求解双曲开普勒方程近似解的新方法
A new method for obtaining approximate solutions of the hyperbolic Kepler's equation
AI总结：
本文提出求解双曲开普勒方程的近似解方法，利用Smale的α理论证明牛顿法在二次速度下收敛，并提供分段定义的近似零函数及简化起始点构造方法。
摘要：
我们为双曲开普勒方程$S - g \, \text{asinh}(S) - L = 0$在$g \in (0,1)$和$L \in [0,\infty)$范围内提供近似零$\widetilde{S}(g,L)$。通过Smale的α理论证明，以该近似零为起点的牛顿法生成的序列以二次速度收敛到实际解$S(g,L)$，即若$S_n$为第$n$次迭代后的值，则$|S_n - S| \leq 0.5^{2^n - 1} |\widetilde{S} - S|$。近似零$\widetilde{S}(g,L)$是一个分段定义的函数，包含多个线性表达式和一个包含立方根和平方根的表达式。在排除$g=1$和$L=0$小邻域的$(0,1) \times [0,\infty)$有界区域内，我们还提供了一种构造仅包含常数的简化起始点的方法。
英文摘要：
We provide an approximate zero $\widetilde{S}(g,L)$ for the hyperbolic Kepler's equation $S-g\, {\rm asinh} (S)-L=0$ for $g\in(0,1)$ and $L\in[0,\infty)$. We prove, by using Smale's $α$-theory, that Newton's method starting at our approximate zero produces a sequence that converges to the actual solution $S(g,L)$ at quadratic speed, i.e. if $S_n$ is the value obtained after $n$ iterations, then $|S_n-S|\leq 0.5^{2^n-1}|\widetilde{S}-S|$. The approximate zero $\widetilde{S}(g,L)$ is a piecewise-defined function involving several linear expressions and one with cubic and square roots. In bounded regions of $(0,1) \times [0,\infty)$ that exclude a small neighborhood of $g=1, L=0$, we also provide a method to construct simpler starters involving only constants.
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