Improved Two-Step High Order Euler-Chebyshev Methods

Improvments of Euler-Chebyshev methods for solving roots of nonlinear equation are given.Their convergence properties are proved.They are at least seventh-order convergence and nineth-order convergence near simple root.In the end,numerical tests are given and compared with other known high order root-finding methods.The results show that the proposed methods have some more advantages than others.They enrich the methods to find the roots of nonlinear equation and they are important in both theory and application.