Abstract

When a sequence or a series of scalars, vectors, matrices, tensors, is converging slowly to its limit, it can be transformed, by a sequence transformation, into a new sequence or a set of new sequences, which under some assumptions converges faster to the same limit. Such a transformation can even be applied to diverging sequences or series, thus providing its analytic continuation. Shanks’ transformation is a well known sequence transformation for accelerating the convergence of scalar sequences. In this survey, we explain its development, its various extensions, and its implementation. Several applications illustrate its effectiveness.

中文翻译：

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香克斯外推法及其应用综述

摘要

当一个序列或一系列标量、向量、矩阵、张量缓慢收敛到其极限时，可以通过序列变换将其转换为一个新序列或一组新序列，在某些假设下，它会更快地收敛到相同的限制。这种变换甚至可以应用于发散序列或序列，从而提供其解析延拓。香克斯变换是众所周知的用于加速标量序列收敛的序列变换。在本次调查中，我们解释了它的发展、它的各种扩展和它的实现。几个应用说明了它的有效性。