改进的割线法及其大范围收敛性

MODIFIED SECANT METHOD & ITSGLOBAL CONVEGENCE

根据常微分方程的Ｌｉａｐｕｎｏｖ渐近稳定性理论和常微分方程数值解的理论提出一种改进的割线法并论证了它的大范围收敛性，大量的数值试验表明，这是解非线性方程的一种行之有效的新算法．

In this paper, a modified Secant method with global convergence is proposed by means of the theory of Liapunov Stability. Quite a number of numerical testsshow that the modified secant method is a very efficient algorithem for solving nonlinearequation.We consider nonlinear equationWithout losing generality, it is assumed that the equation (1) has unique solutioin xin [a,b], f(x)∈C'[a,b], f'(x) ≠0.It is easy to see that the solution x* of equation (1) in [a,b] is equivalent to uniqueminimal point of the fllowing function To find the solution x* of equation (1). we introduce an autonomous system of ordinary differential equation It is clear that the solution x of equation is an equilibrium point of equation (3).Therefore where x(t,x0) is an integral curve of initial value problem (3)-(4).It is obvious that the different numerical methods that are used to find the initial valueproblem (3)-(4) will derive different methods that are used to find the solution of equation(1). For example, if we use Euler method, then we have current formulacorresponding to (6), withand hn=μ,we obtain the modified secant method