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Polynomial Decomposition Algorithms
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Book Synopsis Polynomial Decomposition Algorithms by : Dexter Kozen
Download or read book Polynomial Decomposition Algorithms written by Dexter Kozen and published by . This book was released on 1986 with total page 14 pages. Available in PDF, EPUB and Kindle. Book excerpt: In a recent paper [BZ], Barton and Zippel examine the question of when a polynomial $f(x)$ over a field of characteristic 0 has a nontrivial decomposition $f(x)=g(h(x))$. They give two exponential-time algorithms, both of which require polynomial factorization. We present an $O(s[superscript]{2}r\logr)$ algorithm, where $r$=deg $g$ and $s$ =deg $h$. The algorithm does not use polynomial factorization. We also show that the problem is in $NC$. In addition, we give a new structure theorem for testing decomposibility over any field. We apply this theorem to obtain an $NC$ algorithm for decomposing irreducible polynomials over finite fields and a subexponential algorithm for decomposing irreducible polynomials over any field.
Book Synopsis Polynomial Decomposition Algorithms for Multivariate Polynomials by : Cornell University. Dept. of Computer Science
Download or read book Polynomial Decomposition Algorithms for Multivariate Polynomials written by Cornell University. Dept. of Computer Science and published by . This book was released on 1987 with total page 10 pages. Available in PDF, EPUB and Kindle. Book excerpt: ABSTRACT NOT SUPPLIED
Book Synopsis Polynomial Decomposition Algorithms in Signal Processing by : Guolong Su (S.M.)
Download or read book Polynomial Decomposition Algorithms in Signal Processing written by Guolong Su (S.M.) and published by . This book was released on 2013 with total page 113 pages. Available in PDF, EPUB and Kindle. Book excerpt: Polynomial decomposition has attracted considerable attention in computational mathematics. In general, the field identifies polynomials f(x) and g(x) such that their composition f(g(x)) equals or approximates a given polynomial h(x). Despite potentially promising applications, polynomial decomposition has not been significantly utilized in signal processing. This thesis studies the sensitivities of polynomial composition and decomposition to explore their robustness in potential signal processing applications and develops effective polynomial decomposition algorithms to be applied in a signal processing context. First, we state the problems of sensitivity, exact decomposition, and approximate decomposition. After that, the sensitivities of the composition and decomposition operations are theoretically derived from the perspective of robustness. In particular, we present and validate an approach to decrease certain sensitivities by using equivalent compositions, and a practical rule for parameter selection is proposed to get to a point that is near the minimum of these sensitivities. Then, new algorithms are proposed for the exact decomposition problems, and simulations are performed to make comparison with existing approaches. Finally, existing and new algorithms for the approximate decomposition problems are presented and evaluated using numerical simulations.
Book Synopsis Solving Polynomial Equations by : Alicia Dickenstein
Download or read book Solving Polynomial Equations written by Alicia Dickenstein and published by Springer Science & Business Media. This book was released on 2005-04-27 with total page 433 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a general introduction to modern mathematical aspects in computing with multivariate polynomials and in solving algebraic systems. It presents the state of the art in several symbolic, numeric, and symbolic-numeric techniques, including effective and algorithmic methods in algebraic geometry and computational algebra, complexity issues, and applications ranging from statistics and geometric modelling to robotics and vision. Graduate students, as well as researchers in related areas, will find an excellent introduction to currently interesting topics. These cover Groebner and border bases, multivariate resultants, residues, primary decomposition, multivariate polynomial factorization, homotopy continuation, complexity issues, and their applications.
