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

Anand Louis, Santosh S. Vempala
2016 arXiv   pre-print
We study the problem 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  ... 
arXiv:1511.03186v2 fatcat:6o23i55oqraf7b2a5xfazajtgi

A Derivative-less Approach for Generating Phase Envelopes

Siddiqui F
2015 Oil & Gas Research  
An algorithm 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  ... 
doi:10.4172/2472-0518.1000106 fatcat:binbumsdxnderi3kpsvdbvlbte

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  ... 

GMRES-Accelerated ADMM for Quadratic Objectives [article]

Richard Y. Zhang, Jacob K. White
2018 arXiv   pre-print
We give theoretical justification and numerical evidence that the GMRES-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.  ... 
arXiv:1601.06200v5 fatcat:llf732peqjgjlcbadoovxu3uou

Accelerated Approximation of the Complex Roots and Factors of a Univariate Polynomial [article]

Victor Y. Pan, Elias P. Tsigaridas, Vitaly Zaderman, Liang Zhao
2016 arXiv   pre-print
Moreover the algorithm can be applied to a 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.  ... 
arXiv:1501.05392v3 fatcat:m7kk5aa3yrc6pmjcsv3d6qqd5i

New Progress in Polynomial Root-finding [article]

Victor Y. Pan
2022 arXiv   pre-print
We propose and extensively analyze various novel techniques, which enable significant 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.  ... 
arXiv:1805.12042v27 fatcat:mnybwi34i5gqhhpiypguy5qu5i

New practical advances in polynomial root clustering [article]

Rémi Imbach, Victor Y. Pan
2019 arXiv   pre-print
As in their previous best subdivision algorithms our 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.  ... 
arXiv:1911.06706v1 fatcat:yx5lv7nytfcjrmypn747tn5yru

On the Koopman operator of algorithms [article]

Felix Dietrich, Thomas N. Thiem, Ioannis G. Kevrekidis
2019 arXiv   pre-print
In this paper, we use the Koopman operator framework in the data-driven study 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  ... 
arXiv:1907.10807v2 fatcat:dxtpyagjk5c2bcaijxljnczlwi

Modified Newton-Raphson Method to Tune Feedback Gains of Control System for Standing by Functional Neuromuscular Stimulation Following Spinal Cord Injury

Raviraj Nataraj, Musa L. Audu, Ronald J. Triolo
2014 Applied Bionics and Biomechanics  
feedback 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.  ... 
doi:10.1155/2014/634509 pmid:25684981 pmcid:PMC4326073 fatcat:cx5fx6hjdnfxjbvha4odjdgm5e

Solving secular and polynomial equations: A multiprecision algorithm

Dario A. Bini, Leonardo Robol
2014 Journal of Computational and Applied Mathematics  
Fortune, An 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.  ... 
doi:10.1016/j.cam.2013.04.037 fatcat:7gq257ucxfhdxfzrhlsx3vnz5m

Four ways to compute the inverse of the complete elliptic integral of the first kind

John P. Boyd
2015 Computer Physics Communications  
A fourth is to provide robust "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.  ... 
doi:10.1016/j.cpc.2015.05.006 fatcat:wfymdn76zzf3nm7wq255yrepne

A Geometric Algorithm for Ray/Bézier Surfaces Intersection Using Quasi-Interpolating Control Net

Yohan D. Fougerolle, Sandrine Lanquetin, Marc Neveu, Thierry Lauthelier
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.  ... 
doi:10.1109/sitis.2008.24 dblp:conf/sitis/FougerolleLNL08 fatcat:6saab56pk5fjldf2vzwrkepgwy

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

Janak Raj Sharma, Deepak Kumar, Carlo Cattani
2019 Symmetry  
In this work, we construct a family 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.  ... 
doi:10.3390/sym11081054 fatcat:xtdvi3ydp5gnlmxcnh3rbaxfy4

Recent Advances in Structural Optimization

Yurii Nesterov
2011 Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)  
In this field, we use additional information on the structure 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.  ... 
doi:10.1142/9789814324359_0174 fatcat:ffznkvzerjbnjexfiqa4mgusgm

Root Radii and Subdivision for Polynomial Root-Finding [article]

Rémi Imbach, Victor Y. Pan
2021 arXiv   pre-print
We depart from our approximation 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] .  ... 
arXiv:2102.10821v2 fatcat:z522mpedgnaa7hbnq7hd7m5owm
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