Two New Schemes of Newton′s Interation Method Squares Estimator is BLUE

Two new schemes of Newton′s method,which are Simpson Newton′s method and gemetric mean Newton′s method,are developed and their convergence properties are proved.They are at least third order convergence near simple root and one order convergence near multiple roots.In the end,numerical tests are given and compared with other known Newton′s methods.The results show that the proposed methods have some more advantages than others.They enrich the methods to find the roots of non-linear equations and they are important in both theory and application.