Download or read book Algorithms for Exact Polynomial Root Calculation written by Lee E. Heindel and published by . This book was released on 1970 with total page 334 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Polynomial Root finding and Polynomiography written by Bahman Kalantari and published by World Scientific. This book was released on 2009 with total page 492 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Algorithms for Exact Polynominal Root Calculation written by Lee Edward Heindel and published by . This book was released on 1983 with total page 153 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Initial Approximations and Root Finding Methods written by Nikolay V. Kyurkchiev and published by Wiley-VCH. This book was released on 1998-10-27 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: Polynomials as mathematical objects have been studied extensively for a long time, and the knowledge collected about them is enormous. Polynomials appear in various fields of applied mathematics and engineering, from mathematics of finance up to signal theory or robust control. The calculation of the roots of a polynomial is a basic problems of numerical mathematics. In this book, an update on iterative methods of calculating simultaneously all roots of a polynomial is given: a survey on basic facts, a lot of methods and properties of those methods connected with the classical task of the approximative determination of roots. For the computer determination the choice of the initial approximation is of special importance. Here the authors offers his new ideas and research results of the last decade which facilitate the practical numerical treatment of polynomials.
Download or read book Numerical Methods for Roots of Polynomials Part I written by J.M. McNamee and published by Elsevier. This book was released on 2007-08-17 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerical Methods for Roots of Polynomials - Part I (along with volume 2 covers most of the traditional methods for polynomial root-finding such as Newton’s, as well as numerous variations on them invented in the last few decades. Perhaps more importantly it covers recent developments such as Vincent’s method, simultaneous iterations, and matrix methods. There is an extensive chapter on evaluation of polynomials, including parallel methods and errors. There are pointers to robust and efficient programs. In short, it could be entitled “A Handbook of Methods for Polynomial Root-finding . This book will be invaluable to anyone doing research in polynomial roots, or teaching a graduate course on that topic. First comprehensive treatment of Root-Finding in several decades Gives description of high-grade software and where it can be down-loaded Very up-to-date in mid-2006; long chapter on matrix methods Includes Parallel methods, errors where appropriate Invaluable for research or graduate course
Download or read book Algorithms and Theory of Computation Handbook Volume 1 written by Mikhail J. Atallah and published by CRC Press. This book was released on 2009-11-20 with total page 974 pages. Available in PDF, EPUB and Kindle. Book excerpt: Algorithms and Theory of Computation Handbook, Second Edition: General Concepts and Techniques provides an up-to-date compendium of fundamental computer science topics and techniques. It also illustrates how the topics and techniques come together to deliver efficient solutions to important practical problems. Along with updating and revising many
Download or read book Numerical Methods for Roots of Polynomials Part II written by J.M. McNamee and published by Newnes. This book was released on 2013-07-19 with total page 749 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerical Methods for Roots of Polynomials - Part II along with Part I (9780444527295) covers most of the traditional methods for polynomial root-finding such as interpolation and methods due to Graeffe, Laguerre, and Jenkins and Traub. It includes many other methods and topics as well and has a chapter devoted to certain modern virtually optimal methods. Additionally, there are pointers to robust and efficient programs. This book is invaluable to anyone doing research in polynomial roots, or teaching a graduate course on that topic. First comprehensive treatment of Root-Finding in several decades with a description of high-grade software and where it can be downloaded Offers a long chapter on matrix methods and includes Parallel methods and errors where appropriate Proves invaluable for research or graduate course
Download or read book Algorithms and Computation written by Rudolf Fleischer and published by Springer. This book was released on 2004-12-06 with total page 951 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the 15th Annual International Sym- sium on Algorithms and Computation (ISAAC 2004), held in Hong Kong, 20–22 December, 2004. In the past, it has been held in Tokyo (1990), Taipei (1991), Nagoya (1992), Hong Kong (1993), Beijing (1994), Cairns (1995), Osaka (1996), Singapore (1997), Taejon (1998), Chennai (1999), Taipei (2000), Christchurch (2001), Vancouver (2002), and Kyoto (2003). ISAAC is an annual international symposium that covers a wide range of topics,namelyalgorithmsandcomputation.Themainpurposeofthesymposium is to provide a forum for researchers working in the active research community of algorithms and the theory of computation to present and exchange new ideas. In response to our call for papers we received 226 submissions. The task of selectingthepapersinthisvolumewasdonebyourprogramcommitteeandother referees. After a thorough review process the committee selected 76 papers, the decisions being based on originality and relevance to the ?eld of algorithms and computation. We hope all accepted papers will eventually appear in scienti?c journals in a more polished form. Two special issues, one of Algorithmica and one of the International Journal of Computational Geometry and Applications, with selected papers from ISAAC 2004 are in preparation. Thebeststudentpaperawardwillbegivenfor“Geometricoptimizationpr- lems over sliding windows” by Bashir S. Sadjad and Timothy M. Chan from the University of Waterloo. Two eminent invited speakers, Prof. Erik D. Demaine, MIT, and Prof. David M. Mount, University of Maryland, also contributed to this volume.