Book Synopsis Polynomial Algorithms in Computer Algebra by : Franz Winkler
Download or read book Polynomial Algorithms in Computer Algebra written by Franz Winkler and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt: For several years now I have been teaching courses in computer algebra at the Universitat Linz, the University of Delaware, and the Universidad de Alcala de Henares. In the summers of 1990 and 1992 I have organized and taught summer schools in computer algebra at the Universitat Linz. Gradually a set of course notes has emerged from these activities. People have asked me for copies of the course notes, and different versions of them have been circulating for a few years. Finally I decided that I should really take the time to write the material up in a coherent way and make a book out of it. Here, now, is the result of this work. Over the years many students have been helpful in improving the quality of the notes, and also several colleagues at Linz and elsewhere have contributed to it. I want to thank them all for their effort, in particular I want to thank B. Buchberger, who taught me the theory of Grabner bases nearly two decades ago, B. F. Caviness and B. D. Saunders, who first stimulated my interest in various problems in computer algebra, G. E. Collins, who showed me how to compute in algebraic domains, and J. R. Sendra, with whom I started to apply computer algebra methods to problems in algebraic geometry. Several colleagues have suggested improvements in earlier versions of this book. However, I want to make it clear that I am responsible for all remaining mistakes.
Book Synopsis Polynomial Factorization and Curve Decomposition Algorithms by : Cristina Bertone
Download or read book Polynomial Factorization and Curve Decomposition Algorithms written by Cristina Bertone and published by . This book was released on 2010 with total page 109 pages. Available in PDF, EPUB and Kindle. Book excerpt: Affine algebraic curves are a tool applied in different fields, for instance CAGD. They are defined using polynomials, but they often have several different irreducible components. In this thesis we develop efficient algorithms to decompose a curve defined by rational polynomials. In the first part we present an absolute factorization algorithm for bivariate polynomials (this problem is equivalent to the decomposition of a curve in the plane). We start from the existing algorithm TKTD and we improve the definition of the algebraic extension needed for the factorization, using modular techniques and the LLL algorithm to identify an algebraic number form its p-adic approximation. In the second part we pass to the problem of decomposing a curve in the three-dimensional space: the corresponding technique of the factorization for the case of the plan is the primary decomposition of an ideal for the three-dimensional case. At first, we show some bounds on the degrees of the surfaces separating the different components, using some classical results of algebraic geometry, as the "Lifting problem" or the Castelnuovo-Mumford regularity. After this, we apply consider a classical algorithm of decomposition, which is not efficient for computations, and we apply on it the modular techniques. We obtain a modular algorithm giving the Hilbert function for the reduced components of the curve. The two main algorithms were tested on several examples and compared with the executions times of other softwares.
Book Synopsis Algorithms for Polynomial Factorization by : David R. Musser
Download or read book Algorithms for Polynomial Factorization written by David R. Musser and published by . This book was released on 1971 with total page 376 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A Polynomial Decomposition Algorithm Over Factorial Domains by : Jaime Gutiérrez
Download or read book A Polynomial Decomposition Algorithm Over Factorial Domains written by Jaime Gutiérrez and published by . This book was released on 1989 with total page 10 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Decomposition Methods for Nonlinear Optimization and Data Mining by : Brandon Emmanuel Dutra
Download or read book Decomposition Methods for Nonlinear Optimization and Data Mining written by Brandon Emmanuel Dutra and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We focus on two central themes in this dissertation. The first one is on decomposing polytopes and polynomials in ways that allow us to perform nonlinear optimization. We start off by explaining important results on decomposing a polytope into special polyhedra. We use these decompositions and develop methods for computing a special class of integrals exactly. Namely, we are interested in computing the exact value of integrals of polynomial functions over convex polyhedra. We present prior work and new extensions of the integration algorithms. Every integration method we present requires that the polynomial has a special form. We explore two special polynomial decomposition algorithms that are useful for integrating polynomial functions. Both polynomial decompositions have strengths and weaknesses, and we experiment with how to practically use them. After developing practical algorithms and efficient software tools for integrating a polynomial over a polytope, we focus on the problem of maximizing a polynomial function over the continuous domain of a polytope. This maximization problem is NP-hard, but we develop approximation methods that run in polynomial