Numerical Methods for Roots of Polynomials

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

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 Part II

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.

Numerical Methods for Roots of Polynomials Part II

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

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 Part II

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

Numerical Methods for Roots of Polynomials
Author: J. M. McNamee
Publsiher: Unknown
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

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

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

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.

Some Numerical Methods for Locating Roots of Polynomials

Some Numerical Methods for Locating Roots of Polynomials
Author: Thornton Carle Fry
Publsiher: Unknown
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

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.

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

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 for Roots of Polynomials Part II

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.

Advances in Electronic Commerce Web Application and Communication

Advances in Electronic Commerce  Web Application and Communication
Author: David Jin,Sally Lin
Publsiher: Springer Science & Business Media
Total Pages: 622
Release: 2012-02-24
ISBN 10: 3642286585
ISBN 13: 9783642286582
Language: EN, FR, DE, ES & NL

Advances in Electronic Commerce Web Application and Communication Book Review:

ECWAC2012 is an integrated conference devoted to Electronic Commerce, Web Application and Communication. In the this proceedings you can find the carefully reviewed scientific outcome of the second International Conference on Electronic Commerce, Web Application and Communication (ECWAC 2012) held at March 17-18,2012 in Wuhan, China, bringing together researchers from all around the world in the field.

Polynomial Root finding and Polynomiography

Polynomial Root finding and Polynomiography
Author: Bahman Kalantari
Publsiher: World Scientific
Total Pages: 467
Release: 2009-01
ISBN 10: 9812700595
ISBN 13: 9789812700599
Language: EN, FR, DE, ES & NL

Polynomial Root finding and Polynomiography Book Review:

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.

Inclusion Methods for Nonlinear Problems

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.

Numerical Methods for Roots of Polynomials Part II

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

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.

Applied Numerical Methods for Digital Computation with FORTRAN and CSMP

Applied Numerical Methods for Digital Computation with FORTRAN and CSMP
Author: Merlin L. James,Gerald M. Smith,J. C. Wolford
Publsiher: New York : IEP
Total Pages: 687
Release: 1977
ISBN 10:
ISBN 13: UOM:39076005381178
Language: EN, FR, DE, ES & NL

Applied Numerical Methods for Digital Computation with FORTRAN and CSMP Book Review: