Contents -- Preface -- Introduction -- 1. Approximation of Square-Roots and Their Visualizations -- 1.1 Introduction -- 1.2 A Simple Algebraic Method for Approximation of Square- Roots -- 1.3 High-Order Algebraic Methods for Approximation of Square-Roots -- 1.4 Convergence Analysis -- 1.5 Approximation of Square-Roots from Complex Inputs -- 1.6 The Basic Sequence and Fixed Point Iterations -- 1.7 Determinantal Representation of High-Order Iteration Functions and Basic Sequence -- 1.8 Visualizations in Approximation of Square-Roots -- 1.9 High-Order Methods for Approximation of Cube-Roots -- 1.10 Complexity of Sequential Versus Parallel Algorithms -- 1.11 Extensions -- 2. The Fundamental Theorem of Algebra and a Special Case of Taylor's Theorem -- 2.1 Introduction -- 2.2 Algebraic Derivation of Newton's Method -- 2.3 A Recurrence Relation and the Basic Family -- 2.4 Conclusions -- 3. Introduction to the Basic Family and Polynomiography -- 3.1 Introduction -- 3.2 The Basic Family and its Properties -- 3.3 Polynomiography and Its Applications -- 4. Equivalent Formulations of the Basic Family -- 4.1 Determinantal Formulation of the Basic Family -- 4.2 Properties of a Determinant -- 4.3 Gerlach's Method -- 4.4 Equivalence to the Basic Family -- 4.5 KÄonig's Family and Equivalence to the Basic Family -- 4.6 Notes and Remarks -- 5. Basic Family as Dynamical System -- 5.1 Introduction -- 5.2 Iterations of a Rational Function -- 5.3 Newton's Method and Connections to Mandelbrot Set -- 5.4 Analysis of Infinity as Fixed Point -- 5.5 MÄobius Transformations and Conjugacy -- 5.6 Periodic Points and Cycles of a Rational Function -- 5.7 Critical Points and Their Cardinality -- 5.8 Cardinality of Periodic Points of Different Types -- 5.9 Local Behavior of Iterations Near Fixed Points.

5.10 Local Behavior of Iterations Near General Points: Equicon- tinuity and Normality -- 5.11 Fatou and Julia Sets and Their Basic Properties -- 5.12 Montel Theorem and Characterization of Fatou and Julia Sets -- 5.13 Fatou and Julia Sets as: The Good, The Bad, and The Undesirable -- 5.14 Fatou Components and Their Dynamical Properties -- 5.15 Critical Points and Connection with Periodic Fatou Components -- 5.16 Fatou-Julia and Topological Fatou-Julia Graphs: Analo- gies for Visualization and Conceptualization of Dynamics -- 5.17 Lakes and Waterfalls: Analogy for Dynamics of Rational Maps -- 5.18 General Convergence: Algorithmic Limitation of Iterations -- 5.19 A Summary for the Behavior of Iteration Functions -- 5.20 Undecidability Issues in Rational Functions -- 6. Fixed Points of the Basic Family -- 6.1 Introduction . -- 6.2 Properties of the Fixed Points of the Basic Family -- 6.3 Proof of Main Theorem -- 7. Algebraic Derivation of the Basic Family and Characterizations -- 7.1 Introduction -- 7.2 Algebraic Proof of Existence of the Basic Family -- 7.3 Derivation of Closed Form of the Basic Family -- 7.4 Two Formulas for Generation of Iteration Functions -- 7.5 Deriving the Euler-SchrÄoder Family -- 7.6 Extension to Non-Polynomial Root Finding -- 7.7 Conclusions -- 8. The Truncated Basic Family and the Case of Halley Family -- 8.1 The Halley Family -- 8.2 The Order and Asymptotic Error of Halley Family -- 8.3 The Truncated Basic Family -- 8.4 Applications -- 8.5 Polynomiography with the Truncated Basic Family -- 8.6 Conclusions -- 9. Characterizations of Solutions of Homogeneous Linear Recurrence Relations -- 9.1 Introduction -- 9.2 Homogeneous Linear Recurrence Relations -- 9.3 Explicit Representation of the Fundamental Solution -- 9.4 Explicit Representation Via Characteristic Polynomial -- 9.5 Approximation of Polynomial Roots Using HLRR.

