# Numerical methods for for roots of polynomials

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## Numerical Methods for Roots of Polynomials

Author | : J.M. McNamee |

Publsiher | : Elsevier |

Total Pages | : 354 |

Release | : 2007-08-17 |

ISBN 10 | : 9780080489476 |

ISBN 13 | : 0080489478 |

Language | : EN, FR, DE, ES & NL |

**Numerical Methods for Roots of Polynomials Book Review:**

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

## Numerical Methods for Roots of Polynomials

Author | : J.M. McNamee,Victor Pan |

Publsiher | : Newnes |

Total Pages | : 728 |

Release | : 2013-07-19 |

ISBN 10 | : 008093143X |

ISBN 13 | : 9780080931432 |

Language | : EN, FR, DE, ES & NL |

**Numerical Methods for Roots of Polynomials Book Review:**

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

## Numerical Methods for Roots of Polynomials Part II

Author | : J.M. McNamee,V.Y. Pan |

Publsiher | : Elsevier Inc. Chapters |

Total Pages | : 728 |

Release | : 2013-07-19 |

ISBN 10 | : 0128077018 |

ISBN 13 | : 9780128077016 |

Language | : EN, FR, DE, ES & NL |

**Numerical Methods for Roots of Polynomials Part II Book Review:**

First we consider the Jenkins–Traub 3-stage algorithm. In stage 1 we defineIn the second stage the factor is replaced by for fixed , and in the third stage by where is re-computed at each iteration. Then a root. A slightly different algorithm is given for real polynomials. Another class of methods uses minimization, i.e. we try to find such that is a minimum, where . At this minimum we must have , i.e. . Several authors search along the coordinate axes or at various angles with them, while others move along the negative gradient, which is probably more efficient. Some use a hybrid of Newton and minimization. Finally we come to Lin and Bairstow’s methods, which divide the polynomial by a quadratic and iteratively reduce the remainder to 0. This enables us to find pairs of complex roots using only real arithmetic.

## Numerical Methods for Roots of Polynomials Part II

Author | : J.M. McNamee,V.Y. Pan |

Publsiher | : Elsevier Inc. Chapters |

Total Pages | : 728 |

Release | : 2013-07-19 |

ISBN 10 | : 0128077050 |

ISBN 13 | : 9780128077054 |

Language | : EN, FR, DE, ES & NL |

**Numerical Methods for Roots of Polynomials Part II Book Review:**

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.

## Numerical Methods for Roots of Polynomials Part II

Author | : J.M. McNamee,V.Y. Pan |

Publsiher | : Elsevier Inc. Chapters |

Total Pages | : 728 |

Release | : 2013-07-19 |

ISBN 10 | : 012807700X |

ISBN 13 | : 9780128077009 |

Language | : EN, FR, DE, ES & NL |

**Numerical Methods for Roots of Polynomials Part II Book Review:**

This chapter treats several topics, starting with Bernoulli’s method. This method iteratively solves a linear difference equation whose coefficients are the same as those of the polynomial. The ratios of successive iterates tends to the root of largest magnitude. Special versions are used for complex and/or multiple roots. The iteration may be accelerated, and Aitken’s variation finds all the roots simultaneously. The Quotient-Difference algorithm uses two sequences(with a similar one for ). Then, if the roots are well separated, . Special techniques are used for roots of equal modulus. The Lehmer–Schur method uses a test to determine whether a given circle contains a root or not. Using this test we find an annulus which contains a root, whereas the circle does not. We cover the annulus with 8 smaller circles and test which one contains the roots. We repeat the process until a sufficiently small circle is known to contain the root. We also consider methods using integration, such as by Delves–Lyness and Kravanja et al.

## Numerical Methods for Roots of Polynomials Part II

Author | : J.M. McNamee,V.Y. Pan |

Publsiher | : Elsevier Inc. Chapters |

Total Pages | : 728 |

Release | : 2013-07-19 |

ISBN 10 | : 0128076984 |

ISBN 13 | : 9780128076989 |

Language | : EN, FR, DE, ES & NL |

**Numerical Methods for Roots of Polynomials Part II Book Review:**

We discuss Graeffes’s method and variations. Graeffe iteratively computes a sequence of polynomialsso that the roots of are those of raised to the power . Then the roots of can be expressed in terms of the coefficients of . Special treatment is given to complex and/or multiple modulus roots. A method of Lehmer’s finds the argument as well as the modulus of the roots, while other authors show how to reduce the danger of overflow. Variants such as the Chebyshev-like process are discussed. The Graeffe iteration lends itself well to parallel processing, and two algorithms in that context are described. Error estimates are given, as well as several variants.

