解非线性方程的Newton类方法及其变形

Newton-like methods for solving nonlinear equations

为了求解非线性方程,利用同伦方法推出具有大范围稳定性的连续型方法、进而离散化得到Newton类方法和Steffenson-Newton类方法,分析得出Newton类方法的大范围收敛性,用Taylor展开证明Newton类方法和Steffenson-Newton类方法在弱条件下的二阶收敛性,并得到收敛速度因子。Newton类方法摒弃了f'(x)≠0这一苛刻条件,带有可调整收敛速度的参数,而Steffenson-Newton类方法还不需要调用导数值,它们都优于Newton法和Newton下山法。

A continuous method was developed for solving nonlinear equations with large-scale stability using the homotopy method. Newton-like iterative methods and Steffensen-Newton-like iterative methods were developed by discretization of the method. The large-scale convergence for Newton-like iterative methods has quadratic convergence. The convergence factors for weak convergence for Newton-like iterative methods and Steffensen-Newton-like iterative methods were found using Taylors series expansions. Newton-like iterative methods remove the strict condition f' (x) ≠ 0 imposed on f(x) and have parameters to adjust the convergence rate. The Steffensen-Newton-like iterative methods do not use derivatives, so they have advantages over Newton's method and the Newton-down hill method.