Levenberg-Marquardt Method For Solving Nonlinear Equations

The iterative method is one of the basic method for solving nonlinear equations.The iterative algorithm based on the optimization method owns its unique advantages.Gauss-Newton method and Levenberg-Marquardt method are two important iterative methods.In order to correct the descending direction offset caused by adding items,avoid the waste of information due to the elimination of the second item,reduce the amount of computation in the iterative process and improve the convergence efficiency,this paper proposes a modified two-step Levenberg-Marquardt method and an improved Gauss-Newton method based on RALND function for nonlinear equations.The first part mainly introduces the research background,significance and the re-search status at home and abroad of this paper.In the preliminary knowledge,conver-gence property and convergence speed of the classic Newton method,Gauss-Newton method and Levenberg-Marquardt method are introduced,their advantages and disad-vantages are also analyzed.In the second part,a modified two-step LM method for solving nonlinear equations is proposed.The classical LM method is modified by adding correction terms to reduce the step shift caused by the LM constant guaranteeing non-singularity,and this modified idea is applied into the two-step LM method proposed by Fan in[28]and[29].A modified two-step LM algorithm is proposed and the convergence of the algorithm is proved.Finally,a large number of numerical examples are used into the modified two-step LM method,the classic LM method and the two-step LM method,and numerical results are compared to verify the feasibility and effectiveness of the modified two-step LM method.In the third part,an improved Gauss-Newton method for solving nonlinear equa-tions is proposed.Based on Saheya's new modified RALND function by[42]which is convenient to solve nonlinear equations,an improved Gauss-Newton method is pro-posed.It reduces the waste of information such as function value and gradient val-ue caused by ignoring the second-order information item,and makes the improved Gauss-Newton direction closer to Newton direction.At the same time,the convergence of the method is proved.Numerical results of the improved Gauss-Newton method.Newton-like method([42])and the Gauss-Newton method are compared and we find that the improved Gauss-Newton method has superior characteristics and numerical performance.The fourth part summarizes the main ideas,advantages and disadvantages of the above methods,propose improvement measures for the lack of methods,and forecasts the further work that needs to be done.