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 $x^y=y^x$.

翻译：各种超异方程式的解决方案可以用 Lambert $\ mathrm{W} $ 函数来表示。 $\ mathrm{W} $ 函数, 在应用中经常出现, $\ mathrm{W} 美元功能是一个非元素性功能, 但是现在在所有主要技术计算系统中都执行标准数学函数。 在这项工作中, 我们讨论两个真实分支的近似值, $\ mathrm{W} 和$\ mathrm{W} -1美元。 一方面, 我们用 $\ mathrm{W} 在文献中某些早期结果得到改进的大参数, $\ mathrm} 函数是一个非元素性循环, 一个带有线性, 而另一个则具有二次趋同率。 我们建议, 以二次循环的起始率为合适的起始值, 以确保在两个真实分支的整个定义范围内的趋同。 我们还提供对其趋同速度的先验、 简单、 明确和统一的估计, 其趋同速度可以保证、 $ $\\ 美元 美元 =xxxxxx 的正正率 的正正值 =x 和 等值的正数 。