Polynomial Root-finding and Polynomiography

Bahman Kalantari

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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.

Contents: 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.

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Review:

Bahman Kalantari has created a beautiful new genre of mathematical visual art, that is quite distinct from Fractal Art, and is just as beautiful. Not only is the art beautiful, but the mathematics and the elegant algorithms that generate it. This book can be read on quite a few levels, all very rewarding, and will inspire lots of future research and new gorgeous art. --Doron Zeilberger, Rutgers University, Winner of the Steele Prize

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