Lambert W函数的显式递推估计
Explicit and recursive estimates of the Lambert W function
摘要：
许多超越方程的解都可以用朗伯$\mathrm{W}$函数表示。在应用程序中经常出现的$\mathrm{W}$函数是一个非初等的，但现在在所有主要的技术计算系统中实现的标准数学函数。在这项工作中，我们讨论的两个实分支，$\mathrm{W}_0$和$\mathrm{W}_{-1}$的一些近似。一方面，我们提出了一些分析的上下界$\mathrm{W}_0$大的参数，改善了一些早期的结果在文献中。另一方面，我们分析了两个对数递归，一个是线性的，另一个是二次的收敛速率。我们提出了合适的起始值的递归与二次率，确保收敛的整个定义域的两个真正的分支。我们还提供了一个先验的，简单的，明确的和统一的估计其收敛速度，使保证，高精度的近似$\mathrm{W}_0$和$\mathrm{W}_{-1}$在任何点。最后，作为$\mathrm{W}_0$函数的应用，我们解决了方程$x^y=y^x$的一个关于正非平凡解增长率的猜想。
英文摘要：
Solutions to a wide variety of transcendental equations can be expressed in terms of the Lambert $\mathrm{W}$ function. The $\mathrm{W}$ function, occurring frequently in applications, is a non-elementary, but now standard mathematical function implemented in all major technical computing systems. In this work, we discuss some approximations of the two real branches, $\mathrm{W}_0$ and $\mathrm{W}_{-1}$. On the one hand, we present some analytic lower and upper bounds on $\mathrm{W}_0$ for large arguments that improve on some earlier results in the literature. On the other hand, we analyze two logarithmic recursions, one with linear, and the other with quadratic rate of convergence. We propose suitable starting values for the recursion with quadratic rate that ensure convergence on the whole domain of definition of both real branches. We also provide a priori, simple, explicit and uniform estimates on its convergence speed that enable guaranteed, high-precision approximations of $\mathrm{W}_0$ and $\mathrm{W}_{-1}$ at any point. Finally, as an application of the $\mathrm{W}_0$ function, we settle a conjecture about the growth rate of the positive non-trivial solutions to the equation $x^y=y^x$.
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