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New approximations to the principal real-valued branch of the Lambert W-function

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Abstract

The Lambert W-function is the solution to the transcendental equation W(x)e W(x) = x. It has two real branches, one of which, for x ∈ [−1/e, ∞], is usually denoted as the principal branch. On this branch, the function grows from − 1 to infinity, logarithmically at large x. The present work is devoted to the construction of accurate approximations for the principal branch of the W-function. In particular, a simple, global analytic approximation is derived that covers the whole branch with a maximum relative error smaller than 5 × 10−3. Starting from it, machine precision accuracy is reached everywhere with only three steps of a quadratically convergent iterative scheme, here examined for the first time, which is more efficient than standard Newton’s iteration at large x. Analytic bounds for W are also constructed, for x > e, which are much tighter than those currently available. It is noted that the exponential of the upper bounding function yields an upper bound for the prime counting function π(n) that is better than the well-known Chebyshev’s estimates at large n. Finally, the construction of accurate approximations to W based on Chebyshev spectral theory is discussed; the difficulties involved are highlighted, and methods to overcome them are presented.

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Acknowledgments

The authors wish to thank Rob Corless, David Jeffrey, and an anonymous reviewer for useful comments on the original manuscript, and André LeClair for bringing Ref. [20] to their attention.

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Authors and Affiliations

  1. ENEA, CR Casaccia (SP 118), Via Anguillarese, 301, 00123, Roma, Italy

    Roberto Iacono

  2. Department of Climate and Space Sciences and Engineering, University of Michigan, 2455 Hayward Avenue, Ann Arbor, MI, 48109, USA

    John P. Boyd

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  1. Roberto Iacono
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Correspondence to Roberto Iacono.

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Communicated by: Gitta Kutyniok

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Iacono, R., Boyd, J.P. New approximations to the principal real-valued branch of the Lambert W-function. Adv Comput Math 43, 1403–1436 (2017). https://doi.org/10.1007/s10444-017-9530-3

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  • DOI: https://doi.org/10.1007/s10444-017-9530-3

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