9.6 Basic Sequence and Connection to the Basic Family -- 9.7 The Basic Sequence and the Bernoulli Method -- 9.8 Determinantal Representation of Fundamental Solution -- 9.9 Application to Fibonacci Sequence and Generalizations -- 9.10 Experimental Results Via Polynomiography -- 9.11 A Representation Theorems for Arbitrary Solutions -- 9.12 Applications to Fibonacci and Lucas Numbers -- 9.13 Concluding Remarks -- 10. Generalization of Taylor's Theorem and Newton's Method -- 10.1 Introduction -- 10.2 Taylor's Theorem with Conuent Divided Differences -- 10.2.1 Basic Applications . -- 10.3 The Determinantal Taylor Theorem -- 10.3.1 Determinantal Interpolation Formulas -- 10.4 Proof of Determinantal Taylor Theorem and Equivalent Form -- 10.5 Applications of Determinantal Formulas -- 10.5.1 Infinite Spectrum of Rational Approximation Formulas -- 10.5.2 Infinite Spectrum of Rational Inverse Approximation Formulas -- 10.5.3 Infinite Families of Single and Multipoint Iteration Functions -- 10.5.4 Determinantal Approximation of Roots of Polynomials -- 10.5.5 A Rational Expansion Formula and Connection to Pad e Approximant -- 10.5.6 Algebraic Approximation Formulas -- 10.6 Concluding Remarks -- 11. The Multipoint Basic Family and its Order of Convergence -- 11.1 Introduction -- 11.2 The Multipoint Basic Family -- 11.3 Description of the Order of Convergence -- 11.4 Proof of the Order of Convergence -- 12. A Computational Study of the Multipoint Basic Family -- 12.1 Introduction -- 12.2 The Iteration Functions -- 12.3 The Iteration Complexity -- 12.4 The Experiment -- 12.5 Conclusions -- 13. A General Determinantal Lower Bound -- 13.1 Introduction -- 13.2 An Application in Approximation of Polynomial Root -- 13.3 Conclusions -- 14. Formulas for Approximation of Pi Based on Root-Finding Algorithms -- 14.1 Introduction -- 14.2 Main Results -- 14.3 Auxiliary Results.

14.4 Proof of Main Theorems -- 14.5 Applications in Approximation of -- 14.6 Special Formulas for Approximation of -- 14.7 Approximation of Via the Basic Family -- 14.8 A Formula for Approximation of e -- 14.9 Concluding Remarks -- 15. Bounds on Roots of Polynomials and Analytic Functions -- 15.1 Introduction -- 15.2 Estimate to Zeros of Analytic Functions -- 15.3 The Basic Family for General Analytic Functions -- 15.4 Application of Basic Family in Separation Theorems -- 15.5 Estimate to Nearest Zero and Bounds on Zeros -- 15.6 Applications, Asymptotic Analysis, Computational E - ciency and Comparisons -- 15.7 Concluding Remarks -- 16. A Geometric Optimization and its Algebraic O springs -- 16.1 Introduction -- 16.2 Elementary Proof of the Gauss-Lucas Theorem and the Maximum Modulus Principle -- 16.3 The Gauss Lucas Iteration Function and Extensions of the Maximum Modulus Principle -- 16.4 Conclusions -- 17. Polynomiography: Algorithms for Visualization of Polynomial Equations -- 17.1 A Basic Coloring Algorithm -- 17.2 Basic Family and Variants: The Basis of Polynomiography -- 17.3 Many Polynomiographs of Cubic Roots of Unity -- 18. Visualization of Homogeneous Linear Recurrence Relations -- 18.1 Introduction -- 18.2 The Generalized Fibonacci, the Hyper Fibonacci, and their Polynomiography -- 18.3 The Induced Basic Family and Induced Basic Sequence -- 18.4 The Fibonacci and Lucas Families of Iteration Functions -- 18.5 Visualization of HLRR with Arbitrary Initial Conditions -- 19. Applications of Polynomiography in Art, Education, Science and Mathematics -- 19.1 Polynomiography in Art -- 19.1.1 Polynomiography as a Tool of Art and Design -- 19.1.2 Polynomiography Based on Voronoi Coloring -- 19.1.3 Polynomiography Based on Levels of Convergence -- 19.1.4 Symmetric Designs from Polynomiography -- 19.1.5 Polynomiography of Numbers.

19.1.6 Some Extensions of Polynomiography -- 19.1.7 Glossary of Terms -- 19.2 Polynomiography in Education -- 19.2.1 Polynomiography for Encouraging Creativity in Education -- 19.2.2 Teacher Survey -- 19.2.3 Student Survey -- 19.2.4 Developing Seminars and Courses Based on Polynomiography -- 19.3 Polynomiography in Mathematics and Science -- 19.3.1 Polynomiography for Measuring the Average Performance of Root- nding Algorithms -- 19.4 Conclusions -- 20. Approximation of Square-Roots Revisited -- 20.1 Regular Continued Fractions and the Basic Family -- 20.2 Regular Continued Fraction Convergents Versus Basic Sequence Convergents -- 20.3 Applications of Continued Fractions and Basic Sequence in Factorization -- 20.4 Basic Sequence for Approximation of Higher Roots of a Number and its Factorization -- 21. Further Applications and Extensions of the Basic Family and Polynomiography -- 21.0.1 Extensions to Analytic Functions -- 21.0.2 Extensions to Other Dimensions or Domains -- 21.0.3 Polynomiography for Designing Shapes -- 21.1 Toward a Digital Media Based on Polynomiography -- Bibliography -- Index.