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
Genre: Mathematics
ISBN: 0080489478

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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

Numerical Methods for Roots of Polynomials
Author: J.M. McNamee,Victor Pan
Publsiher: Newnes
Total Pages: 728
Release: 2013-07-19
Genre: Mathematics
ISBN: 9780080931432

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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

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
Genre: Mathematics
ISBN: 9780128076989

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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 Engineers and Scientists

Numerical Methods for Engineers and Scientists
Author: Joe D. Hoffman,Steven Frankel
Publsiher: CRC Press
Total Pages: 840
Release: 2018-10-03
Genre: Mathematics
ISBN: 9781482270600

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Emphasizing the finite difference approach for solving differential equations, the second edition of Numerical Methods for Engineers and Scientists presents a methodology for systematically constructing individual computer programs. Providing easy access to accurate solutions to complex scientific and engineering problems, each chapter begins with objectives, a discussion of a representative application, and an outline of special features, summing up with a list of tasks students should be able to complete after reading the chapter- perfect for use as a study guide or for review. The AIAA Journal calls the book "...a good, solid instructional text on the basic tools of numerical analysis."

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
Genre: Mathematics
ISBN: 9780128077054

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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
Genre: Mathematics
ISBN: 9780128076965

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Introduction to Numerical Analysis Using MATLAB

Introduction to Numerical Analysis Using MATLAB
Author: Butt
Publsiher: Jones & Bartlett Learning
Total Pages: 600
Release: 2009-02-17
Genre: Mathematics
ISBN: 076377376X

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Numerical analysis is the branch of mathematics concerned with the theoretical foundations of numerical algorithms for the solution of problems arising in scientific applications. Designed for both courses in numerical analysis and as a reference for practicing engineers and scientists, this book presents the theoretical concepts of numerical analysis and the practical justification of these methods are presented through computer examples with the latest version of MATLAB. The book addresses a variety of questions ranging from the approximation of functions and integrals to the approximate solution of algebraic, transcendental, differential and integral equations, with particular emphasis on the stability, accuracy, efficiency and reliability of numerical algorithms. The CD-ROM which accompanies the book includes source code, a numerical toolbox, executables, and simulations.

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
Genre: Mathematics
ISBN: 9780128077009

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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

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
Genre: Mathematics
ISBN: 9780128077030

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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.

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
Genre: Mathematics
ISBN: 9780128077016

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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.

Exploring Numerical Methods

Exploring Numerical Methods
Author: Peter Linz,Richard Wang
Publsiher: Jones & Bartlett Learning
Total Pages: 473
Release: 2003
Genre: Computers
ISBN: 0763714992

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Advanced Mathematics

Fundamental Numerical Methods for Electrical Engineering

Fundamental Numerical Methods for Electrical Engineering
Author: Stanislaw Rosloniec
Publsiher: Springer Science & Business Media
Total Pages: 284
Release: 2008-07-17
Genre: Technology & Engineering
ISBN: 9783540795193

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Stormy development of electronic computation techniques (computer systems and software), observed during the last decades, has made possible automation of data processing in many important human activity areas, such as science, technology, economics and labor organization. In a broadly understood technology area, this developmentledtoseparationofspecializedformsofusingcomputersforthedesign and manufacturing processes, that is: – computer-aided design (CAD) – computer-aided manufacture (CAM) In order to show the role of computer in the rst of the two applications m- tioned above, let us consider basic stages of the design process for a standard piece of electronic system, or equipment: – formulation of requirements concerning user properties (characteristics, para- ters) of the designed equipment, – elaboration of the initial, possibly general electric structure, – determination of mathematical model of the system on the basis of the adopted electric structure, – determination of basic responses (frequency- or time-domain) of the system, on the base of previously established mathematical model, – repeated modi cation of the adopted diagram (changing its structure or element values) in case, when it does not satisfy the adopted requirements, – preparation of design and technological documentation, – manufacturing of model (prototype) series, according to the prepared docum- tation, – testing the prototype under the aspect of its electric properties, mechanical du- bility and sensitivity to environment conditions, – modi cation of prototype documentation, if necessary, and handing over the documentation to series production. The most important stages of the process under discussion are illustrated in Fig. I. 1. xi xii Introduction Fig. I.

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
Genre: Mathematics
ISBN: UVA:X004235327

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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.

Object Oriented Implementation of Numerical Methods

Object Oriented Implementation of Numerical Methods
Author: Didier H. Besset
Publsiher: Morgan Kaufmann
Total Pages: 766
Release: 2001
Genre: Computers
ISBN: 1558606793

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"There are few books that show how to build programs of any kind. One common theme is compiler building, and there are shelves full of them. There are few others. It's an area, or a void, that needs filling. this book does a great job of showing how to build numerical analysis programs." -David N. Smith, IBM T J Watson Research Center Numerical methods naturally lend themselves to an object-oriented approach. Mathematics builds high- level ideas on top of previously described, simpler ones. Once a property is demonstrated for a given concept, it can be applied to any new concept sharing the same premise as the original one, similar to the ideas of reuse and inheritance in object-oriented (OO) methodology. Few books on numerical methods teach developers much about designing and building good code. Good computing routines are problem-specific. Insight and understanding are what is needed, rather than just recipes and black box routines. Developers need the ability to construct new programs for different applications. Object-Oriented Implementation of Numerical Methods reveals a complete OO design methodology in a clear and systematic way. Each method is presented in a consistent format, beginning with a short explanation and following with a description of the general OO architecture for the algorithm. Next, the code implementations are discussed and presented along with real-world examples that the author, an experienced software engineer, has used in a variety of commercial applications. Features: Reveals the design methodology behind the code, including design patterns where appropriate, rather than just presenting canned solutions. Implements all methods side by side in both Java and Smalltalk. This contrast can significantly enhance your understanding of the nature of OO programming languages. Provides a step-by-step pathway to new object-oriented techniques for programmers familiar with using procedural languages such as C or Fortran for numerical methods. Includes a chapter on data mining, a key application of numerical methods.

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
Genre: Mathematics
ISBN: 9780128077047

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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.