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

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Abstract

We have developed a new method to compute the real-valued Lambert W-functions, W0(z) and W-1(z). The method is a composite of (1) the series expansions around the branch point, W=-1, and around zero, W=0, and (2) the numerical solution of the modified defining equation, W=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) [16,14] for 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). © 2012 Elsevier B.V. All rights reserved.

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Fukushima, T. (2013). Precise and fast computation of Lambert W-functions without transcendental function evaluations. Journal of Computational and Applied Mathematics, 244(1), 77–89. https://doi.org/10.1016/j.cam.2012.11.021

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