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Guaranteed- and high-precision evaluation of the Lambert W function

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  • Lóczi, Lajos

Abstract

Solutions to a wide variety of transcendental equations can be expressed in terms of the Lambert W function. The 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, W0 and 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 W0 and W−1 at any point.

Suggested Citation

  • Lóczi, Lajos, 2022. "Guaranteed- and high-precision evaluation of the Lambert W function," Applied Mathematics and Computation, Elsevier, vol. 433(C).
    • Handle: RePEc:eee:apmaco:v:433:y:2022:i:c:s0096300322004805
    DOI: 10.1016/j.amc.2022.127406
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    References listed on IDEAS

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    1. H. Vazquez-Leal & M. A. Sandoval-Hernandez & J. L. Garcia-Gervacio & A. L. Herrera-May & U. A. Filobello-Nino, 2019. "PSEM Approximations for Both Branches of Lambert Function with Applications," Discrete Dynamics in Nature and Society, Hindawi, vol. 2019, pages 1-15, March.
    2. Matt Visser, 2018. "Primes and the Lambert W function," Mathematics, MDPI, vol. 6(4), pages 1-6, April.
    3. Pakes, Anthony G., 2018. "The Lambert W function, Nuttall’s integral, and the Lambert law," Statistics & Probability Letters, Elsevier, vol. 139(C), pages 53-60.
    4. Miyajima, Shinya, 2019. "Verified computation for the matrix Lambert W function," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.
    5. 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.
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