Download or read book Algorithms and Theory of Computation Handbook 2 Volume Set written by Mikhail J. Atallah and published by CRC Press. This book was released on 2022-05-30 with total page 1944 pages. Available in PDF, EPUB and Kindle. Book excerpt: Algorithms and Theory of Computation Handbook, Second Edition in a two volume set, provides an up-to-date compendium of fundamental computer science topics and techniques. It also illustrates how the topics and techniques come together to deliver efficient solutions to important practical problems. New to the Second Edition: Along with updating and revising many of the existing chapters, this second edition contains more than 20 new chapters. This edition now covers external memory, parameterized, self-stabilizing, and pricing algorithms as well as the theories of algorithmic coding, privacy and anonymity, databases, computational games, and communication networks. It also discusses computational topology, computational number theory, natural language processing, and grid computing and explores applications in intensity-modulated radiation therapy, voting, DNA research, systems biology, and financial derivatives. This best-selling handbook continues to help computer professionals and engineers find significant information on various algorithmic topics. The expert contributors clearly define the terminology, present basic results and techniques, and offer a number of current references to the in-depth literature. They also provide a glimpse of the major research issues concerning the relevant topics
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 Mathematical Software written by John R. Rice and published by Academic Press. This book was released on 2014-05-10 with total page 399 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematical Software III contains the proceedings of the Symposium on Mathematical Software held in Madison, Wisconsin, on March 28-30, 1977, under the auspices of the Mathematics Research Center at the University of Wisconsin-Madison. The papers focus on software designed for mathematical applications such as LINPACK for the solution of linear systems and least squares problems and ELLPACK for elliptic partial differential equations. Comprised of 14 chapters, this volume begins with an overview of LINPACK, a software package designed to solve linear systems and least squares problems. The reader is then introduced to an extension to the exchange algorithm for solving overdetermined linear equations; infallible calculation of polynomial zeros to specified precision; and representation and approximation of surfaces. Subsequent chapters discuss the ways in which mathematical software and exploratory data analysis should interact to satisfy their respective needs; production of mathematical software; computational aspects of the finite element method; and multi-level adaptive techniques for partial differential equations. The book also describes a realistic model of floating-point computation before concluding with an evaluation of the Block Lanczos method for computing a few of the least or greatest eigenvalues of a sparse symmetric matrix. This monograph should be of considerable interest to students and specialists in the fields of mathematics and computer science.
Download or read book Exact Polynomial System Solving for Robust Geometric Computation written by Koji Ouchi and published by . This book was released on 2006 with total page 168 pages. Available in PDF, EPUB and Kindle. Book excerpt: I describe an exact method for computing roots of a system of multivariate polynomials with rational coefficients, called the rational univariate reduction. This method enables performance of exact algebraic computation of coordinates of the roots of polynomials. In computational geometry, curves, surfaces and points are described as polynomials and their intersections. Thus, exact computation of the roots of polynomials allows the development and implementation of robust geometric algorithms. I describe applications in robust geometric modeling. In particular, I show a new method, called numerical perturbation scheme, that can be used successfully to detect and handle degenerate configurations appearing in boundary evaluation problems. I develop a derandomized version of the algorithm for computing the rational univariate reduction for a square system of multivariate polynomials and a new algorithm for a non-square system. I show how to perform exact computation over algebraic points obtained by the rational univariate reduction. I give a formal description of numerical perturbation scheme and its implementation.