## Numerical Methods for Roots of Polynomials

Author | : J.M. McNamee,Victor Pan |

Publsiher | : Elsevier Science |

Total Pages | : 728 |

Release | : 2013-09-11 |

ISBN 10 | : 9780444527301 |

ISBN 13 | : 0444527303 |

Language | : EN, FR, DE, ES & NL |

**Numerical Methods for Roots of Polynomials Book Review:**

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

## Numerical Methods for Roots of Polynomials Part II

Author | : J.M. McNamee,V.Y. Pan |

Publsiher | : Elsevier Inc. Chapters |

Total Pages | : 728 |

Release | : 2013-07-19 |

ISBN 10 | : 0128076976 |

ISBN 13 | : 9780128076972 |

Language | : EN, FR, DE, ES & NL |

**Numerical Methods for Roots of Polynomials Part II Book Review:**

We discuss the secant method:where are initial guesses. In the Regula Falsi variation we start with initial guesses and such that ; after an iteration similar to the above we replace either a or b by the new value depending on which of or has the same sign as . Often one of the points gets “stuck,” and several variants such as the Illinois or Pegasus methods and variations are used to “unstick” it. We discuss convergence and efficiency of most of the methods considered. We treat methods involving quadratic of higher order interpolation and rational approximation. We also discuss the bisection method where again and we set . We replace a or b by c according to the sign of as in the Regula Falsi method. Various generalizations are described, including some for complex roots. Finally we consider hybrid methods involving two or more of the previously described methods.

## Numerical Methods for Roots of Polynomials Part II

Author | : J.M. McNamee,V.Y. Pan |

Publsiher | : Elsevier Inc. Chapters |

Total Pages | : 728 |

Release | : 2013-07-19 |

ISBN 10 | : 0128076968 |

ISBN 13 | : 9780128076965 |

Language | : EN, FR, DE, ES & NL |

**Numerical Methods for Roots of Polynomials Part II Book Review:**

## Numerical Methods for Roots of Polynomials

Author | : J. M. McNamee |

Publsiher | : Anonim |

Total Pages | : 333 |

Release | : 2007 |

ISBN 10 | : |

ISBN 13 | : OCLC:505137048 |

Language | : EN, FR, DE, ES & NL |

**Numerical Methods for Roots of Polynomials Book Review:**

## Numerical Methods for Roots of Polynomials Part II

Author | : J.M. McNamee,V.Y. Pan |

Publsiher | : Elsevier Inc. Chapters |

Total Pages | : 728 |

Release | : 2013-07-19 |

ISBN 10 | : 0128077026 |

ISBN 13 | : 9780128077023 |

Language | : EN, FR, DE, ES & NL |

**Numerical Methods for Roots of Polynomials Part II Book Review:**

We deal here with low-degree polynomials, mostly closed-form solutions. We describe early and modern solutions of the quadratic, and potential errors in these. Again we give the early history of the cubic, and details of Cardan’s solution and Vieta’s trigonometric approach. We consider the discriminant, which decides what type of roots the cubic has. Then we describe several ways (both old and new) of solving the quartic, most of which involve first solving a “resolvent” cubic. The quintic cannot in general be solved by radicals, but can be solved in terms of elliptic or related functions. We describe an algorithm due to Kiepert, which transforms the quintic into a form having no or term; then into a form where the coefficients depend on a single parameter; and later another similar form. This last form can be solved in terms of Weierstrass elliptic and theta functions, and finally the various transformations reversed.

## Some Numerical Methods for Locating Roots of Polynomials

Author | : Thornton Carle Fry |

Publsiher | : Anonim |

Total Pages | : 17 |

Release | : 1945* |

ISBN 10 | : |

ISBN 13 | : OCLC:43223059 |

Language | : EN, FR, DE, ES & NL |

**Some Numerical Methods for Locating Roots of Polynomials Book Review:**

## Numerical Methods for Roots of Polynomials Part II

Author | : J.M. McNamee,V.Y. Pan |

Publsiher | : Elsevier Inc. Chapters |

Total Pages | : 728 |

Release | : 2013-07-19 |

ISBN 10 | : 0128076992 |

ISBN 13 | : 9780128076996 |

Language | : EN, FR, DE, ES & NL |

**Numerical Methods for Roots of Polynomials Part II Book Review:**

Whereas Newton’s method involves only the first derivative, methods discussed in this chapter involve the second or higher. The “classical” methods of this type (such as Halley’s, Euler’s, Hansen and Patrick’s, Ostrowski’s, Cauchy’s and Chebyshev’s) are all third order with three evaluations, so are slightly more efficient than Newton’s method. Convergence of some of these methods is discussed, as well as composite variations (some of which have fairly high efficiency). We describe special methods for multiple roots, simultaneous or interval methods, and acceleration techniques. We treat Laguerre’s method, which is known to be globally convergent for all-real-roots. The Cluster-Adapted Method is useful for multiple or near-multiple roots. Several composite methods are discussed, as well as methods using determinants or various types of interpolation, and Schroeder’s method.

