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A root-finding algorithm for list decoding of Reed-Muller codes

IEEE TRANSACTIONS ON INFORMATION THEORY 2005年第3期51卷 1190-1196页

作者： Wu, XW Kuijper, M Udaya, P Univ Melbourne Dept Elect & Elect Engn Melbourne Vic 3010 Australia Univ Melbourne Dept Comp Sci & Software Engn Melbourne Vic 3010 Australia

Let F-q [X-1,..., X-m] denote the set of polynomials over F-q in m variables, and F-q[X-1,..., X-m]less than or equal to(u) denote the subset that consists of the polynomials of total degree at most u. Let H(T) be a nontrivial polynomial in T with coefficients in F-q[X-1,...,X-m]. A crucial step in interpolation-based list decoding of q-ary Reed-Muller (RM) codes is finding the roots of H(T) in F-q[X-1,...,X-m]less than or equal to(u). In this correspondence, we present an efficient root-finding algorithm, which finds all the roots of H(T) in F-q[X-1,..., X-m]less than or equal to(u) The algorithm can be used to speed up the list decoding of RM codes.

关键词： q-ary Reed-Muller (RM) codes list decoding Reed-Solomon (RS) codes root-finding algorithm

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Absorbing boundary condition for scalar-wave propagation problems in infinite media based on a root-finding algorithm

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2018年第Mar.1期330卷 207-219页

作者： Lee, Jin Ho Tassoulas, John L. Pukyong Natl Univ Dept Ocean Engn 45 Yongso Ro Busan 48513 South Korea Univ Texas Dept Civil Architectural & Environm Engn Austin TX 78712 USA

In this study, we propose a new approach for development of absorbing boundary conditions for scalar-wave propagation problems in infinite media based on a root-finding algorithm for the solution of the exact wave dispersion relation. We select the Newton-Raphson method as the root-finding algorithm in the present study and assess the accuracy of the newly developed boundary condition by estimating its reflection coefficient. Furthermore, we evaluate and verify the stability of the boundary condition. We apply our development to various scalar-wave propagation problems and demonstrate that the proposed approach leads to accurate and stable computations (C) 2017 Elsevier B.V. All rights reserved.

关键词： root-finding absorbing boundary condition (RFABC) Absorbing boundary condition Wave propagation Scalar wave Infinite medium root-finding algorithm

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Iterative root-finding algorithm for Accurate Parameter Extraction of Solar Photovoltaic Cells

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GAZI UNIVERSITY JOURNAL OF SCIENCE 2024年第4期37卷 1770-1789页

作者： Douiri, Moulay Rachid Cadi Ayyad Univ Fac Sci & Technol Dept Appl Phys Marrakech 40000 Morocco

The performance of photovoltaic models depends significantly on the accuracy of their parameters, which are determined by the chosen method and objective function. Extracting these parameters accurately under different environmental conditions is essential to enhance reliability, accuracy, and minimize system costs. In this research, a novel technique is proposed for extracting the electrical parameters of the solar cell single diode model, including saturation current, serial resistance, parallel resistance, and ideality factor. To overcome the challenges posed by the chaotic behavior of the I-V curve equation, an improved Iterative root-finding algorithm is introduced. This algorithm acts as an optimization tool, increasing the likelihood of obtaining highly accurate solutions by minimizing the quadratic error between experimental and theoretical characteristics in a shorter time frame. The numerical and experimental results demonstrate the effectiveness of this approach in solar module modeling, showing squared errors approaching zero. This study opens new possibilities for improving the accuracy and reliability of photovoltaic models, leading to more efficient solar energy systems.

关键词： Parameters extraction root-finding algorithm Single diode model Solar cells

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Efficient root-finding algorithm with application to list decoding of algebraic-geometric codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2001年第6期47卷 2579-2587页

作者： Wu, XW Siegel, PH Univ Calif San Diego Dept Elect & Comp Engn La Jolla CA 92093 USA Univ Calif San Diego Ctr Magnet Recording Res La Jolla CA 92093 USA

A list decoding for an error-correcting code is a decoding algorithm that generates a list of codewords within a Hamming distance t from the received vector, where t can be greater than the error-correction bound. In [18], a list-decoding procedure for Reed-Solomon codes [19] was generalized to algebraic-geometric codes. A recent work [8] gives improved list decodings for Reed-Solomon codes and algebraic-geometric codes that work for all rates and have many applications. However, these list-decoding algorithms are rather complicated. In [17], Roth and Ruckenstein proposed an efficient implementation of the list decoding of Reed-Solomon codes. In this correspondence, extending Roth and Ruckenstein's fast algorithm for finding roots of univariate polynomials over polynomial rings, i.e., the Reconstruct algorithm, we will present an efficient algorithm for finding the roots of univariate polynomials over function fields. Based on the extended algorithm, we give an efficient list-decoding algorithm for algebraic-geometric codes.

