Tham khảo tài liệu 'lập trình c# all chap "numerical recipes in c" part 20', công nghệ thông tin phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | ConvergentMethods for NonlinearSystems ofEquations 383 such methods can still occasionally fail by coming to rest on a local minimum of F they often succeed where a direct attack via Newton s method alone fails. The next section deals with these methods. CITED REFERENCES AND FURTHER READING Acton . 1970 Numerical Methods That Work 1990 corrected edition Washington Mathematical Association of America Chapter 14. 1 Ostrowski . 1966 Solutions of Equations and Systems of Equations 2nd ed. New York Academic Press . Ortega J. and Rheinboldt W. 1970 Iterative Solution of Nonlinear Equations in Several Variables New York Academic Press . Globally ConvergentMethods forNonlinear Systems of Equations We have seen that Newton s method for solving nonlinear equations has an unfortunate tendency to wander off into the wild blue yonder if the initial guess is not sufficiently close to the root. A global method is one that converges to a solution from almost any starting point. In this section we will develop an algorithm that combines the rapid local convergence of Newton s method with a globally convergent strategy that will guarantee some progress towards the solution at each iteration. The algorithm is closely related to the quasi-Newton method of minimization which we will describe in . Recall our discussion of the Newton step for the set of equations is where F x 0 xnew Xold Sx Sx -J-1 F Here J is the Jacobian matrix. How do we decide whether to accept the Newton step Sx A reasonable strategy is to require that the step decrease F 2 F F. This is the same requirement we would impose if we were trying to minimize f 2 f f The 2 is for later convenience. Every solution to minimizes but there may be local minima of that are not solutions to . Thus as already mentioned simply applying one of our minimum finding algorithms from Chapter 10 to is not a good idea. To develop a better strategy note that