## Numerical Methods for Roots of Polynomials Part II

Author | : J.M. McNamee,V.Y. Pan |

Publsiher | : Elsevier Inc. Chapters |

Total Pages | : 728 |

Release | : 2013-07-19 |

ISBN 10 | : 0128077034 |

ISBN 13 | : 9780128077030 |

Language | : EN, FR, DE, ES & NL |

**Numerical Methods for Roots of Polynomials Part II Book Review:**

We consider proofs that every polynomial has one zero (and hence n) in the complex plane. This was proved by Gauss in 1799, although a flaw in his proof was pointed out and fixed by Ostrowski in 1920, whereas other scientists had previously made unsuccessful attempts. We give details of Gauss’ fourth (trigonometric) proof, and also more modern proofs, such as several based on integration, or on minimization. We also treat the proofs that polynomials of degree 5 or more cannot in general be solved in terms of radicals. We define groups and fields, the set of congruence classes mod p (x), extension fields, algebraic extensions, permutations, the Galois group. We quote the fundamental theorem of Galois theory, the definition of a solvable group, and Galois’ criterion (that a polynomial is solvable by radicals if and only if its group is solvable). We prove that for the group is not solvable. Finally we mention that a particular quintic has Galois group , which is not solvable, and so the quintic cannot be solved by radicals.

## Initial Approximations and Root Finding Methods

Author | : Nikolay V. Kyurkchiev |

Publsiher | : Wiley-VCH |

Total Pages | : 180 |

Release | : 1998-10-27 |

ISBN 10 | : |

ISBN 13 | : UVA:X004235327 |

Language | : EN, FR, DE, ES & NL |

**Initial Approximations and Root Finding Methods Book Review:**

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.

## Numerical Methods for Roots of Polynomials Part II

Author | : J.M. McNamee,V.Y. Pan |

Publsiher | : Elsevier Inc. Chapters |

Total Pages | : 728 |

Release | : 2013-07-19 |

ISBN 10 | : 0128077042 |

ISBN 13 | : 9780128077047 |

Language | : EN, FR, DE, ES & NL |

**Numerical Methods for Roots of Polynomials Part II Book Review:**

In considering the stability of mechanical systems we are led to the characteristic equation . Continuous-time systems are stable if all the roots of this equation are in the left half-plane (Hurwitz stability), while discrete-time systems require all (Schur stability). Hurwitz stability has been treated by the Cauchy index and Sturm sequences, leading to various determinantal criteria and Routh’s array, and several other methods. We also have to consider the question of robust stability, i.e. whethera system remains stable when its coefficients vary. In the Hurwitz case Kharitonov’s theorem reduces the answer to the consideration of 4 extreme polynomials, and other authors consider cases where the coefficients depend on parameters in various ways. Schur stability is notably dealt with by the Schur–Cohn algorithm, which constructs a sequence of polynomials and tests whether all their constant terms are negative. Methods are described which reduce overflow in this process. Robust Schur stability is harder to deal with than Hurwitz, but several partial solutions are described.

