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

2002
*
IEEE Transactions on Signal Processing
*

We review

doi:10.1109/tsp.2002.801912
fatcat:lsadwmiw65gr3puopq5hstnckm
*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*. ...##
###
Comments on "numerical evaluation of the Lambert W function and application to generation of generalized Gaussian noise with exponent 1/2"

2004
*
IEEE Transactions on Signal Processing
*

ACKNOWLEDGMENT

doi:10.1109/tsp.2004.826154
fatcat:zr3lw224c5gu5kwvdyxdpqfzqy
*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. ...##
###
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

doi:10.35940/ijitee.k1772.0981119
fatcat:hsys4wltojfhppwt5dmvorqkkm
*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. ...##
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Approximate Solutions of Linearized Delay Differential Equations Arising from a Microbial Fermentation Process Using the Matrix Lambert Function

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. ...

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

2006
*
2006 American Control Conference
*

To

doi:10.1109/acc.2006.1656585
fatcat:az4nc2opuzejlfgbkm2lqhwjmy
*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*. ...##
###
Simulation techniques for generalized Gaussian densities

2009
*
Journal of Statistical Computation and Simulation
*

This contribution deals with Monte Carlo simulation

doi:10.1080/00949650802290912
fatcat:7mthgkyngrey5mg6kjnlwdmduu
*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*...##
###
Implementation of the Green-Ampt Infiltration Model: Comparative between different numerical solutions

2021
*
Trends in Computational and Applied Mathematics
*

Thus, this article aims to implement

doi:10.5540/tcam.2021.022.04.00645
fatcat:7yxbecqtynabna7hhgx4kbwjfy
*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*. ...##
###
On the simulation of the Hawkes process via Lambert-W functions
[article]

2019
*
arXiv
*
pre-print

This manuscript shows that

arXiv:1907.09162v1
fatcat:hhavmmxmjjf3thzg7gfhsbsk2u
*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*). ...##
###
The Ricker wavelet and the Lambert W function

2014
*
Geophysical Journal International
*

Since

doi:10.1093/gji/ggu384
fatcat:wrgp5tll7raedeyo3utehxugui
*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. ...##
###
Some physical applications of generalized Lambert functions

2016
*
European journal of physics
*

In this paper we review

doi:10.1088/0143-0807/37/6/065802
fatcat:k4qqzmg2sjgm7gvcrge2tbdi2y
*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*...##
###
PSEM Approximations for Both Branches of Lambert W Function with Applications

2019
*
Discrete Dynamics in Nature and Society
*

Transcendental

doi:10.1155/2019/8267951
fatcat:go5komfidzb2to4qhfud4rukmi
*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. ...##
###
Solutions for Series of Exponential Equations in Terms of Lambert-W Function and Fundamental Constants

2018
*
Journal of Applied Mathematics and Physics
*

Series

doi:10.4236/jamp.2018.64065
fatcat:a7hjfwfdknegvni73pf4k5gbp4
*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*. ...##
###
Analysis and Control of Time Delayed Systems via the Lambert W Function

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*. ...

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

2015
*
Automatica
*

By studying a particular, yet common, second order system, we show that in

doi:10.1016/j.automatica.2015.01.016
fatcat:rap64fykjjebhiuvbl2gtimkra
*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. ...##
###
Finding the Dominant Roots of a Time Delay System without Using the Principal Branch of the Lambert W Function
[article]

2017
*
arXiv
*
pre-print

This brief note complements some results regarding a recently developed technique for

arXiv:1708.03606v1
fatcat:pt2hb6qldfcgfexjudmmgtjx4m
*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*. ...
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