Precise and fast computation of Lambert WW-functions without transcendental function evaluations

We have developed a new method to compute the real-valued Lambert WW-functions, W0(z)W0(z) and W−1(z)W−1(z). The method is a composite of (1) the series expansions around the branch point, W=−1W=−1, and around zero, W=0W=0, and (2) the numerical solution of the modified defining equation, W=ze−WW=ze−W. In the latter process, we (1) repeatedly duplicate a test interval until it brackets the solution, (2) conduct bisections to find an approximate solution, and (3) improve it by a single application of the fifth-order formula of Schröder’s method. The first two steps are accelerated by preparing auxiliary numerical constants beforehand and utilizing the addition theorem of the exponential function. As a result, the new method requires no call of transcendental functions such as the exponential function or the logarithm. This makes it around twice as fast as existing methods: 1.7 and 2.0 times faster than the methods of Fritsch et al. (1973) and Veberic (2012) [16] and [14] for W0(z)W0(z) and 1.8 and 2.0 times faster than the methods of Veberic (2012) [14] and Chapeau-Blondeau and Monir (2002) [13] for W−1(z)W−1(z).

► We provide a new method to compute Lambert WW-functions, W0(z)W0(z) and W(−1)(z)W(−1)(z). ► The new method uses bisections and a fifth-order improvement formula. ► The new method requires no call of transcendental functions including the exponential function. ► The new method runs around twice as fast as existing methods.