牛顿法在放松连续性假设下的改进收敛定理
An improved convergence theorem for the Newton method under relaxed continuity assumptions
AI总结：
本文在主要化技术框架下，提出更宽松的收敛条件，改进了牛顿法在弱连续性假设下的半局部收敛性，通过重构'第一积分'近似，得到更弱的收敛条件和更优的解位置界。
摘要：
在主要化技术框架下，本文提出了一种改进的条件，用于在弱假设下（即涉及算子F(x)的导数连续）的牛顿法的半局部收敛性。我们的起点是Argyros对牛顿序列相邻成员间距离最优上界的表示。我们提案的主要创新点是优化重构了定义标量主要化序列的递推关系的'第一积分'近似。与Argyros之前的成果相比，这使得能够获得更弱的收敛条件，从而得到方程F(x)=0的解位置的更优界，并允许更宽的初始猜测选择。在最简单的情况下，即Lipschitz连续导数算子时，找到了一个显式的限制，保证新的收敛条件改进了著名的Kantorovich条件。
英文摘要：
In the framework of the majorization technique, an improved condition is proposed for the semilocal convergence of the Newton method under the mild assumption that the derivative of the involved operator F(x) is continuous. Our starting point is the Argyros representation of the optimal upper bound for the distance between the adjacent members of the Newton sequence. The major novel element of our proposal is the optimally reconstructed 'first integral' approximation to the recurrence relation defining the scalar majorizing sequence. Compared to the previous results of Argyros, it enables one to obtain a weaker convergence condition that leads to a better bound on the location of the solution of the equation F(x)=0 and allows for a wider choice of initial guesses. In the simplest case of the Lipschitz continuous derivative operator, an explicit restriction is found which guarantees that the new convergence condition improves the famous Kantorovich condition.
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