time when the dimension is fixed. Moreover, our algorithm for approximating the maximum of a polynomial over a polytope is related to integrating the polynomial over the polytope. We show how the integration methods can be used for optimization. We then change topics slightly and consider a problem in combinatorics. Specifically, we seek to compute the function E(t) that counts the number of nonnegative integer solutions to the equation [alpha]1x1 + ··· + [alpha][subscript n]x[subscript n] = t where the [alpha][subscript i] are given positive integers. It is known that this function is a quasi-polynomial function, and computing every term is #P-hard. Instead of computing every term, we compute the top k terms of this function in polynomial time in varying dimension when k is fixed. We review some applications and places where this counting function appears in mathematics. Our new algorithm for computing the largest order terms of E(t) is based on the polyhedral decomposition methods we used in integration and optimization. We also use an additional polyhedral decomposition: Barvinok's fast decomposition of a polyhedral cone into unimodular cones. The second central topic in this dissertation is on problems in data science. We first consider a heuristic for mixed-integer linear optimization. We show how many practical mixed-integer linear have a special substructure containing set partition constraints. We then describe a nice data structure for finding feasible zero-one integer solutions to systems of set partition constraints. Finally, we end with an applied project using data science methods in medical research. The focus is on identifying how T-cells and nervous-system cells interact in the spleen during inflammation. To study this problem, we apply topics in data science and computational geometry to clean data and model the problem. We then use clustering algorithms and develop models for identifying when a spleen sample is responding to inflammation. This project's lifetime surpasses the author's involvement in it. Nevertheless, we focus on the author's contributions, and on the future steps.
Book Synopsis Computer Algebra and Polynomials by : Jaime Gutierrez
Download or read book Computer Algebra and Polynomials written by Jaime Gutierrez and published by Springer. This book was released on 2015-01-20 with total page 222 pages. Available in PDF, EPUB and Kindle. Book excerpt: Algebra and number theory have always been counted among the most beautiful mathematical areas with deep proofs and elegant results. However, for a long time they were not considered that important in view of the lack of real-life applications. This has dramatically changed: nowadays we find applications of algebra and number theory frequently in our daily life. This book focuses on the theory and algorithms for polynomials over various coefficient domains such as a finite field or ring. The operations on polynomials in the focus are factorization, composition and decomposition, basis computation for modules, etc. Algorithms for such operations on polynomials have always been a central interest in computer algebra, as it combines formal (the variables) and algebraic or numeric (the coefficients) aspects. The papers presented were selected from the Workshop on Computer Algebra and Polynomials, which was held in Linz at the Johann Radon Institute for Computational and Applied Mathematics (RICAM) during November 25-29, 2013, at the occasion of the Special Semester on Applications of Algebra and Number Theory.
Book Synopsis A Polynomial Decomposition Algorithm Over Factorial Domains by : Gutiérrez Gutiérrez Gutiérrez
Download or read book A Polynomial Decomposition Algorithm Over Factorial Domains written by Gutiérrez Gutiérrez Gutiérrez and published by . This book was released on 1989 with total page 16 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Effective Polynomial Computation by : Richard Zippel
Download or read book Effective Polynomial Computation written by Richard Zippel and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 364 pages. Available in PDF, EPUB and Kindle. Book excerpt: Effective Polynomial Computation is an introduction to the algorithms of computer algebra. It discusses the basic algorithms for manipulating polynomials including factoring polynomials. These algorithms are discussed from both a theoretical and practical perspective. Those cases where theoretically optimal algorithms are inappropriate are discussed and the practical alternatives are explained. Effective Polynomial Computation provides much of the mathematical motivation of the algorithms discussed to help the reader appreciate the mathematical mechanisms underlying the algorithms, and so that the algorithms will not appear to be constructed out of whole cloth. Preparatory to the discussion of algorithms for polynomials, the first third of this book discusses related issues in elementary number theory. These results are either used in later algorithms (e.g. the discussion of lattices and Diophantine approximation), or analogs of the number theoretic algorithms are used for polynomial problems (e.g. Euclidean algorithm and p-adic numbers). Among the unique features of Effective Polynomial Computation is the detailed material on greatest common divisor and factoring algorithms for sparse multivariate polynomials. In addition, both deterministic and probabilistic algorithms for irreducibility testing of polynomials are discussed.
