牛顿法在新的仿射逆变条件下的半局部收敛性分析

Semi-local Convergence for Newton Method under New Affine Contravariant Conditions

非线性方程及非线性方程组的数值求解一直是计算数学所关注的问题,公认的经典算法是牛顿法,对于它的局部收敛性已有很多研究.在经典牛顿法的半局部收敛Kantorovich定理的基础上引入仿射逆变性,研究了牛顿法在仿射逆变Lipschitz条件和仿射逆变Holder条件下的半局部收敛性.简化了牛顿法的收敛行为,得到了相应的半局部收敛性定理及误差估计.推广并改进了相关文献的结果,表明了该方法的有效性.

Numerical solutions for nonlinear equations and systems of nonlinear equations are always appealing greatly to people.There are many papers discussing the local convergence of Newton method,a universally acknowledged classical algorithm.In the paper,on the basis of the Kantorovich theorem about the classic semi-local convergence of Newton method,affine contravariant conditions were introduced and the semi-local convergence of Newton method under affine contravariant Lipschitz conditions and affine contravariant Holder conditions was analysed.The convergence behavior of Newton method was simplified.The semi-local convergence theorems were presented and the estimations of the iterative residual were obtained.The results extend and improve the results in the related paper,which show that the method is efficient.