BDF formulas are among the most efficient methods for numerical integration, in particular of stiff equations (see e.g. Gear in Numerical initial value problems in ordinary differential equations, Prentice Hall, Upper Saddle River, 1971). Their excellent stability properties are known for precisely half a century, from the first calculation of their angles of $$A(\alpha )$$-stability by Norsett (BIT, 9:259–263, 1969). Later, more insight was gained and more precise values were calculated numerically (see for example Hairer and Wanner in Solving ordinary differential equations, Springer, New York, 1996, Sect. V.2). This was the state-of-the-art, when Akrivis and Katsoprinakis (BIT, 2019. https://doi.org/10.1007/s10543-019-00768-1) discovered exact values for these angles. In this note we simplify the derivation and results by using Maple.

中文翻译：

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带枫木的精确 BDF 稳定角

BDF 公式是最有效的数值积分方法之一，尤其是刚性方程（参见例如齿轮在常微分方程中的数值初始值问题，Prentice Hall，Upper Saddle River，1971）。从 Norsett (BIT, 9:259–263, 1969) 第一次计算 $$A(\alpha )$$-stability 的角度开始，它们卓越的稳定性特性已经为人所知半个世纪了。后来，获得了更深入的了解，并以数值方式计算了更精确的值（例如，参见 Hairer 和 Wanner 在 Solvingordinary Different equations, Springer, New York, 1996, Sect. V.2 中）。这是最先进的技术，当时 Akrivis 和 Katsoprinakis（BIT，2019 年。https://doi.org/10.1007/s10543-019-00768-1）发现了这些角度的精确值。在本笔记中，我们使用 Maple 简化了推导和结果。