Lambert-W Function

By A. Adler. 2015

2,154 citations recorded by the source corpus.

Subfields: Numerical Methods & Scientific Computing

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Citations in the corpus

Stochastic Polyak Step-size for SGD: An Adaptive Learning Rate for Fast Convergence (2020, methodology): “In the regularized case, f∗i can be pre-computed in closed form for each i using the Lambert W function (Corless et al., 1996) (see Appendix D); while f∗i is simply zero in the unregularized case.”
Avoiding pathologies in very deep networks (2014, background): “(26) The solution to this recurrence is related to the Lambert W-function [Corless et al., 1996] and has”
Gravitational waves from a universe filled with primordial black holes (2020, neutral): “14) The equation ΩGW,tot = 1 can be solved by means of the Lambert function [59], and one obtains Ωmax PBH,f = [ −3μ 16−1 ( − 16 3μ − 16κ 3 )]−3/16 , (4.”
Risk management with expectiles (2017, neutral): “The Lambert W function arises in many applied problems, typically in connection with the solution of transcendental equations; for a detailed review of its properties see, for example, Corless et al. (1996).”
Oscillation criteria for second-order neutral delay differential equations (2017, background): “where W standardly denotes the principal branch of the Lambert function, see [5] for details.”
The Exponentially Convergent Trapezoidal Rule (2014, neutral): “For its properties and applications, see [32].”
Loss Functions for Top-k Error: Analysis and Insights (2015, methodology): “The Lambert W function is defined to be the inverse of the function w 7→ we and is widely used in many fields [13, 17, 45].”
Alpha power transformed extended exponential distribution: properties and applications (2018, methodology): “[3] and Jodra [10]) the Lambert W function has been applied to solve several problems in mathematics, physics and engineering.”
Automatic model construction with Gaussian processes (2014, background): “The solution to this recurrence is related to the Lambert W function (Corless et al., 1996) and has no closed form.”
$f$ -Divergence Inequalities (2015, background): “with r in (397) is given in (43), and W−1 in (398) denotes the secondary real branch of the Lambert W function [19].”
Image Resolution in Optical Nanoscopy (2013, background): “Barry, D.A. et al. Analytical approximations for real values of the Lambert W-function.”
The Lipschitz Constant of Self-Attention (2020, methodology): “…on Lip ∞ ( F ) : ∞ and the following bound on Lip 2 ( F ) : where φ ( x ) := x exp( x + 1) is an invertible univariate function on x > 0 , and N is the input sequence length. where W 0 is the Lambert W -function, which grows sub-logarithmically as O (log N − log log N ) (Corless et al., 1996).”
Frequencies of the Ricker wavelet (2015, methodology): “The solution of this equation is a special function known as the Lambert W function (Lambert, 1758, 1772; Euler, 1779; Corless et al., 1996).”
Bayesian Recovery of the Initial Condition for the Heat Equation (2011, background): “By Corless et al. (1996), W x ∼ log x , which completes the proof.”
Giant acoustic atom: A single quantum system with a deterministic time delay (2016, background): “Here, W (z) is the Lambert W function [39] defined by the equation z = W (z)e (z), which in general is a multivalued function with branches Wk(z), k ∈ Z.”
On f-Divergences: Integral Representations, Local Behavior, and Inequalities (2018, positive): “tγ = −γ W−1 ( − 1 γ e − 1 γ ) (154) where W−1 in (154) denotes the secondary real branch of the Lambert W function [5].”