Kalantari B. Polynomial Root-finding and Polynomiography

Singapore: World Scientific Publishing Company, 2008. - 492p.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.Approximation of Square-Roots and Their Visualizations;
The Fundamental Theorem of Algebra and a Special Case of Taylor s Theorem;
Introduction to the Basic Family and Polynomiography;
Equivalent Formulations of the Basic Family;
Basic Family as Dynamical System;
Fixed Points of the Basic Family;
Algebraic Derivation of the Basic Family and Characterizations;
The Truncated Basic Family and the Case of Halley Family;
Characterizations of Solutions of Homogeneous Linear Recurrence Relations;
Generalization of Taylor s Theorem and Newton s Method;
The Multipoint Basic Family and Its Order of Convergence;
A Computational Study of the Multipoint Basic Family;
A General Determinantal Lower Bound;
Formulas for Approximation of Pi Based on Root-Finding Algorithms;
Bounds on Roots of Polynomials and Analytic Functions;
A Geometric Optimization and Its Algebraic Offsprings;
Polynomiography: Algorithms for Visualization of Polynomial Equations;
Visualization of Homogeneous Linear Recurrence Relations;
Applications of Polynomiography in Art, Education, Science and Mathematics;
Approximation of Square-Roots Revisited;
Further Applications and Extensions of the Basic Family and Polynomiography.