Several Algorithms For Systems Of Nonlinear Equations

In recent years,more and more fields of scientific and engineering calculation ap-pear the nonlinear equations.Such as machine learning,artificial intelligence,financial computing,petroleum geological exploration,satellite orbit prediction and other fields involving the nonlinear equations,how to effectively and quickly to solve various non-linear equations has been widely concerned.This paper mainly proposes a modified quasi-Newton method,several modified Newton-GPSS methods and Newton-SGPSS method for solving nonlinear equations,the details are as follows:The first chapter mainly introduces the research background and significance,re-search status at home and abroad and the main research content of this paper.The preliminary knowledge introduces the classic Newton method,quasi-Newton method and Newton-GPSS method of nonlinear equations.In second chapter,based on the quasi-Newton method for solving nonlinear e-quations proposed by[26],we construct an approximate Jacobian matrix by using a quadratic interpolation relationship between the last three iteration points.The modi-fied quasi-Newton method is given and its convergence behavior is analyzed.Numeri-cal test results indicate that the modified quasi-Newton method has superior character-istics and numerical performance.In third chapter,firstly,by utilizing the modified Newton method instead of the classic Newton method as an outer solver of inexact Newton method,a modified Newton-GPSS method for solving nonlinear equations with non-Hermitian positive Ja-cobian matrices is proposed and we discuss the local convergence property.Further,by the successive-overrelaxation technique,an accelerated modified Newton-GPSS method is proposed and its convergence behavior is analyzed.Moreover,by using the multi-step modified Newton method as an outer solver of the inexact Newton method,a multi-step modified Newton-GPSS method is proposed and its local convergence be-havior is analyzed.Finally,numerous numerical test results demonstrate that the three methods are significantly better than the Newton-GPSS method in terms of CPU time and number of iterations.In fourth chapter,by utilizing the generalized positive-definite and skew-Hermitian splitting iteration technique,we establish a SGPSS method for non-Hermitian positive linear systems and its convergence behavior is analyzed.The S-GPSS method avoids solving a linear subsystem with coefficient matrix+₂.In addition,in order to improve the computation efficiency,by utilizing classic Newton method and modified Newton method as the outer solver of the inexact Newton method and SGPSS method as the inner,Newton-SGPSS method and modified Newton-SGPSS method for solving nonlinear equations with non-Hermitian positive Jacobian matrices are proposed,respectively.We discuss the local convergence properties.Finally,nu-merous numerical test results verify the feasibility and effectiveness of the proposed method.In fifth chapter,summarize the work of this paper and propose some prospects for further research.There are totally 2 figures,34 tables and 54 references in this paper.