Acceleration Algorithom For Nonlinear Equations

In recent decades, with the development of studying mathematical and the emergent of large-scale computers, a variety of non-linear problem is becoming increasingly attracted scientists and engineers. Recently the non-linear science research is in vigorous development at home and abroad,non-linear numerical analysis and methods are playing an increasingly important role. Solving non-linear mathematics, physics issues (including ODE, partial differential equation micro-boundary value problem, integral equations, differential equations, etc.), non-linear mechanics, nonlinear optimization, numerical problems of economics are the fundamental issue of non-linear science, but these issues were eventually reduced to solving nonlinear equations. Therefore, Studying equation has great significance. The creation of calculus in the seventeenth century, Newton and Halley invented iterative method with their name, now there are many experts and scholars committed to finding iterative methods for solving nonlinear equations.This article discusses the accelerated algorithm for solving nonlinear equations. In engineering applications and scientific computing, we often simplify the problem and summarized to get a a mathematical expression with numerical result, this expression is often expressed as a mathematical equation. Solving equation f ( x ) = 0is an important issue in applied mathematics. In Numerical analysis the basis for all numerical method solving nonlinear equations is a simple iterative method, it repeatedly through a iterative method to make the approximation root more precise step by step until to meet the reservation of the exact value. But the methods of the numerical analysis for solving nonlinear equations (for bisection method, linear interpolation method, Newton's method, etc.) have their own shortcomings. So the effective and accelerated algorithms needed to design to solve the issues of life and science.The text is divided into five parts .The first part introduces the research background, significance and the status at home and abroad of solving nonlinear equations, points out that solving the nonlinear equations is the research hotspot in today's scientific work.The second part describes some prior knowledge, including relevant definitions and theorems.The third section discusses the convergence behavior of power mean Newton's for multiples roots. Using this method, it converges linearly to multiple roots and if we known the multiplicity, it converges quadratically to multiple roots, Moreover, it is shown that if you using this method to get the multiple roots of nonlinear equation,its power exponent should be smaller, then it can convergent more quickly.The fourth part, we show three steps iterative method to find the simple roots of nonlinear equation which accelerate a existing Pth-order iterative method to (P+3)th-order Newton's method by increasing an iterative step. Because it increased the order of convergence, several numerical examples can these methods are efficient in their performance.Part V is a summary of the content, summed up the aforementioned iterative method as well as something to think about for solving nonlinear equations, it also pointed out that we should pay attention to the problems in subsequent research and gave some research directions...