A New Levenberg-Marquardt Algorithm For Solving Nonlinear Equations And Its Theoretical Analysis

Solving nonlinear problems is an active research topic in the field of optimization,the calculations of Jacobin matrix is expensive in the solution of nonlinear equations.In order to save the calculation of Jacobin matrix,two methods are proposed in this paper.First,we put forward an accelerating multi step LevenbergMarquardt method by using LM step and four approximate LM steps,which used the previous calculated Jacobian are computed,second the high order accelerated multi step Levenberg-Marquardt algorithm is proposed based on accelerating multi step Levenberg-Marquardt algorithm.The accelerating multi step Levenberg-Marquardt algorithm,At every iteration,this metohd save the amount of calculation and improve the calculation efficiency.After the improved multi step Levenberg-Marquardt algorithm,the Levenberg-Marquardt algorithm with higher order acceleration is obtained,which increases the LM step by step,so as to reduce the computation of Jacobin matrix.The global convergence and local convergence is given by the trust region technique and singular value decomposition method,Numerical result shows that two kinds of LM algorithm performs very well and save many Jacobian calculations.