标量与向量外推的综述

AI总结：

本文综述了标量与向量外推方法，涵盖经典算法及现代应用，旨在提供统一视角。

摘要：

标量外推和收敛加速方法是数值分析中的核心工具，用于提高迭代算法的效率和缓慢收敛级数的求和效率。这些方法构建了收敛更快但趋向相同极限的转换序列，而不会改变底层的迭代过程，从而降低计算成本并提高数值精度。历史上，此类技术的起源可以追溯到AlKhwarizmi的古典代数方法和Newton早期的级数加速技术，而系统的方法在20世纪初由Aitken过程和Richardson外推法等出现。后来的发展，包括Shanks变换和Wynn epsilon算法，提供了能够消除多个主导误差成分的一般框架，与Pade近似和生成函数的有理近似有深刻联系。本文对经典的标量外推方法进行了全面回顾，包括Richardson外推法、Aitken过程、Shanks变换、Wynn epsilon算法及其他算法。我们探讨了它们的理论基础、渐近误差模型、收敛性质、数值稳定性以及实际实现考虑。本文的第二部分专注于向量外推方法：基于多项式的以及扩展到向量序列的epsilon算法通用化。此外，我们还突出现代发展，如其在迭代求解器、Krylov子空间方法和大规模计算模拟中的应用。本文的目的是提供对标量和向量外推技术的统一视角，连接历史起源、理论洞察和当代计算应用。

英文摘要：

Scalar extrapolation and convergence acceleration methods are central tools in numerical analysis for improving the efficiency of iterative algorithms and the summation of slowly convergent series. These methods construct transformed sequences that converge more rapidly to the same limit without altering the underlying iterative process, thereby reducing computational cost and enhancing numerical accuracy. Historically, the origins of such techniques can be traced back to classical algebraic methods by AlKhwarizmi and early series acceleration techniques by Newton, while systematic approaches emerged in the 20th century with Aitken process and Richardson extrapolation. Later developments, including the Shanks transformation and Wynn epsilon algorithm, provided general frameworks capable of eliminating multiple dominant error components, with deep connections to Pade approximants and rational approximations of generating functions. This paper presents a comprehensive review of classical scalar extrapolation methods, including Richardson extrapolation, Aitken process, Shanks transformation, Wynn epsilon algorithm, and other algorithms. We examine their theoretical foundations, asymptotic error models, convergence properties, numerical stability, and practical implementation considerations. The second part of this work is dedicated to vector extrapolation methods: polynomial based ones and epsilon algorithm generalizations to vector sequences. Additionally, we highlight modern developments such as their applications to iterative solvers, Krylov subspace methods, and large-scale computational simulations. The aim of this review is to provide a unified perspective on scalar and vector extrapolation techniques, bridging historical origins, theoretical insights, and contemporary computational applications.

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