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Bahman Kalantari - One of the best experts on this subject based on the ideXlab platform.
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Polynomial Root-Finding and Polynomiography
2020Co-Authors: Bahman KalantariAbstract:This book offers fascinating and modern perspectives into the theory and practice of the historical subject of polynomial Root-Finding, rejuvenating the field via polynomiography, a creative and novel computer visualization that renders spectacular images of a polynomial equation. Polynomiography will not only pave the way for new applications of polynomials in science and mathematics, but also in art and education. The book presents a thorough development of the basic family, arguably the most fundamental family of iteration functions, deriving many surprising and novel theoretical and practical applications such as: algorithms for approximation of roots of polynomials and analytic functions, polynomiography, bounds on zeros of polynomials, formulas for the approximation of Pi, and characterizations or visualizations associated with a homogeneous linear recurrence relation. These discoveries and a set of beautiful images that provide new visions, even of the well-known polynomials and recurrences, are the makeup of a very desirable book. This book is a must for mathematicians, scientists, advanced undergraduates and graduates, but is also for anyone with an appreciation for the connections between a fantastically creative art form and its ancient mathematical foundations. Contents: Approximation of Square-Roots and Their Visualizations; The Fundamental Theorem of Algebra and a Special Case of Taylor s Theorem; Introduction to the Basic Family and Polynomiography; Equivalent Formulations of the Basic Family; Basic Family as Dynamical System; Fixed Points of the Basic Family; Algebraic Derivation of the Basic Family and Characterizations; The Truncated Basic Family and the Case of Halley Family; Characterizations of Solutions of Homogeneous Linear Recurrence Relations; Generalization of Taylor s Theorem and Newton s Method; The Multipoint Basic Family and Its Order of Convergence; A Computational Study of the Multipoint Basic Family; A General Determinantal Lower Bound; Formulas for Approximation of Pi Based on Root-Finding Algorithms; Bounds on Roots of Polynomials and Analytic Functions; A Geometric Optimization and Its Algebraic Offsprings; Polynomiography: Algorithms for Visualization of Polynomial Equations; Visualization of Homogeneous Linear Recurrence Relations; Applications of Polynomiography in Art, Education, Science and Mathematics; Approximation of Square-Roots Revisited; Further Applications and Extensions of the Basic Family and Polynomiography.
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Combinatorics of polynomial Root-Finding algorithms
2020Co-Authors: Bahman KalantariAbstract:In this dissertation, we apply combinatorial methods to the ancient subject of polynomial Root-Finding. First, we introduce symmetric functions to the study of polynomial Root-Finding algorithms. We reveal a symmetric algebraic structure of a fundamental family of Root-Finding iteration functions known as the Basic Family, which gives rise to simple combinatorial proofs of many important properties of this family and two of its variants. The first variant maintains high order of convergence for any multiple root. The second variant, called the Truncated Basic Family, is an infinite family of m-th order methods for every m ≥ 3, using only the first m - 1 derivatives. Our algebraic combinatorial perspective not only sheds new insight into existing algorithms, but also gives rise to a powerful tool for constructing new high order methods. In particular, we develop an efficient algorithm and an explicit formula for another variant of the Basic Family for roots of a fixed multiplicity, and derive a family of iteration functions that converge faster to multiple roots than to simple roots. When applying our high order methods for finding multiple roots to rational functions and more generally, meromorphic functions, we observe that they also find the poles of these functions with the same high orders of convergence. While searching for an explanation of this phenomenon, we rediscovered supersymmetric functions which first emerged in representation theory of Lie superalgebras. Next, we study Kalantari's infinite family of lower and upper bounds on moduli of zeros of complex polynomials. We give a simple characterization of these bounds which leads to an efficient algorithm for their computation. For a polynomial of degree n our algorithm computes the first m bounds in Kalantari's family in O(mn) operations. We further prove that for every complex polynomial these lower and upper bounds converge to the tightest annulus containing the roots. Finally, we touch on the dynamics of Root-Finding iteration functions. We study the global convergence property of the Basic Family for extracting radicals. The first member of the Basic Family, i.e., Newton's iteration function, is well-known to be generally convergent for solving xn - c = 0. We extend this result to the next two members of the Basic Family. With the aid of polynomiography, techniques for the visualization of polynomial Root-Finding, we further conjecture the general convergence of all members of the Basic Family when extracting radicals. Using the computer algebra system Maple, we obtain some partial results toward the proof of our conjecture.
