Polynomial Root-finding and Polynomiography
Polynomial Root-finding and Polynomiography

Author: Bahman Kalantari

Publisher: World Scientific

Published: 2009

Total Pages: 492

ISBN-13: 9812700595

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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.

Numerical Methods for Roots of Polynomials - Part II
Numerical Methods for Roots of Polynomials - Part II

Author: J.M. McNamee

Publisher: Elsevier Inc. Chapters

Published: 2013-07-19

Total Pages: 150

ISBN-13: 0128076976

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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.

Approximations for Digital Computers
Approximations for Digital Computers

Author: Cecil Hastings Jr.

Publisher: Princeton University Press

Published: 2015-12-08

Total Pages: 212

ISBN-13: 1400875595

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Numerical analysts and computer operators in all fields will welcome this publication in book form of Cecil Hastings' well-known approximations for digital computers, formerly issued in loose sheets and available only to a limited number of specialists. In a new method that combines judgment and intuition with mathematics, Mr. Hasting has evolved a set of approximations which far surpasses in simplicity earlier approximations developed by conventional methods. Part I of this book introduces the collection of useful and illustrative approximations, each of which is presented with a carefully drawn error curve in Part II. Originally published in 1955. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Numerical Methods for Roots of Polynomials - Part I
Numerical Methods for Roots of Polynomials - Part I

Author: J.M. McNamee

Publisher: Elsevier

Published: 2007-08-17

Total Pages: 354

ISBN-13: 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