Two Iterative Algorithms For Solving Nonlinear Equations

With the rapid development of science and technology,the application of nonlinear science has involved various industries,such as meteorological data analysis,aircraft,automobile and ship design,petroleum geology,computational biochemistry,Aerospace industry and orbit design,Information rescue,etc.There are a large number of practical problems,all of which need to be described with the help of nonlinear models,which can ultimately be reduced to the problem of solving nonlinear equations and systems of nonlinear equations.However,for algebraic equations whose order is greater than 4 times,its exact solution can no longer be obtained by analytical methods.At this time,if you want to require an approximate solution of the equation,you can only find a certain numerical method,and it is more difficult to solve systems of nonlinear equations.Therefore,it is very important to use numerical methods to solve nonlinear equations and systems of nonlinear equations in both theories and practical applications.The first chapter,The research background and significance of nonlinear equations are introduced in detail,and the importance of numerical methods in solving nonlinear equations and systems of nonlinear equations is clarified.To solve this problem,Many scholars at home and abroad continue to explore more effective numerical solutions of nonlinear equations,and introduce several common numerical solutions and convergence analysis in the article..The second chapter,A 32-step iterative algorithm for solving unary nonlinear equations is proposed.Newton's iteration method is the most classic method for solving nonlinear equations.Newton's method converges fast and reaches second-order convergence,but each iteration needs to calculate the derivative,which increases the amount of calculation and reduces the efficiency index.In view of this shortcoming of Newton's method,an improved Newton's method is constructedwhen solving a one-variable nonlinear equation.Based on the Newton iteration method as the main function,the interpolation method is skillfully combined with it,which reduces the calculation amount,improves the efficiency index,An optimal iterative method with a 32-order convergence speed is constructed,the convergence of the method is proved,and numerical examples are analyzed to verify the effectiveness of the algorithm.The third chapter,In an entirely new way,an improved Newton method for solving nonlinear equations is proposed.It mainly improves the classical Newton method for solving nonlinear equations,and a modified Newton method is constructed.The improved method converges faster under the same function and derivative evaluation times as the Newton method.And the convergence of the method is proved in theory.Finally,numerical examples verify the effectiveness of the method.