## The Numerical Solution of Systems of Polynomials Arising in Engineering and Science

Author | : Andrew J Sommese,Charles W Wampler II |

Publsiher | : World Scientific |

Total Pages | : 424 |

Release | : 2005-03-21 |

ISBN 10 | : 9814480886 |

ISBN 13 | : 9789814480888 |

Language | : EN, FR, DE, ES & NL |

**The Numerical Solution of Systems of Polynomials Arising in Engineering and Science Book Review:**

' Written by the founders of the new and expanding field of numerical algebraic geometry, this is the first book that uses an algebraic-geometric approach to the numerical solution of polynomial systems and also the first one to treat numerical methods for finding positive dimensional solution sets. The text covers the full theory from methods developed for isolated solutions in the 1980's to the most recent research on positive dimensional sets. Contents:Background:Polynomial SystemsHomotopy ContinuationProjective SpacesGenericity and Probability OnePolynomials of One VariableOther MethodsIsolated Solutions:Coefficient-Parameter HomotopyPolynomial StructuresCase StudiesEndpoint EstimationChecking Results and Other Implementation TipsPositive Dimensional Solutions:Basic Algebraic GeometryBasic Numerical Algebraic GeometryA Cascade Algorithm for Witness SupersetsThe Numerical Irreducible DecompositionThe Intersection of Algebraic SetsAppendices:Algebraic GeometrySoftware for Polynomial ContinuationHomLab User's Guide Readership: Graduate students and researchers in applied mathematics and mechanical engineering. Keywords:Polynomial Systems;Numerical Methods;Homotopy Methods;Mechanical Engineering;Numerical Algebraic Geometry;Kinematics;RoboticsKey Features:Useful introduction to the field for graduate students and researchers in related areasIncludes exercises suitable for classroom use and self-studyIncludes Matlab software to illustrate the methodIncludes many graphical illustrationsIncludes a detailed summary of useful results from algebraic geometryReviews:“The text is written in a very smooth and intelligent form, yielding a readable book whose contents are accessible to a wide class of readers, even to undergraduate students, provided that they accept that some delicate points of some of the proofs could be omitted. Its readability and fast access to the core of the book makes it recommendable as a pleasant read.”Mathematical Reviews “This is an excellent book on numerical solutions of polynomials systems for engineers, scientists and numerical analysts. As pioneers of the field of numerical algebraic geometry, the authors have provided a comprehensive summary of ideas, methods, problems of numerical algebraic geometry and applications to solving polynomial systems. Through the book readers will experience the authors' original ideas, contributions and their techniques in handling practical problems … Many interesting examples from engineering and science have been used throughout the book. Also the exercises are well designed in line with the content, along with the algorithms, sample programs in Matlab and author's own software 'HOMLAB' for polynomial continuation. This is a remarkable book that I recommend to engineers, scientists, researchers, professionals and students, and particularly numerical analysts who will benefit from the rapid development of numerical algebraic geometry.”Zentralblatt MATH '

## Handbook of Numerical Methods for the Solution of Algebraic and Transcendental Equations

Author | : V. L. Zaguskin |

Publsiher | : Elsevier |

Total Pages | : 216 |

Release | : 2014-05-12 |

ISBN 10 | : 1483225674 |

ISBN 13 | : 9781483225678 |

Language | : EN, FR, DE, ES & NL |

**Handbook of Numerical Methods for the Solution of Algebraic and Transcendental Equations Book Review:**

Handbook of Numerical Methods for the Solution of Algebraic and Transcendental Equations provides information pertinent to algebraic and transcendental equations. This book indicates a well-grounded plan for the solution of an approximate equation. Organized into six chapters, this book begins with an overview of the solution of various equations. This text then outlines a non-traditional theory of the solution of approximate equations. Other chapters consider the approximate methods for the calculation of roots of algebraic equations. This book discusses as well the methods for making roots more accurate, which are essential in the practical application of Berstoi's method. The final chapter deals with the methods for the solution of simultaneous linear equations, which are divided into direct methods and methods of successive approximation. This book is a valuable resource for students, engineers, and research workers of institutes and industrial enterprises who are using mathematical methods in the solution of technical problems.

## Numerical Methods With Programs In C

Author | : Veerarajan & Ramachandran |

Publsiher | : Tata McGraw-Hill Education |

Total Pages | : 329 |

Release | : 2005-11-01 |

ISBN 10 | : 9780070601611 |

ISBN 13 | : 0070601615 |

Language | : EN, FR, DE, ES & NL |

**Numerical Methods With Programs In C Book Review:**

## Inclusion Methods for Nonlinear Problems

Author | : Jürgen Herzberger |

Publsiher | : Springer Science & Business Media |

Total Pages | : 244 |

Release | : 2012-12-06 |

ISBN 10 | : 3709160332 |

ISBN 13 | : 9783709160336 |

Language | : EN, FR, DE, ES & NL |

**Inclusion Methods for Nonlinear Problems Book Review:**

This workshop was organized with the support of GAMM, the International Association of Applied Mathematics and Mechanics, on the occasion of J. Herzberger's 60th birthday. GAMM is thankful to him for all the time and work he spent in the preparation and holding of the meeting. The talks presented during the workshop and the papers published in this volume are part of the field of Verification Numerics. The important subject is fostered by GAMM already since a number of years, especially also by the GAMM FachausschuB (special interest group) "Rechnerarithmetik und Wissenschaft liches Rechnen". GiHz Alefeld Karlsruhe, Dezember 2001 (President of GAMM) Preface At the end of the year 2000, about 23 scientists from many countries gathered in the beautiful city of Munich on the occasion of the International GAMM Workshop on "Inclusion Methods for Nonlinear Problems with Applications in Engineering, Economics and Physics" from December 15 to 18. The purpose of this meeting was to bring together representatives of research groups from Austria, Bulgaria, China, Croatia, Germany, Japan, Russia, Ukraine and Yugoslavia who in a wider sense work in the field of calculating numerical solutions with error-bounds. Most of those participants have already known each other from earlier occasions or closely cooperated in the past. Representatives from three Academies of Sciences were among the speakers of this conference: from the Bulgarian Academy, the Russian Academy and the Ukrainian Academy of Sciences.