Download or read book Quantifier Elimination and Cylindrical Algebraic Decomposition written by Bob F. Caviness and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 455 pages. Available in PDF, EPUB and Kindle. Book excerpt: George Collins’ discovery of Cylindrical Algebraic Decomposition (CAD) as a method for Quantifier Elimination (QE) for the elementary theory of real closed fields brought a major breakthrough in automating mathematics with recent important applications in high-tech areas (e.g. robot motion), also stimulating fundamental research in computer algebra over the past three decades. This volume is a state-of-the-art collection of important papers on CAD and QE and on the related area of algorithmic aspects of real geometry. It contains papers from a symposium held in Linz in 1993, reprints of seminal papers from the area including Tarski’s landmark paper as well as a survey outlining the developments in CAD based QE that have taken place in the last twenty years.
Download or read book Error Estimate of Solving Polynomial Equations and the Modified Durand Kerner Iteration written by Haiyang Zhu and published by . This book was released on 2019 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: "In this thesis, we study the accuracy of a computed polynomial root and we construct a modification of the Durand-Kerner method for computing multiple roots. For the first topic, we focus on the error estimate of solving polynomial equations. An important notion of the numerical solution is its backward error along with the condition number and forward error. The basic tenet of the backward error analysis may be summarized in one sentence: A stable algorithm calculates the exact solution of a nearby problem or the same problem at nearby data. We formulate the backward error as a constrained minimization problem and apply the classical method of Lagrange multipliers. By solving this optimization problem, we obtain a precise formula of the backward error. Using this formula, we can estimate the accuracy of a computed root of a polynomial and decide if it is an acceptable solution. For the second objective, we concentrate on developing a new algorithm for computing multiple roots. The Durand-Kerner iteration is one of the widely used root-finding methods due to its simplicity and the theoretical global convergence. From our experiment, however, the Durand-Kerner iteration is inaccurate and inefficient when the polynomial possesses multiple roots. We construct a new algorithm to compute multiple roots accurately by using a similar approach for developing the Durand-Kerner iteration. We assume the multiplicities of the roots are known in the Vieta's equation and use only the distinct roots as variables. The resulting Vieta's equation is an overdetermined nonlinear system. The Gauss-Newton algorithm is then applied to solve for the least squares solution. In this way, we obtain a modified Durand-Kerner iteration method for finding the polynomial roots. From our computing experiment on polynomials possesses multiple roots, it appears that our new iteration is substantially more accurate than the original Durand-Kerner iteration."--
Download or read book Numerical Recipes in C written by William H. Press and published by . This book was released on 2002 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Now the acclaimed Second Edition of Numerical Recipes is available in the C++ object-oriented programming language. Including and updating the full mathematical and explanatory contents of Numerical Recipes in C, this new version incorporates completely new C++ versions of the more than 300 Numerical Recipes routines that are widely recognized as the most accessible and practical basis for scientific computing. The product of a unique collaboration among four leading scientists in academic research and industry, Numerical Recipes is a complete text and reference book on scientific computing. In a self-contained manner it proceeds from mathematical and theoretical considerations to actual practical computer routines. Highlights include linear algebra, interpolation, special functions, random numbers, nonlinear sets of equations, optimization, eigensystems, Fourier methods and wavelets, statistical tests, ODEs and PDEs, integral equations and inverse theory. The authors approach to C++ preserves the efficient execution that C users expect, while simultaneously employing a clear, object-oriented interface to the routines. Tricks and tips for scientific computing in C++ are liberally included. The routines, in ANSI/ISO C++ source code, can thus be used with almost any existing C++ vector/matrix class library, according to user preference. A simple class library for stand-alone use is also included in the book. Both scientific programmers new to C++, and experienced C++ programmers who need access to the Numerical Recipes routines, can benefit from this important new version of an invaluable, classic text.
Download or read book Effective Polynomial Computation written by Richard Zippel and published by Springer Science & Business Media. This book was released on 1993-07-31 with total page 386 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.
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 14 pages. Available in PDF, EPUB and Kindle. Book excerpt:
