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Numerical evaluation of the Lambert W function and application to generation of generalized Gaussian noise with exponent 1/2

F. Chapeau-Blondeau, A. Monir
2002 IEEE Transactions on Signal Processing  
We review the main methods for numerical evaluation of the relevant branch of the (multivalued) Lambert W function with controlled accuracy and complement them with an original rational function approximation  ...  We show that this transformation is expressable in terms of a special function known under the name of the Lambert W function.  ...  ACKNOWLEDGMENT The authors would like to thank R. Corless for valuable informations on the Lambert W function.  ... 
doi:10.1109/tsp.2002.801912 fatcat:lsadwmiw65gr3puopq5hstnckm

Comments on "numerical evaluation of the Lambert W function and application to generation of generalized Gaussian noise with exponent 1/2"

D.A. Barry, L. Li, D.-S. Jeng
2004 IEEE Transactions on Signal Processing  
ACKNOWLEDGMENT The authors thank the Associate Editor, Prof. Fredrik Gustafsson, and the three anonymous reviewers for their helpful and constructive comments and suggestions.  ...  Comments on "Numerical Evaluation of the Lambert Function and Application to Generation of Generalized Abstract-The Lambert function appears in a wide variety of circumstances, including the recent application  ...  In [3] , a new use of the lower real branch of the Lambert W function W 01 is described; it arises in the inverse distribution function of generalized Gaussian noise with power 1/2.  ... 
doi:10.1109/tsp.2004.826154 fatcat:zr3lw224c5gu5kwvdyxdpqfzqy

Numerical Computation of Delay Differential Equation using Laplace Transform and Lambert W Function

2019 VOLUME-8 ISSUE-10, AUGUST 2019, REGULAR ISSUE  
In this paper a novel approach using Laplace transform for the solution of delay differential equation with a single delay based on Lambert W function has been investigated.  ...  An obtained result is extended to the nth order DDEs. Numerical examples have been provided to illustrate the obtained result.  ...  In 2003, an analytical method on the basis of Lambert W function was developed to find the solution of DDEs by Ulsoy et.al., [2] . F.  ... 
doi:10.35940/ijitee.k1772.0981119 fatcat:hsys4wltojfhppwt5dmvorqkkm

Approximate Solutions of Linearized Delay Differential Equations Arising from a Microbial Fermentation Process Using the Matrix Lambert Function

Agus Yodi Gunawan, Kasbawati Kasbawati, Kuntjoro Adji Sidarto
2016 Journal of Mathematical and Fundamental Sciences  
The simulations are worked out by taking the principal branch of the matrix Lambert function as the most dominant mode.  ...  In this paper we present approximate solutions of linearized delay differential equations using the matrix Lambert function.  ...  The second author would also like to thank DIKTI for the support received through her BPPDN scholarship.  ... 
doi:10.5614/j.math.fund.sci.2016.48.1.3 fatcat:do5545qnw5hhzdu5f2hxolxhpu

Solution of a system of linear delay differential equations using the matrix Lambert function

Sun Yi, A.G. Ulsoy
2006 2006 American Control Conference  
To generalize the Lambert function method for scalar DDEs, we introduce a new matrix, Q when the coefficient matrices in a system of DDEs do not commute.  ...  The solution has the form of an infinite series of modes written in terms of the matrix Lambert functions.  ...  Every function W (h), such that W (h)e W (h) = h, is called a Lambert function.  ... 
doi:10.1109/acc.2006.1656585 fatcat:az4nc2opuzejlfgbkm2lqhwjmy

Simulation techniques for generalized Gaussian densities

Martina Nardon, Paolo Pianca
2009 Journal of Statistical Computation and Simulation  
This contribution deals with Monte Carlo simulation of generalized Gaussian random variables.  ...  For values of the shape parameter α within a certain range, the distribution presents heavy tails. In particular, the cases α = 1/3 and α = 1/2 are considered.  ...  The Lambert W function The Lambert W function, specified implicitly as the root of the equation W (z)e W (z) = z , (14) is a multivalued function defined in general for z complex and assuming values W  ... 
doi:10.1080/00949650802290912 fatcat:7mthgkyngrey5mg6kjnlwdmduu

Implementation of the Green-Ampt Infiltration Model: Comparative between different numerical solutions

T. A. Mendes, S. F. Sousa Júnior, S. A. S. Pereira
2021 Trends in Computational and Applied Mathematics  
Thus, this article aims to implement the Green-Ampt model using two numerical methods - Newton-Raphson method and W-Lambert function - to determine soil permeability parameters - K and matric potential  ...  The Green-Ampt model adjusted well to the data measured from the rain simulator, with a determination coefficient of 0.978 for the Newton-Raphson method and 0.984 for the W-Lambert function.  ...  W-Lambert function The W-Lambert function, in its generalized mode, is presented in Equation 2.4, and does not have a defined inverse function.  ... 
doi:10.5540/tcam.2021.022.04.00645 fatcat:7yxbecqtynabna7hhgx4kbwjfy

On the simulation of the Hawkes process via Lambert-W functions [article]

Martin Magris
2019 arXiv   pre-print
This manuscript shows that the ITS approach can be conveniently discussed in terms of Lambert-W functions.  ...  Several methods have been developed for the simulation of the Hawkes process.  ...  The inverse function f −1 (ze z ) is the so-called Lambert-W function: z = f −1 (ze z ) = W (ze z ) Given a general problem in the form ze z = w, its solution (z * ) is therefore z * = W (w).  ... 
arXiv:1907.09162v1 fatcat:hhavmmxmjjf3thzg7gfhsbsk2u

The Ricker wavelet and the Lambert W function

Yanghua Wang
2014 Geophysical Journal International  
Since the Ricker wavelet is the second derivative of a Gaussian function and its spectrum is a single-valued smooth curve, numerical evaluation of the Lambert W function can be implemented by a stable  ...  Moreover, the Lambert W function is a variation of the Ricker wavelet amplitude spectrum.  ...  Since the spectrum (14) of the Ricker wavelet, the second derivative of a Gaussian, is a single-valued smooth curve, we can numerically evaluate the Lambert W function by interpolation.  ... 
doi:10.1093/gji/ggu384 fatcat:wrgp5tll7raedeyo3utehxugui

Some physical applications of generalized Lambert functions

István Mező, Grant Keady
2016 European journal of physics  
In this paper we review the physical applications of the generalized Lambert function recently defined by the first author.  ...  We also point out that the inverse Langevin function is nothing else but a specially parametrized generalized Lambert function.  ...  A simple mathematical application connects W to the distribution of primes via the Prime Number Theorem [36] . 2 The generalized Lambert function Definition Now we recall the recent generalization of  ... 
doi:10.1088/0143-0807/37/6/065802 fatcat:k4qqzmg2sjgm7gvcrge2tbdi2y

PSEM Approximations for Both Branches of Lambert W Function with Applications

H. Vazquez-Leal, M. A. Sandoval-Hernandez, J. L. Garcia-Gervacio, A. L. Herrera-May, U. A. Filobello-Nino
2019 Discrete Dynamics in Nature and Society  
Transcendental functions are a fundamental building block of science and engineering. Among them, a relatively new function denominated as Lambert W is highlighted.  ...  The importance of such function relies on the fact that it can perform novel isolation of variables.  ...  He also acknowledges the financial support from the Secretary of Public Education of México (SEP) through Grant 2703 E476300.0275652.  ... 
doi:10.1155/2019/8267951 fatcat:go5komfidzb2to4qhfud4rukmi

Solutions for Series of Exponential Equations in Terms of Lambert-W Function and Fundamental Constants

S. Gnanarajan
2018 Journal of Applied Mathematics and Physics  
Series of exponential equations in the form of  ...  The analytical solution was derived in terms of Lambert-W function. A general numerical solution for any y is found in terms of n or in base y.  ...  In this paper, we considered series of exponential equations and solved them graphically, numerically, and analytically in terms of Lambert-W function.  ... 
doi:10.4236/jamp.2018.64065 fatcat:a7hjfwfdknegvni73pf4k5gbp4

Analysis and Control of Time Delayed Systems via the Lambert W Function

Sun Yi, Patrick W. Nelson, A. Galip Ulsoy
2008 IFAC Proceedings Volumes  
The solution has the form of an infinite series of modes written in terms of the matrix Lambert W function.  ...  W function.  ...  The solution to DDEs in terms of the Lambert W function shows a formal semblance to that of ODEs ], using monotonicity of the Lambert W function with resect to the branch of the Lambert W function.  ... 
doi:10.3182/20080706-5-kr-1001.02272 fatcat:6pj44tfcanaeziv25u3ygxfnbu

Some special cases in the stability analysis of multi-dimensional time-delay systems using the matrix Lambert W function

Rudy Cepeda-Gomez, Wim Michiels
2015 Automatica  
By studying a particular, yet common, second order system, we show that in general there is no one to one correspondence between the branches of the matrix Lambert W function and the characteristic roots  ...  This paper revisits a recently developed methodology based on the matrix Lambert W function for the stability analysis of linear time invariant, time delay systems.  ...  Acknowledgements This paper is the result of a research visit that R.  ... 
doi:10.1016/j.automatica.2015.01.016 fatcat:rap64fykjjebhiuvbl2gtimkra

Finding the Dominant Roots of a Time Delay System without Using the Principal Branch of the Lambert W Function [article]

Rudy Cepeda-Gomez
2017 arXiv   pre-print
This brief note complements some results regarding a recently developed technique for the stability analysis of linear time-invariant, time delay systems using the matrix Lambert W function.  ...  By means of a numeric example, it is shown that there are cases for which the dominant roots of the system can be found without using the principal branch of this multi-valued function, contradicting the  ...  The characteristic roots of the system are found analytically in terms of the Lambert W function.  ... 
arXiv:1708.03606v1 fatcat:pt2hb6qldfcgfexjudmmgtjx4m
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