AI 中文总结
针对牛顿迭代因导数接近零可能无法收敛的问题，提出修正稳定的加速牛顿迭代算法，分析其收敛性以提升效率与稳定性。
AI 中文摘要
牛顿迭代是求根与方程组数值解的基础工具，它能快速将初始近似值迭代至精确根，通常收敛速度为二次。由于该方法每次迭代都需要计算函数值及其导数，在某些情况下可能因导数接近零而无法收敛。本文提出了一种修正且稳定的牛顿迭代算法，解决了上述问题，降低了计算成本并提升了效率；此外，还分析了该修正方法的收敛性，以验证其有效性。
英文摘要
Newton's iteration is a fundamental tool for root-finding and numerical solutions of systems of equations. The iteration rapidly refines the initial approximation to the exact root, and in general the convergence is quadratic. Since the method requires finding the function value and its derivative at each iteration, in some cases, it may not converge. This is because the value of the derivative gets close to zero. In this paper, we introduce a modified and stable algorithm of Newton's iteration method that addresses this issue, reduces computational cost, and improves efficiency. In addition. we analyze the convergence properties of the modified method to demonstrate its effectiveness.
