标量 Levin 型序列变换
Scalar Levin-Type Sequence Transformations
AI总结：
本文系统回顾并统一了Levin型序列变换的理论框架，探讨其构造方法，推导两个新变换，并给出收敛加速、稳定性性质及应用指南。
摘要：
序列变换是加速缓慢收敛的标量序列或级数收敛以及求和发散级数的重要工具。那些不仅依赖于序列元素或部分和 $s_n$，还依赖于所谓的余项估计 $\omega_n$ 的辅助序列的变换，如果它们关于 $s_n$ 是线性的，而关于 $\omega_n$ 是非线性的，则属于 Levin 型。本文回顾了已知的 Levin 型序列变换，并将其置于一个统一的理论框架中。讨论了如何通过模型序列方法或通过简单变换的迭代来构造此类变换。作为说明，推导了两个新的序列变换。给出了关于收敛加速和稳定性的共同性质与结果。对于重要的特殊情况，给出了普遍结果的推广。此外，还讨论了 Levin 型序列变换的应用指南，并给出了一些数值示例。
英文摘要：
Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. Transformations that depend not only on the sequence elements or partial sums $s_n$ but also on an auxiliary sequence of so-called remainder estimates $ω_n$ are of Levin-type if they are linear in the $s_n$, and nonlinear in the $ω_n$. Known Levin-type sequence transformations are reviewed and put into a common theoretical framework. It is discussed how such transformations may be constructed by either a model sequence approach or by iteration of simple transformations. As illustration, two new sequence transformations are derived. Common properties and results on convergence acceleration and stability are given. For important special cases, extensions of the general results are presented. Also, guidelines for the application of Levin-type sequence transformations are discussed, and a few numerical examples are given.
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。
