FP//LINSPACE evaluation of real Lambert W-function $W_0$

In the paper, we construct FP//LINSPACE algorithmic analog of real Lambert W-function $W_0(x)$ on segment $[-(re)^{-1},(re)^{-1}]$ of FP//LINSPACE algorithmic real numbers, where $r$ is a rational number, $r>4/3$ (any such rational number is suitable). To construct algorithmic analog of real Lambert W-function $W_0(x)$, we consider algorithm WLE for the evaluation of dyadic rational approximations of the function on segment $[-(re)^{-1},(re)^{-1}]$ on Turing machine using polynomial time and linear space. Algorithm WLE is based on the Taylor series expansion of the function; it is shown that the Taylor series of real Lambert W-function $W_0(x)$ on segment $[-(re)^{-1},(re)^{-1}]$ converges linearly. This fact is used in the algorithm.