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Accelerated Newton Iteration: Roots of Black Box Polynomials and Matrix Eigenvalues
[article]

2016
*
arXiv
*
pre-print

We study the problem

arXiv:1511.03186v2
fatcat:6o23i55oqraf7b2a5xfazajtgi
*of*computing the largest*root**of*a real*rooted**polynomial*p(x) to within error ε given only*black**box*access to it, i.e.,*for*any x ∈ R, the algorithm can query an oracle*for*the value ... A folklore result*for*this problem is that the largest*root**of*a*polynomial*can be computed in O(n (1/ε ))*polynomial*queries using the*Newton**iteration*. ... We are grateful to Ryan O' Donnell*for*helpful discussions, and to Yin Tat Lee, Prasad Raghavendra, Aaron Schild and Aaron Sidford*for*pointing us to the finite difference method*for*approximating higher ...##
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A Derivative-less Approach for Generating Phase Envelopes

2015
*
Oil & Gas Research
*

An algorithm

doi:10.4172/2472-0518.1000106
fatcat:binbumsdxnderi3kpsvdbvlbte
*for*generating a complete phase envelope using computer modeling was proposed by Michelsen. ... These modifications allowed running the model successfully*for*different hydrocarbon systems without any convergence issues. ... This algorithm can also provide initializing K-values to the*black*-*box*EOS to aid in convergence*of*flash calculation and also an overall*acceleration**of*computation because extrapolated K-values will ...##
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Page 3926 of Mathematical Reviews Vol. , Issue 92g
[page]

1992
*
Mathematical Reviews
*

Applications

*for*these problems include numerical continuation methods, where a “*black**box*” solver is available*for*A. ... Summary: “We show that the relaxed*Newton*method*for*finding the*roots**of*a cubic with one double*root*is conjugate, by a linear fractional transformation on the Riemann sphere, to the*iterations**of*the ...##
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GMRES-Accelerated ADMM for Quadratic Objectives
[article]

2018
*
arXiv
*
pre-print

We give theoretical justification and numerical evidence that the GMRES-

arXiv:1601.06200v5
fatcat:llf732peqjgjlcbadoovxu3uou
*accelerated*variant consistently solves the same problem in O(κ^1/4)*iterations**for*an order-*of*-magnitude reduction in*iterations*... frequently arise as the*Newton*subproblem*of*interior-point methods. ... A large part*of*the paper was written during R.Y. Zhang's visit to UC Berkeley as postdoctoral scholar, and he would like to thank his faculty mentor Javad Lavaei*for*his warm accommodation. ...##
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Accelerated Approximation of the Complex Roots and Factors of a Univariate Polynomial
[article]

2016
*
arXiv
*
pre-print

Moreover the algorithm can be applied to a

arXiv:1501.05392v3
fatcat:m7kk5aa3yrc6pmjcsv3d6qqd5i
*polynomial*given by a*black**box**for*its evaluation (even if its coefficients are not known); it promises to be*of*practical value*for**polynomial**root*-finding ... Furthermore our algorithm is nearly optimal*for*the approximation*of*the*roots*isolated in a fixed disc, square or another region on the complex plane rather than all complex*roots**of*a*polynomial*. ... given by a*black**box*subroutine*for*its evaluation. ...##
###
New Progress in Polynomial Root-finding
[article]

2022
*
arXiv
*
pre-print

We propose and extensively analyze various novel techniques, which enable significant

arXiv:1805.12042v27
fatcat:mnybwi34i5gqhhpiypguy5qu5i
*acceleration**of*the known algorithms, particularly where a*polynomial*is given by a subroutine*for*its evaluation rather ... Our auxiliary algorithms and estimates, e.g.,*for*the approximation*of**root*radii, that is, the distances from a complex point to the*roots*, can be*of*independent interest. ...*for*their pointers to [4, 55, 118, 14, 124, 120] , comments on their research, and on the challenge*of*handling wild*polynomial**roots*that appear in Ehrlich's and Newton's*iterations*. ...##
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New practical advances in polynomial root clustering
[article]

2019
*
arXiv
*
pre-print

As in their previous best subdivision algorithms our

arXiv:1911.06706v1
fatcat:yx5lv7nytfcjrmypn747tn5yru
*root*-finders are robust even*for*multiple*roots**of*a*polynomial*given by a*black**box**for*the approximation*of*its coefficients, and their complexity ... We report an ongoing work on clustering algorithms*for*complex*roots**of*a univariate*polynomial*p*of*degree d with real or complex coefficients. ... Erhlich-Aberth (simultaneous*Newton*-like)*iterations*. ...##
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On the Koopman operator of algorithms
[article]