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Algorithms for quaternion polynomial Root-Finding
Journal of Complexity, 2013Co-Authors: Bahman KalantariAbstract:Abstract In 1941 Niven pioneered Root-Finding for a quaternion polynomial P ( x ) , proving the fundamental theorem of algebra (FTA) and proposing an algorithm, practical if the norm and trace of a solution are known. We present novel results on theory, algorithms and applications of quaternion Root-Finding. Firstly, we give a new proof of the FTA resulting in explicit formulas for both exact and approximate quaternion roots of P ( x ) in terms of exact and approximate complex roots of the real polynomial F ( x ) = P ( x ) P ¯ ( x ) , where P ¯ ( x ) is the conjugate polynomial. In particular, if | F ( c ) | ≤ ϵ , then for a computable quaternion conjugate q of c , | P ( q ) | ≤ ϵ . Consequences of these include relevance of Root-Finding methods for complex polynomials, computation of bounds on zeros, and algebraic solution of special quaternion equations. Secondly, working directly in the quaternion space, we develop Newton and Halley methods and analyze their local behavior. Surprisingly, even for a quadratic quaternion polynomial Newton’s method may not converge locally. Finally, we derive an analogue of the Bernoulli method in the quaternion space for computing the dominant root in certain cases. This requires the development of an independent theory for the solution of quaternion homogeneous linear recurrence relations. These results also lay a foundation for quaternion polynomiography.
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Voronoi Diagrams and Polynomial Root-Finding
2009 Sixth International Symposium on Voronoi Diagrams, 2009Co-Authors: Bahman KalantariAbstract:Voronoi diagram of points in the Euclidean plane and its computation is foundational to computational geometry. Polynomial Root-Finding is the origin of fundamental discoveries in all of mathematics and sciences. There is an intrinsic connection between polynomial Root-Finding in the complex plane and the approximation of Voronoi cells of its roots via a fundamental family of iteration functions, the basic family. For instance, the immediate basin of attraction of a root of a complex polynomial under Newton's method is a rough approximation to its Voronoi cell. We formally introduce these connections through the Basic Family of iteration functions, its properties with respect to Voronoi diagrams, and a corresponding visualization called polynomiography. Polynomiography is a medium for art, math, education and science. By making use of the Basic Family we introduce a layering of the points within each Voronoi cell of a polynomial root and study its properties and potential applications. In particular, we prove some novel results about the basic family in connection with Voronoi diagrams.
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ISVD - Voronoi Diagrams and Polynomial Root-Finding
2009 Sixth International Symposium on Voronoi Diagrams, 2009Co-Authors: Bahman KalantariAbstract:Voronoi diagram of points in the Euclidean plane and its computation is foundational to computational geometry. Polynomial Root-Finding is the origin of fundamental discoveries in all of mathematics and sciences. There is an intrinsic connection between polynomial Root-Finding in the complex plane and the approximation of Voronoi cells of its roots via a fundamental family of iteration functions, the Basic Family. For instance, the immediate basin of attraction of a root of a complex polynomial under Newton's method is a rough approximation to its Voronoi cell. We formally introduce these connections through the Basic Family of iteration functions, its properties with respect to Voronoi diagrams, and a corresponding visualization called polynomiography. Polynomiography is a medium for art, math, education and science. By making use of the Basic Family we introduce a layering of the points within each Voronoi cell of a polynomial root and study its properties and potential applications. In particular, we prove some novel results about the Basic Family in connection with Voronoi diagrams.
Vitaly Zaderman - One of the best experts on this subject based on the ideXlab platform.
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CASC - Root-Finding with Implicit Deflation
Computer Algebra in Scientific Computing, 2019Co-Authors: Rémi Imbach, Ilias S. Kotsireas, Vitaly ZadermanAbstract:Functional iterations such as Newton’s are a popular tool for polynomial Root-Finding. We consider realistic situation where some (e.g., better-conditioned) roots have already been approximated and where further computations is directed to the approximation of the remaining roots. Such a situation is also realistic for root by means of subdivision iterations. A natural approach of applying explicit deflation has been much studied and recently advanced by one of the authors of this paper, but presently we consider the alternative of implicit deflation combined with the mapping of the variable and reversion of an input polynomial. We also show another unexplored direction for substantial further progress in this long and extensively studied area. Namely we dramatically increase the local efficiency of Root-Finding by means of the incorporation of fast algorithms for multipoint polynomial evaluation and Fast Multipole Method.
