Solutions to a wide variety of transcendental equations can be expressed in terms of the Lambert $\mathrm{W}$ function. The $\mathrm{W}$ function, also occurring frequently in many branches of science, is a non-elementary but now standard mathematical function implemented in all major technical computing systems. In this work, we analyze an efficient logarithmic recursion with quadratic convergence rate to approximate its two real branches, ${\mathrm{W}}_0$ and ${\mathrm{W}}_{-1}$. We propose suitable starting values that ensure monotone convergence on the whole domain of definition of both branches. Then, we provide a priori, simple, explicit and uniform estimates on the convergence speed, which enable guaranteed, high-precision approximations of ${\mathrm{W}}_0$ and ${\mathrm{W}}_{-1}$ at any point.

各种超越方程的解可以用 Lambert 表示$\mathrm{W}$功能。这$\mathrm{W}$函数，也经常出现在许多科学分支中，是一种非基本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，我们分析了一个有效的具有二次收敛速度的对数递归来逼近它的两个实数分支，${\mathrm{W}}_0$和${\mathrm{W}}_{-1}$. 我们提出了合适的起始值，以确保在两个分支的整个定义域上单调收敛。然后，我们提供了关于收敛速度的先验、简单、明确和统一的估计，这使得有保证的高精度近似成为可能${\mathrm{W}}_0$和${\mathrm{W}}_{-1}$在任何时候。