2019
*
arXiv
*
pre-print

In this paper, we use the Koopman operator framework in the data-driven study

arXiv:1907.10807v2
fatcat:dxtpyagjk5c2bcaijxljnczlwi
*of*such algorithms and discuss benefits*for*analysis and*acceleration**of*numerical computation. ... A systematic mathematical framework*for*the study*of*numerical algorithms would allow comparisons, facilitate conjugacy arguments, as well as enable the discovery*of*improved,*accelerated*, data-driven ... At the end*of*section 4, we analyze the*Newton*-Raphson method*for**root*finding*of**polynomials*on the complex plane in section 4.4, with explicit construction*of*the spectrum and eigenfunctions, as well ...##
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Modified Newton-Raphson Method to Tune Feedback Gains of Control System for Standing by Functional Neuromuscular Stimulation Following Spinal Cord Injury

2014
*
Applied Bionics and Biomechanics
*

feedback

doi:10.1155/2014/634509
pmid:25684981
pmcid:PMC4326073
fatcat:cx5fx6hjdnfxjbvha4odjdgm5e
*of*total body center*of*mass*acceleration*to modulate stimulation levels to targeted paralyzed musculature*of*the lower extremities and trunk.Methods: Gains*for*this control system were tuned ... An algorithm based on a modified form*of*the*Newton*-Raphson method was employed to find the optimal feedback gains with lower subject effort than that to determine the original tuning curves.Results: This ... The authors would further like to thank the study participant and the people and facilities*of*the Cleveland APT and FES Centers*of*Excellence. ...##
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Solving secular and polynomial equations: A multiprecision algorithm

2014
*
Journal of Computational and Applied Mathematics
*

Fortune, An

doi:10.1016/j.cam.2013.04.037
fatcat:7gq257ucxfhdxfzrhlsx3vnz5m
*iterated*eigenvalue algorithm*for*approximating the*roots**of*univariate*polynomials*, J. Symbolic Comput. 33 (5) (2002) 627-646]. ...*For*certain*polynomials*, like the Mandelbrot or the partition*polynomials*the*acceleration*is dramatic. The algorithm exploits the parallel architecture*of*the computing platform. ... Acknowledgments The authors wish to thank the anonymous referees*for*providing useful suggestions that helped to improve the presentation*of*the paper. ...##
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Four ways to compute the inverse of the complete elliptic integral of the first kind

2015
*
Computer Physics Communications
*

A fourth is to provide robust "

doi:10.1016/j.cpc.2015.05.006
fatcat:wfymdn76zzf3nm7wq255yrepne
*black**box*" software*for*computing this function. ... The Matlab/*Newton*code is recommended*for*numerical purposes. ... Only four*Newton**iterations*are employed because further*iterations*produced no improvement. ...##
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A Geometric Algorithm for Ray/Bézier Surfaces Intersection Using Quasi-Interpolating Control Net

2008
*
2008 IEEE International Conference on Signal Image Technology and Internet Based Systems
*

*for*very simple projections and 2D intersection tests to determine sub-patches containing a potential intersection. ... The novelty

*of*our approach resides in the use

*of*bounds

*of*the difference between a Bézier patch and its quasi-interpolating control net. ... Nishita et al. determine the

*roots*

*of*these two equations using

*Newton*

*iterations*and by

*iteratively*clipping away the parts

*of*the patches that do not to intersect the ray. ...

##
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An Efficient Class of Weighted-Newton Multiple Root Solvers with Seventh Order Convergence

2019
*
Symmetry
*

In this work, we construct a family

doi:10.3390/sym11081054
fatcat:xtdvi3ydp5gnlmxcnh3rbaxfy4
*of*seventh order*iterative*methods*for*finding multiple*roots**of*a nonlinear function. ... The scheme consists*of*three steps,*of*which the first is Newton's step and last two are the weighted-*Newton*steps. Hence, the name*of*the scheme is 'weighted-*Newton*methods'. ... In particular, Geum et al. in [22, 23] have proposed two-and three-point*Newton*-like methods with convergence order six*for*finding multiple*roots*. ...##
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Recent Advances in Structural Optimization

2011
*
Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)
*

In this field, we use additional information on the structure

doi:10.1142/9789814324359_0174
fatcat:ffznkvzerjbnjexfiqa4mgusgm
*of*specific problem instances*for**accelerating*standard*Black*-*Box*methods. ... We show that the proper use*of*problem structure can provably*accelerate*these methods by the order*of*magnitudes. ... Is it possible to use this structure*for**accelerating*the*Black*-*Box*schemes? Intuitively we always hope that this is true. ...##
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Root Radii and Subdivision for Polynomial Root-Finding
[article]

2021
*
arXiv
*
pre-print

We depart from our approximation

arXiv:2102.10821v2
fatcat:z522mpedgnaa7hbnq7hd7m5owm
*of*2000*of*all*root*radii*of*a*polynomial*, which has readily extended Schönhage's efficient algorithm*of*1982*for*a single*root*radius. ... This saving relies on our novel recipes*for*the initialization*of**root*-finding*iterations**of*independent interest. ... In the verification step*of**Newton**iterations*, use the T * -test*of*[2] . ...
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