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Root-Finding with Implicit Deflation
arXiv: Numerical Analysis, 2016Co-Authors: Ilias S. Kotsireas, Vitaly ZadermanAbstract:We were initially motivated by the paper [SS17] by Schleicher and Stoll about initialization of Newton's iterations for univariate polynomial Root-Finding. Given a black box subroutine for the evaluation of the Newton's ratio of a polynomial and its derivative, their algorithm very fast approximates all roots of a univariate polynomial except for a small fraction of them. The challenge of fast approximation of the remaining roots motivated our present work, but our recipes for this task should have much broader independent interest for implicit deflation in Root-Finding for univariate or multivariate polynomials, rational, and analytic functions. These recipes can be also an example of synergy of the combination of various methods for Root-Finding towards enhancing their power, in particular towards faster convergence of functional iterations in a larger domain.
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Implicit Deflation for Univariate Polynomial Root-Finding
2016Co-Authors: Ilias S. Kotsireas, Vitaly ZadermanAbstract:We were initially motivated by the paper by Schleicher and Stoll of 2017 about the initialization of Newton's iterations. Given a black box subroutine for the evaluation of the Newton's ratio of a polynomial and its derivative, their algorithm very fast approximates all roots of a univariate polynomial except for a small fraction of them. The challenge of fast approximation of the remaining roots motivated our present work, but our recipes for this task should have independent and much broader interest for implicit deflation in polynomial Root-Finding. They can be also an example of synergy of the combination of various methods of polynomial Root-Finding towards enhancing their power, in particular towards faster convergence of functional iterations in a larger domain.
Tekle Gemechu - One of the best experts on this subject based on the ideXlab platform.
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ROOT FINDING FOR NONLINEAR EQUATIONS
Mathematical theory and modeling, 2020Co-Authors: Tekle GemechuAbstract:Nonlinear equations /systems appear in most science and engineering models. For example, when solving eigen value problems, optimization problems, differential equations, in circuit analysis, analysis of state equations for a real gas, in mechanical motions /oscillations, weather forecasting, integral equations, image processing and many other fields of engineering designing processes. Nonlinear systems /problems are difficult to solve manually but they occur naturally in fluid motions, heat transfer, wave motions, chemical reactions, etc. This study deals with construction of iterative methods for nonlinear root finding, applying Taylor’s series approximation of a nonlinear function f(x) combined with a new correction term in a quadratic or cubic model. Competent iterative algorithms of higher order were investigated. For test of convergence and efficiency, we applied basic theorems and solved some equations in C++. Keywords – nonlinear equations, Taylor’s approximation, iterative algorithms for roots, error correction
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Some Root Finding With Extensions to Higher Dimensions
Mathematical theory and modeling, 2020Co-Authors: Tekle GemechuAbstract:Root finding is an issue in scientific computing. Because most nonlinear problems in science and engineering can be considered as the root finding problems, directly or indirectly. The research in numerical modeling for root finding is still going on. In this study, fixed point iterative methods for solving simple real roots of nonlinear equations, which improve convergence of some existing methods, are thorough. Derivative estimations up to the third order (in root finding, some recent ideas) are applied in Taylor’s approximation of a nonlinear equation by a cubic model to achieve efficient iterative methods. We may also discuss possible extensions to two dimensions and consider Newton’s method and Halley’s method in 1D and 2D problem solving. Several examples for test of efficiency and convergence analyses using C++ are offered. And some engineering applications of root finding are conferred. Graphical demonstrations are supported with matlab basic tools. Keywords: engineering applications, derivative estimations, iterative methods, simple roots, Taylor’s approximation.
