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Numerical Methods For For Roots Of Polynomials

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Numerical Methods for Roots of Polynomials

Numerical Methods for Roots of Polynomials   Book
Author : J.M. McNamee,Victor Pan
Publisher : Newnes
Release : 2013-07-19
ISBN : 008093143X
Language : En, Es, Fr & De

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Book Description :

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   Book
Author : J.M. McNamee
Publisher : Elsevier
Release : 2007-08-17
ISBN : 9780080489476
Language : En, Es, Fr & De

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Book Description :

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 Book
Author : J.M. McNamee,V.Y. Pan
Publisher : Elsevier Inc. Chapters
Release : 2013-07-19
ISBN : 0128076968
Language : En, Es, Fr & De

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Book Description :

Download Numerical Methods for Roots of Polynomials Part II book written by J.M. McNamee,V.Y. Pan, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Numerical Methods for Roots of Polynomials Part II

Numerical Methods for Roots of Polynomials   Part II Book
Author : J.M. McNamee,V.Y. Pan
Publisher : Elsevier Inc. Chapters
Release : 2013-07-19
ISBN : 0128077018
Language : En, Es, Fr & De

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Book Description :

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

Numerical Methods for Roots of Polynomials Book
Author : J. M. McNamee
Publisher : Unknown
Release : 2007
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Download Numerical Methods for Roots of Polynomials book written by J. M. McNamee, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Numerical Methods for Roots of Polynomials

Numerical Methods for Roots of Polynomials   Book
Author : J.M. McNamee,Victor Pan
Publisher : Elsevier Science
Release : 2013-09-11
ISBN : 9780444527301
Language : En, Es, Fr & De

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Book Description :

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 Book
Author : J.M. McNamee,V.Y. Pan
Publisher : Elsevier Inc. Chapters
Release : 2013-07-19
ISBN : 012807700X
Language : En, Es, Fr & De

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Book Description :

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 Book
Author : J.M. McNamee,V.Y. Pan
Publisher : Elsevier Inc. Chapters
Release : 2013-07-19
ISBN : 0128077050
Language : En, Es, Fr & De

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Book Description :

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.

Initial Approximations and Root Finding Methods

Initial Approximations and Root Finding Methods Book
Author : Nikolay V. Kyurkchiev
Publisher : Wiley-VCH
Release : 1998-10-27
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

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 Engineers and Scientists

Numerical Methods for Engineers and Scientists Book
Author : Joe D. Hoffman,Steven Frankel
Publisher : CRC Press
Release : 2018-10-03
ISBN : 1482270609
Language : En, Es, Fr & De

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Book Description :

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 Book
Author : J.M. McNamee,V.Y. Pan
Publisher : Elsevier Inc. Chapters
Release : 2013-07-19
ISBN : 0128077034
Language : En, Es, Fr & De

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Book Description :

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 Book
Author : J.M. McNamee,V.Y. Pan
Publisher : Elsevier Inc. Chapters
Release : 2013-07-19
ISBN : 0128076976
Language : En, Es, Fr & De

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Book Description :

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

Numerical Methods for Roots of Polynomials   Part II Book
Author : J.M. McNamee,V.Y. Pan
Publisher : Elsevier Inc. Chapters
Release : 2013-07-19
ISBN : 0128076984
Language : En, Es, Fr & De

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Book Description :

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.

Some Numerical Methods for Locating Roots of Polynomials

Some Numerical Methods for Locating Roots of Polynomials Book
Author : Thornton Carle Fry
Publisher : Unknown
Release : 1945*
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Download Some Numerical Methods for Locating Roots of Polynomials book written by Thornton Carle Fry, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Numerical Methods for Roots of Polynomials Part II

Numerical Methods for Roots of Polynomials   Part II Book
Author : J.M. McNamee,V.Y. Pan
Publisher : Elsevier Inc. Chapters
Release : 2013-07-19
ISBN : 0128077042
Language : En, Es, Fr & De

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Book Description :

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.

Numerical Methods for Roots of Polynomials Part II

Numerical Methods for Roots of Polynomials   Part II Book
Author : J.M. McNamee,V.Y. Pan
Publisher : Elsevier Inc. Chapters
Release : 2013-07-19
ISBN : 0128077026
Language : En, Es, Fr & De

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Book Description :

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.

Exploring Numerical Methods

Exploring Numerical Methods Book
Author : Peter Linz,Richard Wang
Publisher : Jones & Bartlett Learning
Release : 2003
ISBN : 9780763714994
Language : En, Es, Fr & De

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Book Description :

Advanced Mathematics

Object Oriented Implementation of Numerical Methods

Object Oriented Implementation of Numerical Methods Book
Author : Didier H. Besset
Publisher : Morgan Kaufmann
Release : 2001
ISBN : 9781558606791
Language : En, Es, Fr & De

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Book Description :

"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 Book
Author : J.M. McNamee,V.Y. Pan
Publisher : Elsevier Inc. Chapters
Release : 2013-07-19
ISBN : 0128076992
Language : En, Es, Fr & De

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Book Description :

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

Numerical Methods Book
Author : M. K. Jain,Satteluri R. K. Iyengar,Rajinder Kumar Jain
Publisher : New Age International
Release : 2007-01-01
ISBN : 8122415342
Language : En, Es, Fr & De

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Book Description :

Is An Outline Series Containing Brief Text Of Numerical Solution Of Transcendental And Polynomial Equations, System Of Linear Algebraic Equations And Eigenvalue Problems, Interpolation And Approximation, Differentiation And Integration, Ordinary Differential Equations And Complete Solutions To About 300 Problems. Most Of These Problems Are Given As Unsolved Problems In The Authors Earlier Book. User Friendly Turbo Pascal Programs For Commonly Used Numerical Methods Are Given In The Appendix. This Book Can Be Used As A Text/Help Book Both By Teachers And Students.