Several New Iterative Methods For Solving Nonlinear Equations

In scientific and engineering computations, many problems can be attributed to the nonlinear equation f(χ)=0. While solving the non-linear equation it generally does not have ready-made formula,. So, finding the root of nonlinear equations is usually adapting numerical methods, which gives an approximate root by the method to reach a certain accuracy. One of the most commonly used method is the famous iterative method, the iterative method is basic issue involved in the construction of iterative format. However, whether the iterative scheme we construct is meaningful, depends on whether it is iterative convergence. For a. convergent iterative scheme, its value will depend on the iterative process of convergence speed and computational efficiency. This paper mainly concerns several new iterative methods, and gives several numerical examples to demonstrate the effectiveness of the presented methods by comparing with some other methods. The paper is divided into six parts, specified as followings:Chapter1introduces the background and present situation of the study of the itera-tive methods for solving nonlinear equation and several commonly used iterative methods.Chapter2introduces the classical Chebyshev-Halley iteration methods of the re-search background and current situation, and on this basis we construct new family of modified Chebyshev-Halley iteration methods free from the second derivative for solving the nonlinear equation, and finally several numerical examples were given to demonstrate the effectiveness of the presented methods by comparing with some other methods.Chapter3is based on the second chapter2, it introduces the classical Hansen-Patrick iteration methods, and on this basis we construct new family of modified Hansen-Patrick iteration methods free from the second derivative for solving the nonlinear equation, and finally several numerical examples were given to demonstrate the effectiveness of the presented methods by comparing with some other methods.In chapter4, it, was introduced the well-known Newton iteration method and St.ef-fensen iteration method of research background and status, and presented two new modifi-cations of Steffensen methods free from derivative. The latter also gives several numerical examples to demonstrate the effectiveness of the presented methods by comparing with some other methods. In chapter5, it was introduced the well-known three-step iterative methods of re-search background and status, and presented a new three-step iterative methods.The latter gives several numerical examples to demonstrate the effectiveness of the presented methods by comparing with some other methods.In chapter6. we introduce the famous modified Newton iteration method for solving the multiple roots of the nonlinear equation of the research background and current situation, and on this basis, we give a new family of third-order modified Newton iteration method.