论文标题
Lambert W功能的显式和递归估计
Explicit and recursive estimates of the Lambert W function
论文作者
论文摘要
可以用兰伯特$ \ mathrm {w} $函数来表示多种超越方程的解决方案。 $ \ mathrm {w} $函数在应用程序中经常发生，是一种非元素，但现在在所有主要技术计算系统中实现的标准数学函数。在这项工作中，我们讨论了两个真实分支的一些近似值，即$ \ mathrm {w} _0 $和$ \ mathrm {w} _ { - 1} $。一方面，我们在$ \ mathrm {w} _0 $上介绍了一些分析性下限和上限，以改善文献中一些早期结果。另一方面，我们分析了两个对数递归，一个递归是线性的，另一个具有二次收敛速率。我们建议使用二次率的递归递归开始值，以确保两个真实分支的定义的整个领域收敛。我们还对其收敛速度提供了先验，简单，显式和统一的估计，该估计值$ \ mathrm {w} _0 $和$ \ mathrm {w} _ { - 1} $在任何时候都可以保证保证的高精度近似值。最后，作为$ \ mathrm {w} _0 $函数的应用，我们解决了关于正式非平凡解决方案的增长率$ x^y = y = y^x $的猜想。
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 $x^y=y^x$.
