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Numerical Methods For Roots Of Polynomials Part Ii

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Numerical Methods For Roots Of Polynomials Part Ii

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Author : J.M. McNamee
language : en
Publisher: Elsevier Inc. Chapters
Release Date : 2013-07-19

Numerical Methods For Roots Of Polynomials Part Ii written by J.M. McNamee and has been published by Elsevier Inc. Chapters this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-07-19 with Mathematics categories.

Numerical Methods For Roots Of Polynomials Part Ii

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Author : J.M. McNamee
language : en
Publisher: Elsevier Inc. Chapters
Release Date : 2013-07-19

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

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Author : J.M. McNamee
language : en
Publisher: Elsevier Inc. Chapters
Release Date : 2013-07-19

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 I

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Author : J.M. McNamee
language : en
Publisher: Elsevier
Release Date : 2007-08-17

Numerical Methods For Roots Of Polynomials Part I written by J.M. McNamee and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-08-17 with Mathematics categories.

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

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Author : J.M. McNamee
language : en
Publisher: Elsevier Inc. Chapters
Release Date : 2013-07-19

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

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Author : J.M. McNamee
language : en
Publisher: Elsevier Inc. Chapters
Release Date : 2013-07-19

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.

Computational Methods In Physics

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Author : Simon Širca
language : en
Publisher: Springer
Release Date : 2018-06-21

Computational Methods In Physics written by Simon Širca and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-06-21 with Science categories.

This book is intended to help advanced undergraduate, graduate, and postdoctoral students in their daily work by offering them a compendium of numerical methods. The choice of methods pays significant attention to error estimates, stability and convergence issues, as well as optimization of program execution speeds. Numerous examples are given throughout the chapters, followed by comprehensive end-of-chapter problems with a more pronounced physics background, while less stress is given to the explanation of individual algorithms. The readers are encouraged to develop a certain amount of skepticism and scrutiny instead of blindly following readily available commercial tools. The second edition has been enriched by a chapter on inverse problems dealing with the solution of integral equations, inverse Sturm-Liouville problems, as well as retrospective and recovery problems for partial differential equations. The revised text now includes an introduction to sparse matrix methods, the solution of matrix equations, and pseudospectra of matrices; it discusses the sparse Fourier, non-uniform Fourier and discrete wavelet transformations, the basics of non-linear regression and the Kolmogorov-Smirnov test; it demonstrates the key concepts in solving stiff differential equations and the asymptotics of Sturm-Liouville eigenvalues and eigenfunctions. Among other updates, it also presents the techniques of state-space reconstruction, methods to calculate the matrix exponential, generate random permutations and compute stable derivatives.

Numerical Methods For Roots Of Polynomials Part Ii

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Author : J.M. McNamee
language : en
Publisher: Elsevier Inc. Chapters
Release Date : 2013-07-19

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.

Algorithms And Complexity

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Author : Marios Mavronicolas
language : en
Publisher: Springer Nature
Release Date : 2023-04-24

Algorithms And Complexity written by Marios Mavronicolas and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-04-24 with Computers categories.

This book constitutes the refereed proceedings of the 13th International Conference on Algorithms and Complexity, CIAC 2023, which took place in Larnaca, Cyprus, during June 13–16, 2023. The 25 full papers included in this book were carefully reviewed and selected from 49 submissions. They cover all important areas of research on algorithms and complexity such as algorithm design and analysis; sequential, parallel and distributed algorithms; data structures; computational and structural complexity; lower bounds and limitations of algorithms; randomized and approximation algorithms; parameterized algorithms and parameterized complexity classes; smoothed analysis of algorithms; alternatives to the worst-case analysis of algorithms (e.g., algorithms with predictions), on-line computation and competitive analysis, streaming algorithms, quantum algorithms and complexity, algorithms in algebra, geometry, number theory and combinatorics, computational geometry, algorithmic game theory and mechanism design, algorithmic economics (including auctions and contests), computational learning theory, computational biology and bioinformatics, algorithmic issues in communication networks, algorithms for discrete optimization (including convex optimization) and algorithm engineering.

Scientific Computing

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Author : Michael T. Heath
language : en
Publisher: SIAM
Release Date : 2018-11-14

Scientific Computing written by Michael T. Heath and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-11-14 with Science categories.

This book differs from traditional numerical analysis texts in that it focuses on the motivation and ideas behind the algorithms presented rather than on detailed analyses of them. It presents a broad overview of methods and software for solving mathematical problems arising in computational modeling and data analysis, including proper problem formulation, selection of effective solution algorithms, and interpretation of results.? In the 20 years since its original publication, the modern, fundamental perspective of this book has aged well, and it continues to be used in the classroom. This Classics edition has been updated to include pointers to Python software and the Chebfun package, expansions on barycentric formulation for Lagrange polynomial interpretation and stochastic methods, and the availability of about 100 interactive educational modules that dynamically illustrate the concepts and algorithms in the book. Scientific Computing: An Introductory Survey, Second Edition is intended as both a textbook and a reference for computationally oriented disciplines that need to solve mathematical problems.

Numerical Methods For Roots Of Polynomials Part Ii

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