四次多项式中牛顿法与哈雷法同时全局收敛失败的构造
Simultaneous Global Convergence Failure of Newton's and Halley's Methods in Degree Four
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中文总结 AI 辅助
本文构造显式四次多项式，证明牛顿法与哈雷法在非空开集上同时不收敛，填补四次次数空白，并给出另一多项式的数值证据表明三种方法同时失败。
中文摘要 AI 辅助
我们构造了一个显式的四次多项式，对于该多项式，牛顿法和哈雷法的求根方法在非空的初始值开集上均无法收敛到根。该示例可通过在一族四次多项式（设计用于展现预设的牛顿二周期）中进行可复现的计算搜索而找到。随后，我们利用两区间压缩论证，解析地证明了牛顿法具有一个超吸引二周期，而哈雷法则具有一个不同的吸引性外生二周期。这填补了牛顿-哈雷同时失败问题中的四次次数空白。计算方法还识别出另一个四次多项式，其数值证据表明牛顿法、哈雷法和施罗德法会同时失败。
英文摘要
We construct an explicit quartic polynomial for which Newton's and Halley's root-finding methods both fail to converge to a root on nonempty open sets of initial values. The example can be found through a reproducible computational search within a one-parameter family of quartics designed to exhibit a prescribed Newton two-cycle. We then prove analytically that Newton's method has a superattracting two-cycle and that Halley's method has a distinct attracting extraneous two-cycle, using a two-interval contraction argument. This fills a degree-four gap in the simultaneous Newton--Halley failure problem. The computational approach also identifies another quartic with numerical evidence of simultaneous failure of Newton's, Halley's, and Schröder's methods.
发表机构
- “G. d’Annunzio” University of Chieti–Pescara（吉·丹农齐奥基耶蒂-佩斯卡拉大学）
- Trinity College Dublin（都柏林三一学院）
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