非线性序列变换用于加速收敛与发散级数求和
Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series
AI总结：
本报告系统探讨了多种非线性序列变换，用于加速收敛和求和发散级数，提出了新变换及高效算法，并通过理论分析和数值测试验证了其性能。
摘要：
在科学问题的数学处理中，缓慢收敛的级数和序列以及发散级数相当频繁地出现。本报告讨论了大量主要用于加速收敛和发散级数求和的非线性序列变换。本报告中的一些序列变换，例如Wynn的$\epsilon$算法或Levin序列变换，在收敛加速文献中已得到充分确立，但其中大多数是新的。我们推导了评估这些变换的高效算法。分析了序列变换在收敛加速和求和过程中的理论性质。最后，通过将本报告的序列变换应用于某些缓慢收敛和发散级数来测试其性能，这些级数有望成为科学问题和应用数学中可能出现的许多缓慢收敛或发散级数的现实模型。
英文摘要：
Slowly convergent series and sequences as well as divergent series occur quite frequently in the mathematical treatment of scientific problems. In this report, a large number of mainly nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series are discussed. Some of the sequence transformations of this report as for instance Wynn's $ε$ algorithm or Levin's sequence transformation are well established in the literature on convergence acceleration, but the majority of them is new. Efficient algorithms for the evaluation of these transformations are derived. The theoretical properties of the sequence transformations in convergence acceleration and summation processes are analyzed. Finally, the performance of the sequence transformations of this report are tested by applying them to certain slowly convergent and divergent series, which are hopefully realistic models for a large part of the slowly convergent or divergent series that can occur in scientific problems and in applied mathematics.
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。
