|Forlag||World Scientific Publishing Co|
|Emne||Algebra; Applied mathematics|
|Se flere detaljer|
Om Polynomial Root-Finding and Polynomiogra
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.Contents:Approximation of Square-Roots and Their VisualizationsThe Fundamental Theorem of Algebra and a Special Case of Taylor's TheoremIntroduction to the Basic Family and PolynomiographyEquivalent Formulations of the Basic FamilyBasic Family as Dynamical SystemFixed Points of the Basic FamilyAlgebraic Derivation of the Basic Family and CharacterizationsThe Truncated Basic Family and the Case of Halley FamilyCharacterizations of Solutions of Homogeneous Linear Recurrence RelationsGeneralization of Taylor's Theorem and Newton's MethodThe Multipoint Basic Family and Its Order of ConvergenceA Computational Study of the Multipoint Basic FamilyA General Determinantal Lower BoundFormulas for Approximation of Pi Based on Root-Finding AlgorithmsBounds on Roots of Polynomials and Analytic FunctionsA Geometric Optimization and Its Algebraic OffspringsPolynomiography: Algorithms for Visualization of Polynomial EquationsVisualization of Homogeneous Linear Recurrence RelationsApplications of Polynomiography in Art, Education, Science and MathematicsApproximation of Square-Roots RevisitedFurther Applications and Extensions of the Basic Family and PolynomiographyReadership: Researchers in numerical and computational mathematics, complex systems and approximation theory.