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Numerical Methods For Roots Of Polynomials Part Ii
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Book Synopsis Numerical Methods for Roots of Polynomials - Part I by : J.M. McNamee
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
Book Synopsis Numerical Methods for Roots of Polynomials - by : J.M. McNamee
Download or read book Numerical Methods for Roots of Polynomials - written by J.M. McNamee and published by Newnes. This book was released on 2013-07-19 with total page 728 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
Book Synopsis Numerical Methods for Roots of Polynomials - Part II by : J.M. McNamee
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 728 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Numerical Methods for Roots of Polynomials by : J. M. McNamee
Download or read book Numerical Methods for Roots of Polynomials written by J. M. McNamee and published by . This book was released on 2007 with total page 364 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book (along with volume 2 covers most of the traditional methods for polynomial root-finding such as Newtons, as well as numerous variations on them invented in the last few decades. Perhaps more importantly it covers recent developments such as Vincents 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. P - 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. P.
Book Synopsis Numerical Methods for Roots of Polynomials - Part II by : J.M. McNamee
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 728 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.
Book Synopsis Numerical Methods for Roots of Polynomials - Part II by : J.M. McNamee
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 728 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Numerical Methods for Roots of Polynomials - Part II by : J.M. McNamee
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 728 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Numerical Methods for Roots of Polynomials - Part II by : J.M. McNamee
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 728 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Numerical Methods for Roots of Polynomials - Part II by : J.M. McNamee
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 728 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Numerical Methods for Roots of Polynomials - Part II by : J.M. McNamee
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 728 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Numerical Methods for Roots of Polynomials - Part II by : J.M. McNamee
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 728 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Numerical Methods for Roots of Polynomials - Part II by : J.M. McNamee
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 728 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Numerical Methods for Roots of Polynomials - Part II by : J.M. McNamee
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 728 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Initial Approximations and Root Finding Methods by : Nikolay V. Kyurkchiev
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.
Book Synopsis Numerical Methods that Work by : Forman S. Acton
Download or read book Numerical Methods that Work written by Forman S. Acton and published by American Mathematical Soc.. This book was released on 2020-07-31 with total page 549 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A Graduate Introduction to Numerical Methods by : Robert M. Corless
Download or read book A Graduate Introduction to Numerical Methods written by Robert M. Corless and published by Springer Science & Business Media. This book was released on 2013-12-12 with total page 869 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an extensive introduction to numerical computing from the viewpoint of backward error analysis. The intended audience includes students and researchers in science, engineering and mathematics. The approach taken is somewhat informal owing to the wide variety of backgrounds of the readers, but the central ideas of backward error and sensitivity (conditioning) are systematically emphasized. The book is divided into four parts: Part I provides the background preliminaries including floating-point arithmetic, polynomials and computer evaluation of functions; Part II covers numerical linear algebra; Part III covers interpolation, the FFT and quadrature; and Part IV covers numerical solutions of differential equations including initial-value problems, boundary-value problems, delay differential equations and a brief chapter on partial differential equations. The book contains detailed illustrations, chapter summaries and a variety of exercises as well some Matlab codes provided online as supplementary material. “I really like the focus on backward error analysis and condition. This is novel in a textbook and a practical approach that will bring welcome attention." Lawrence F. Shampine A Graduate Introduction to Numerical Methods and Backward Error Analysis” has been selected by Computing Reviews as a notable book in computing in 2013. Computing Reviews Best of 2013 list consists of book and article nominations from reviewers, CR category editors, the editors-in-chief of journals, and others in the computing community.
Book Synopsis The Numerical Solution Of Systems Of Polynomials Arising In Engineering And Science by : Andrew J Sommese
Download or read book The Numerical Solution Of Systems Of Polynomials Arising In Engineering And Science written by Andrew J Sommese and published by World Scientific. This book was released on 2005-03-21 with total page 425 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