关键词： algebraic-geometric codes list decoding root-finding algorithm

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Variance reduction for root-finding problems

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MATHEMATICAL PROGRAMMING 2023年第1期197卷 375-410页

作者： Davis, Damek Cornell Univ Sch Operat Res & Informat Engn Ithaca NY 16850 USA

Minimizing finite sums of smooth and strongly convex functions is an important task in machine learning. Recent work has developed stochastic gradient methods that optimize these sums with less computation than methods that do not exploit the finite sum structure. This speedup results from using efficiently constructed stochastic gradient estimators, which have variance that diminishes as the algorithm progresses. In this work, we ask whether the benefits of variance reduction extend to fixed point and root-finding problems involving sums of nonlinear operators. Our main result shows that variance reduction offers a similar speedup when applied to a broad class of root-finding problems. We illustrate the result on three tasks involving sums of n nonlinear operators: averaged fixed point, monotone inclusions, and nonsmooth common minimizer problems. In certain "poorly conditioned regimes," the proposed method offers an n-fold speedup over standard methods.

关键词： Stochastic algorithm Variance reduction root-finding algorithm Operator splitting Monotone inclusions Saddle-point problems

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On the simultaneous refinement of the zeros of H-palindromic polynomials

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2014年 272卷 293-303页

作者： Brugiapaglia, Simone Gemignani, Luca Univ Pisa Dipartimento Informat I-56127 Pisa Italy Politecn Milan Dipartimento Matemat F Brioschi MOX I-20133 Milan Italy

关键词： Generalized eigenvalue problem root-finding algorithm Palindromic polynomials

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The Pade iterations for the matrix sign function and their reciprocals are optimal

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LINEAR ALGEBRA AND ITS APPLICATIONS 2012年第3期436卷 472-477页

作者： Greco, Federico Iannazzo, Bruno Poloni, Federico Univ Perugia Dipartimento Matemat & Informat I-06123 Perugia Italy Scuola Normale Super Pisa I-56126 Pisa Italy

It is proved that among the rational iterations locally converging with order s > 1 to the sign function, the Fade iterations and their reciprocals are the unique with the lowest sum of the degrees of numerator and... 详细信息

关键词： Rational iterations Matrix functions Matrix sign function Local convergence Fade approximation root-finding algorithm

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Newton-like methods and polynomiographic visualization of modified Thakur processes

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2021年第5期98卷 1049-1068页

作者： Usurelu, Gabriela Ioana Bejenaru, Andreea Postolache, Mihai Univ Politehn Bucuresti Dept Math & Informat Bucharest 060042 Romania China Med Univ Ctr Gen Educ Taichung 40402 Taiwan Romanian Acad Gh Mihoc C Iacob Inst Math Stat & Appl Math Bucharest 050711 Romania

The content of this paper is twofold. First, it aims to provide some new Newton-like methods for solving the root-finding problem in the complex plane. Moreover a convergence test for the resulted methods is phrased and proved. The pseudo-Newton method of Kalantari for finding the maximum modulus of complex polynomials arises as particular case of the newly proposed procedures. Secondly, a recently introduced Thakur iterative process is used in connection with the newly described methods. Its stability and data dependence is subject to analysis. Ultimately, an illustrative analysis regarding some modified Thakur iteration procedures, is obtained via polynomiographic techniques.

关键词： Iterative process root-finding algorithm polynomiography Newton's method simulation

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AN ITERATION METHOD WITH GENERALLY CONVERGENT PROPERTY FOR CUBIC POLYNOMIALS

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INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS 2009年第1期19卷 395-401页

作者： Tang, Peipei Wang, Xinghua Zhejiang Univ Dept Math Hangzhou 310027 Peoples R China

In this paper, we first establish a rational iteration method which can be used as a root-finding algorithm for almost every polynomial. It has no nonrepelling extraneous fixed point in the complex plane and is generally convergent for both quadratic and cubic polynomials. Then some properties of this algorithm are given. By the aid of computer, we produce pictures of the Julia sets for the iterations of some polynomials. Numerical results show that it is a root-finding method with convergence order the same as Halley's method.

关键词： root-finding algorithm rational iteration generally convergent fixed point Julia set

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Self-triggered stabilizing controllers for linear continuous-time systems

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INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL 2020年第16期30卷 6502-6517页

作者： Zobiri, Fairouz Meslem, Nacim Bidegaray-Fesquet, Brigitte Katholieke Univ Leuven 10 Kasteelpk Arenberg B-3001 Heverlee Belgium Energyville Genk Belgium Univ Grenoble Alpes Grenoble INP CNRS GIPSA Lab Grenoble France Univ Grenoble Alpes Grenoble INP CNRS LJK Grenoble France

Self-triggered control is an improvement on event-triggered control methods. Unlike the latter, self-triggered control does not require monitoring the behavior of the system constantly. Instead, self-triggered algorithms predict the events at which the control law has to be updated before they happen, relying on system model and past information. In this work, we present a self-triggered version of an event-triggered control method in which events are generated when a pseudo-Lyapunov function (PLF) associated with the system increases up to a certain limit. This approach has been shown to considerably decrease the communications between the controller and the plant, while maintaining system stability. To predict the intersections between the PLF and the upper limit, we use a simple and fast root-finding algorithm. The algorithm mixes the global convergence properties of the bisection and the fast convergence properties of the Newton-Raphson method. Moreover, to ensure the convergence of the method, the initial iterate of the algorithm is found through a minimization algorithm.

关键词： Lyapunov function root-finding algorithm self-triggered control

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