by Bahman Kalantari (Rutgers University, USA)

Table of Contents (290k)
Preface (213k)
Chapter 1: Introduction (202k)

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