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Some Multiple and Simple Real Root Finding Methods
Mathematical theory and modeling, 2020Co-Authors: Tekle GemechuAbstract:Solving nonlinear equations with root finding is very common in science and engineering models. In particular, one applies it in mathematics, physics, electrical engineering and mechanical engineering. It is a researchable area in numerical analysis. This present work focuses on some iterative methods of higher order for multiple roots. New and existing novel multiple and simple root finding techniques are discussed. Methods independent of a multiplicity m of a root r, which function very well for both simple and multiple roots, are also presented. Error-correction and variatonal technique with some function estimations are used for the constructions. For the analysis of orders of convergence, some basic theorems are applied. Ample test examples are provided (in C++) for test of efficiencies with suitable initial guesses. And convergence of some methods to a root is shown graphically using matlab applications. Keywords :Iterative algorithms, error-correction, variational methods, multiple roots, applications
Ilias S. Kotsireas - One of the best experts on this subject based on the ideXlab platform.
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CASC - Root-Finding with Implicit Deflation
Computer Algebra in Scientific Computing, 2019Co-Authors: Rémi Imbach, Ilias S. Kotsireas, Vitaly ZadermanAbstract:Functional iterations such as Newton’s are a popular tool for polynomial Root-Finding. We consider realistic situation where some (e.g., better-conditioned) roots have already been approximated and where further computations is directed to the approximation of the remaining roots. Such a situation is also realistic for root by means of subdivision iterations. A natural approach of applying explicit deflation has been much studied and recently advanced by one of the authors of this paper, but presently we consider the alternative of implicit deflation combined with the mapping of the variable and reversion of an input polynomial. We also show another unexplored direction for substantial further progress in this long and extensively studied area. Namely we dramatically increase the local efficiency of Root-Finding by means of the incorporation of fast algorithms for multipoint polynomial evaluation and Fast Multipole Method.
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Root-Finding with Implicit Deflation
arXiv: Numerical Analysis, 2016Co-Authors: Ilias S. Kotsireas, Vitaly ZadermanAbstract:We were initially motivated by the paper [SS17] by Schleicher and Stoll about initialization of Newton's iterations for univariate polynomial Root-Finding. Given a black box subroutine for the evaluation of the Newton's ratio of a polynomial and its derivative, their algorithm very fast approximates all roots of a univariate polynomial except for a small fraction of them. The challenge of fast approximation of the remaining roots motivated our present work, but our recipes for this task should have much broader independent interest for implicit deflation in Root-Finding for univariate or multivariate polynomials, rational, and analytic functions. These recipes can be also an example of synergy of the combination of various methods for Root-Finding towards enhancing their power, in particular towards faster convergence of functional iterations in a larger domain.
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Implicit Deflation for Univariate Polynomial Root-Finding
2016Co-Authors: Ilias S. Kotsireas, Vitaly ZadermanAbstract:We were initially motivated by the paper by Schleicher and Stoll of 2017 about the initialization of Newton's iterations. Given a black box subroutine for the evaluation of the Newton's ratio of a polynomial and its derivative, their algorithm very fast approximates all roots of a univariate polynomial except for a small fraction of them. The challenge of fast approximation of the remaining roots motivated our present work, but our recipes for this task should have independent and much broader interest for implicit deflation in polynomial Root-Finding. They can be also an example of synergy of the combination of various methods of polynomial Root-Finding towards enhancing their power, in particular towards faster convergence of functional iterations in a larger domain.
Dian Darina Indah Daruis - One of the best experts on this subject based on the ideXlab platform.
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The Used of Pedal-Pad Effective Amplitude Transmissibility Value to Predict Size of Comfortable Pedal-Pads
Applied Mechanics and Materials, 2013Co-Authors: A. R. Yusoff, Deros, Dian Darina Indah DaruisAbstract:Vibration transmissibility from a car body to pedal-pad could result to drivers discomfort and fatigue, which may lead to health problems. Vibration transmissibility occurs when the vehicle engine is turned on and vibration is transmitted from the car body to pedal-pad in this research Pedal-pad effective Amplitude Transmissibility (PEAT) value was used to predict suitable size of the pedal-pad that could provide comfortable operation of the pedal. In this study, the variables are the three different sizes of pedal-pads and data was recorded while the car was moving on the road at constant speed with three different sizes of pedal-pads. The data was measured in root mean square (r.m.s) unit, of the frequency weighted acceleration (m/s2) for every minute. The finding of the study shows that percentage PEATr.m.s of the three different sizes of pedal-pad is greater than 100%. It shows that vibrations to pedal-pads are greater than vibration from the car body consequentially it means that the foot on the pedal is exposed to higher vibration transmissibility than the foot on the floor.
