通过组合非线性凝聚变换加速收敛
Convergence Acceleration via Combined Nonlinear-Condensation Transformations
AI总结：
本文提出结合Van Wijngaarden凝聚变换与非线性序列变换加速单调级数收敛，并通过特殊函数和球贝塞尔函数乘积级数示例展示其效率与稳定性。
摘要：
本文描述了一种数值评估缓慢收敛单调级数的方法。首先，我们对原始级数应用Van Wijngaarden凝聚变换，将原始单调级数转化为交错级数。第二步，借助已知对交错级数特别有效的合适非线性序列变换，加速变换后级数的收敛。我们讨论了该方法的一些理论方面。通过多个例子展示了组合非线性凝聚变换的效率、数值稳定性和广泛适用性。我们讨论了在收敛圆边界上或接近边界处，甚至在奇点附近特殊函数的评估。我们还考虑了一个球贝塞尔函数乘积的级数，该级数可作为量子电动力学束缚态计算中出现的分波展开的模型。
英文摘要：
A method of numerically evaluating slowly convergent monotone series is described. First, we apply a condensation transformation due to Van Wijngaarden to the original series. This transforms the original monotone series into an alternating series. In the second step, the convergence of the transformed series is accelerated with the help of suitable nonlinear sequence transformations that are known to be particularly powerful for alternating series. Some theoretical aspects of our approach are discussed. The efficiency, numerical stability, and wide applicability of the combined nonlinear-condensation transformation is illustrated by a number of examples. We discuss the evaluation of special functions close to or on the boundary of the circle of convergence, even in the vicinity of singularities. We also consider a series of products of spherical Bessel functions, which serves as a model for partial wave expansions occurring in quantum electrodynamic bound state calculations.
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