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Global approximations to the principal real-valued branch of the Lambert \(W\)-function. (English) Zbl 0940.65018

Summary: \(W(z)\) is defined implicitly as the root of \(W\exp(W)= z\). It is shown that a simple analytic approximation has a relative error of less than 5% over the whole domain \(z\in [-\exp(-1),\infty]\) of the principle branch – sufficiently accurate so that four Newton iterations refine this approximation to a relative error smaller than 1.E-12. As a second form of global approximation, the \(W\)-function is expanded as a series of rational Chebyshev functions \(TB_j\) in a shifted, logarithmic coordinate with an error that decreases exponentially fast with the series trunation.

MSC:

65D20 Computation of special functions and constants, construction of tables
33C67 Hypergeometric functions associated with root systems

Software:

WAPR; Algorithm 443
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Full Text: DOI

References:

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