全线性多步方法作为求根方法
Full linear multistep methods as root-finders
AI总结：
本文研究全线性多步方法作为求根方法的稳定性与收敛性，通过逆多项式插值分析其收敛速率限制，并展示数值实验表明其优越性能，同时提供基于Brent方法的稳健实现。
摘要：
基于全线性多步方法（LMMs）的求根方法利用先前的函数值、导数和根估计进行迭代求根。作为ODE求解器，全LMM通常不零稳定，但作为求根方法时，插值点的收敛性规避了稳定性问题。本文基于逆多项式插值进行一般分析，证明了任何基于LMM的方法的收敛速率存在根本限制。通过数值实验展示全LMM方法的卓越性能。最后，提供基于Brent方法的稳健实现，确保收敛。
英文摘要：
Root-finders based on full linear multistep methods (LMMs) use previous function values, derivatives and root estimates to iteratively find a root of a nonlinear function. As ODE solvers, full LMMs are typically not zero-stable. However, used as root-finders, the interpolation points are convergent so that such stability issues are circumvented. A general analysis is provided based on inverse polynomial interpolation, which is used to prove a fundamental barrier on the convergence rate of any LMM-based method. We show, using numerical examples, that full LMM-based methods perform excellently. Finally, we also provide a robust implementation based on Brent's method that is guaranteed to converge.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
