore-cast, petroleum exploration, nonlinear finite elements problems, calculation of electricpower system, all the problems can eventually translate into large-scale or even oversizenumerical solution of nonlinear equations. This paper studies the method of incompleteJacobian Newton method and incomplete Newton-Moser’s method for large-scale non-linear problems, specific divided into the following sections:In chapter1, the Introduction introduces an overview of the nonlinear equations,research background and significance, research status and the main research content ofthe thesis. Preliminary knowledge about nonlinear mapping, iterative method, New-ton method and some modified Newton methods, homotopic mapping is also underdiscussion.The main work of this paper is the2nd,3rd and4th chapters. In chapter2, westudies a new constructing policy of incomplete Jacobian matrix and proves the con-vergence, which makes the origin incomplete Jacobian Newton method more generallyapplicable. In chapter3, we construct a new incomplete Newton-Moser’s method byintroducing the idea of incomplete Jacobian. The R-linear convergence of the algorithmis also proved. In chapter4, the homotopy algorithm for incomplete Jacobian Newtonmethod will be studied and enrich the research of large-scale nonlinear problems.In chapter5, numerical experiments are given. By analyzing the results of thenumerical experiments, we can see that the the modified incomplete Jacobian Newtonmethod has the same calculation efficiency with original incomplete Jacobian Newtonmethod, but more convenience and effective; the incomplete Newton-Moser’s methodand incomplete newtonial continuation algorithm are more suitable for solving largescale nonlinear problems, and the establishment of the new algorithm provides a goodverification for solving large scale nonlinear problems with incomplete Jacobian.In chapter6, all the algorithms developed are summarized and some prospectswhich are worth of in-depth studies are proposed.