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Analytical approximations for real values of the Lambert W-function

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  • Barry, D.A
  • Parlange, J.-Y
  • Li, L
  • Prommer, H
  • Cunningham, C.J
  • Stagnitti, F

Abstract

The Lambert W is a transcendental function defined by solutions of the equation Wexp(W)=x. For real values of the argument, x, the W-function has two branches, W0 (the principal branch) and W−1 (the negative branch). A survey of the literature reveals that, in the case of the principal branch (W0), the vast majority of W-function applications use, at any given time, only a portion of the branch viz. the parts defined by the ranges −1≤W0≤0 and 0≤W0. Approximations are presented for each portion of W0, and for W−1. It is shown that the present approximations are very accurate with relative errors down to around 0.02% or smaller. The approximations can be used directly, or as starting values for iterative improvement schemes.

Suggested Citation

  • Barry, D.A & Parlange, J.-Y & Li, L & Prommer, H & Cunningham, C.J & Stagnitti, F, 2000. "Analytical approximations for real values of the Lambert W-function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 53(1), pages 95-103.
    • Handle: RePEc:eee:matcom:v:53:y:2000:i:1:p:95-103
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    References listed on IDEAS

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    1. Beccaria, M. & Soliani, G., 1998. "Mathematical properties of models of the reaction–diffusion type," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 260(3), pages 301-337.
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    Cited by:

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    3. John Wiedenhoeft & Eric Brugel & Alexander Schliep, 2016. "Fast Bayesian Inference of Copy Number Variants using Hidden Markov Models with Wavelet Compression," PLOS Computational Biology, Public Library of Science, vol. 12(5), pages 1-28, May.
    4. Lóczi, Lajos, 2022. "Guaranteed- and high-precision evaluation of the Lambert W function," Applied Mathematics and Computation, Elsevier, vol. 433(C).
    5. Kai Han & Yuntian He & Alex X. Liu & Shaojie Tang & He Huang, 2020. "Differentially Private and Budget-Limited Bandit Learning over Matroids," INFORMS Journal on Computing, INFORMS, vol. 32(3), pages 790-804, July.
    6. Jodrá, P., 2009. "A closed-form expression for the quantile function of the Gompertz–Makeham distribution," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(10), pages 3069-3075.
    7. Wafo Tekam, Raoul Blaise & Kengne, Jacques & Djuidje Kenmoe, Germaine, 2019. "High frequency Colpitts’ oscillator: A simple configuration for chaos generation," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 351-360.
    8. Jodrá, P., 2010. "Computer generation of random variables with Lindley or Poisson–Lindley distribution via the Lambert W function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(4), pages 851-859.
    9. Jose Miguel Riquelme-Dominguez & Sergio Martinez, 2020. "A Photovoltaic Power Curtailment Method for Operation on Both Sides of the Power-Voltage Curve," Energies, MDPI, vol. 13(15), pages 1-17, July.
    10. Jiménez, F. & Jodrá, P., 2009. "On the computer generation of the Erlang and negative binomial distributions with shape parameter equal to two," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(5), pages 1636-1640.

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