Book Synopsis Numerical Methods for Roots of Polynomials - Part II by : J.M. McNamee
Download or read book Numerical Methods for Roots of Polynomials - Part II written by J.M. McNamee and published by Elsevier Inc. Chapters. This book was released on 2013-07-19 with total page 94 pages. Available in PDF, EPUB and Kindle. Book excerpt: The zeros of a polynomial can be readily recovered from its linear factors. The linear factors can be approximated by first splitting a polynomial numerically into the product of its two nonconstant factors and then recursively splitting every computed nonlinear factor in similar fashion. For both the worst and average case inputs the resulting algorithms solve the polynomial factorization and root-finding problems within fixed sufficiently small error bounds by using nearly optimal arithmetic and Boolean time, that is using nearly optimal numbers of arithmetic and bitwise operations; in the case of a polynomial with integer coefficients and simple roots we can immediately extend factorization to root isolation, that is to computing disjoint covering discs, one for every root on the complex plane. The presented algorithms compute highly accurate approximations to all roots nearly as fast as one reads the input coefficients. Furthermore, our algorithms allow processor efficient parallel acceleration, which enables root-finding, factorization, and root isolation in polylogarithmic arithmetic and Boolean time. The chapter thoroughly covers the design and analysis of these algorithms, including auxiliary techniques of independent interest. At the end we compare the presented polynomial root-finders with alternative ones, in particular with the popular algorithms adopted by users based on supporting empirical information. We also comment on some promising directions to further progress.
Download or read book Polynomials written by Maurice Mignotte and published by Springer. This book was released on 1999-05 with total page 326 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook gives a well-balanced presentation of the classic procedures of polynomial algebra which are computationally relevant and some algorithms developed during the last decade. The first chapter discusses the construction and the representation of polynomials. The second chapter focuses on the computational aspects of the analytical theory of polynomials. Polynomials with coefficients in a finite field are then described in chapter three, and the final chapetr is devoted to factorization of polynomials with integral coefficients. The book is primarily aimed at graduate students taking courses in Polynomial Algebra, with a prerequisite knowledge of set theory, usual fields and basic algebra. Fully worked out examples, hints and references complement the main text, and details concerning the implementation of algorithms as well as indicators of their efficiency are provided. The book is also useful as a supplementary text for courses in scientific computing, analysis of algorithms, computational polynomial factorization, and computational geometry of polynomials.
Book Synopsis Sequencing by Modular Decomposition by : Jeffrey B. Sidney
Download or read book Sequencing by Modular Decomposition written by Jeffrey B. Sidney and published by . This book was released on 1985 with total page 17 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Polynomial-time Algorithms for the Factorization of Polynomials by : Arjen Klaas Lenstra
Download or read book Polynomial-time Algorithms for the Factorization of Polynomials written by Arjen Klaas Lenstra and published by . This book was released on 1984 with total page 132 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Structured Matrix Based Methods for Approximate Polynomial GCD by : Paola Boito
Download or read book Structured Matrix Based Methods for Approximate Polynomial GCD written by Paola Boito and published by Springer Science & Business Media. This book was released on 2012-03-13 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: Defining and computing a greatest common divisor of two polynomials with inexact coefficients is a classical problem in symbolic-numeric computation. The first part of this book reviews the main results that have been proposed so far in the literature. As usual with polynomial computations, the polynomial GCD problem can be expressed in matrix form: the second part of the book focuses on this point of view and analyses the structure of the relevant matrices, such as Toeplitz, Toepliz-block and displacement structures. New algorithms for the computation of approximate polynomial GCD are presented, along with extensive numerical tests. The use of matrix structure allows, in particular, to lower the asymptotic computational cost from cubic to quadratic order with respect to polynomial degree.
