Several Improvements On Newton And Quasi-newton Algorithm

Newton and Quasi-Newton algorithms have been proved to be one of the most commonly used and most effective ones for solving nonlinear equations and unconstrained optimization problem.In this paper,we analyze the Newton and Quasi-Newton equation further based on the previous works of Newton and Quasi-Newton algorithm and put forward the modified Newton and Quasi-Newton equation to improve the Newton and Quasi-Newton algorithm combining with correction formula and search criteria.There are five parts in this thesis:Firstly,we introduced the related content of Newton and Quasi-Newton algorithms for solving nonlinear equations and unconstrained optimization problem.This introduced the algorithms,the choice of the step,the line search methods and the convergence,etc.Secondly,we used auxiliary function to get one kind of effective new Newton algorithm.Under some suitable conditions,we proved the global convergence.At last,through numerical experiments,our algorithm is efficient for solving nonlinear equations problem.Thirdly,we let the step factor where we get from the CF and PCG search conditions combined with Newton method to get one kind of effective new Newton algorithm.Under some suitable conditions,we proved the global convergence.Fourthly,we used auxiliary function to get one kind of effective new Quasi-Broyden algorithm.Under some suitable conditions,we proved the global convergence.Finally,by adding parameters,we generalized the existing Quasi-Newton equation.And under the non-monotone linear search technique and some suitable conditions,we proved the global convergence of the new non-monotone quasi-newton algorithm.