Numerics of the generalized Lambert
            <i>W</i>
            function

Scott, Tony; Fee, G. J.; Grotendorst, Johannes; Zhang, W. Z.

doi:10.1145/2644288.2644298

Cited by 14 publications

(14 citation statements)

References 15 publications

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“…We recall that G ∞ is known in terms of k by Eq. (32). The two partial approximants meet exactly at τ = τ 0 specified by Eq.…”

Section: Discussionmentioning

confidence: 89%

“…with f k defined in (13), over the range of all possible values for G ∈ [0, G ∞ ] and k ∈ [0, 1], where G ∞ is given by (32). For the special case of G = G 0 , we thus also address the peak time τ 0 = τ (G 0 ) introduced in Section III G. The f k in the denominator of the integral is semipositive, has its maximum at x m = − ln(k), and reaches zero at G ∞ .…”

Section: Discussionmentioning

confidence: 99%

“…(F3) with τ = t 0 a(t )dt . Lambert's W function 26,27 with several applications in physics [28][29][30][31][32][33] solves the equation z = W (z)e W (z) . It can be used to calculate the solution of transcendental equations of the form e −cx = a 0 (x − r)…”

Section: Remark: Non-monotonous D S/dτmentioning

confidence: 99%

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Analytical Solution of the SIR-Model for the Temporal Evolution of Epidemics. Part A: Time-Independent Reproduction Factor

Kröger

Schlickeiser

2020

Preprint

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We revisit the Susceptible-Infectious-Recovered/Removed (SIR) model which is one of the simplest compartmental models. Many epidemological models are derivatives of this basic form. While an analytic solution to the SIR model is known in parametric form for the case of a time-independent infection rate, we derive an analytic solution for the more general case of a time-dependent infection rate, that is not limited to a certain range of parameter values. Our approach allows us to derive several exact analytic results characterizing all quantities, and moreover explicit, non-parametric, and accurate analytic approximants for the solution of the SIR model for time-independent infection rates. We relate all parameters of the SIR model to a measurable, usually reported quantity, namely the cumulated number of infected population and its first and second derivatives at an initial time t=0, where data is assumed to be available. We address the question on how well the differential rate of infections is captured by the Gauss model (GM). To this end we calculate the peak height, width, and position of the bell-shaped rate analytically. We find that the SIR is captured by the GM within a range of times, which we discuss in detail. We prove that the SIR model exhibits an asymptotic behavior at large times that is different from the logistic model, while the difference between the two models still decreases with increasing reproduction factor. This part A of our work treats the original SIR model to hold at all times, while this assumption will be released in part B. Releasing this assumption allows to formulate initial conditions incompatible with the original SIR model.

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“…We recall that G ∞ is known in terms of k by Eq. (32). The two partial approximants meet exactly at τ = τ 0 specified by Eq.…”

Section: Discussionmentioning

confidence: 89%

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Analytical Solution of the SIR-Model for the Temporal Evolution of Epidemics. Part A: Time-Independent Reproduction Factor

Kröger

Schlickeiser

2020

Preprint

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“…Independently of our study on the generalized W function [23], Scott et al [28,29] studied the number of solutions of equations of the form (3).…”

Section: Bose-fermi Mixturesmentioning

confidence: 99%

Some physical applications of generalized Lambert functions

Mező¹,

Keady²

2016

Eur. J. Phys.

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In this paper we review the physical applications of the generalized Lambert function recently defined by the first author. Among these applications we mention the eigenstate anomaly of the H + 2 ion, the two dimensional two-body problem in general relativity, the stability analysis of delay differential equations and water-wave applications. We also point out that the inverse Langevin function is nothing else but a specially parametrized generalized Lambert function.

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“…Consequently, a wild amount of papers related to these problems can be found in the literature, here we only cite a small sample of them, for instance, [8,10,15,21,25,26,36] and, [18]. The relevance of the Lambert equation lead also to the researchers to look for good approximations and tight bounds on its solutions, see for example, [1,7,19,20,32,33] and, [30].…”

Section: Introductionmentioning

confidence: 99%

On the convergence of infinite towers of powers and logarithms for general initial data: applications to Lambert W function sequences

Toledo

2022

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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The objective of this paper is to provide the exact sets of initial data ensuring the convergence or divergence of a special class of real towers of powers and logarithms. All the terms forming these towers have a common value except the cusp element, that is indeed the initial data of the sequences defining the towers. The results obtained will be applied to some Lambert W function sequences, providing also the whole set of initial data which ensure their convergence or divergence.

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Numerics of the generalized Lambert W function

Cited by 14 publications

References 15 publications

Analytical Solution of the SIR-Model for the Temporal Evolution of Epidemics. Part A: Time-Independent Reproduction Factor

Analytical Solution of the SIR-Model for the Temporal Evolution of Epidemics. Part A: Time-Independent Reproduction Factor

Some physical applications of generalized Lambert functions

On the convergence of infinite towers of powers and logarithms for general initial data: applications to Lambert W function sequences

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