Book contents
- Frontmatter
- Note to the Reader
- Preface
- Contents
- 1 Transformations and their Iteration
- 2 Arithmetic and Geometric Means
- 3 Isoperimetric Inequality for Triangles
- 4 Isoperimetric Quotient
- 5 Colored Marbles
- 6 Candy for School Children
- 7 Sugar Rather Than Candy
- 8 Checkers on a Circle
- 9 Decreasing Sets of Positive Integers
- 10 Matrix Manipulations
- 11 Nested Triangles
- 12 Morley's Theorem and Napoleon's Theorem
- 13 Complex Numbers in Geometry
- 14 Birth of an IMO Problem
- 15 Barycentric Coordinates
- 16 Douglas-Neumann Theorem
- 17 Lagrange Interpolation
- 18 The Isoperimetric Problem
- 19 Formulas for Iterates
- 20 Convergent Orbits
- 21 Finding Roots by Iteration
- 22 Chebyshev Polynomials
- 23 Sharkovskii's Theorem
- 24 Variation Diminishing Matrices
- 25 Approximation by Bernstein Polynomials
- 26 Properties of Bernstein Polynomials
- 27 Bézier Curves
- 28 Cubic Interpolatory Splines
- 29 Moving Averages
- 30 Approximation of Surfaces
- 31 Properties of Triangular Patches
- 32 Convexity of Patches
- Appendix A Approximation
- Appendix B Limits and Continuity
- Appendix C Convexity
- Bibliography
- Hints and Solutions
- Index
This chapter is part of a book that is no longer available to purchase from Cambridge Core
19 - Formulas for Iterates
Book contents
- Frontmatter
- Note to the Reader
- Preface
- Contents
- 1 Transformations and their Iteration
- 2 Arithmetic and Geometric Means
- 3 Isoperimetric Inequality for Triangles
- 4 Isoperimetric Quotient
- 5 Colored Marbles
- 6 Candy for School Children
- 7 Sugar Rather Than Candy
- 8 Checkers on a Circle
- 9 Decreasing Sets of Positive Integers
- 10 Matrix Manipulations
- 11 Nested Triangles
- 12 Morley's Theorem and Napoleon's Theorem
- 13 Complex Numbers in Geometry
- 14 Birth of an IMO Problem
- 15 Barycentric Coordinates
- 16 Douglas-Neumann Theorem
- 17 Lagrange Interpolation
- 18 The Isoperimetric Problem
- 19 Formulas for Iterates
- 20 Convergent Orbits
- 21 Finding Roots by Iteration
- 22 Chebyshev Polynomials
- 23 Sharkovskii's Theorem
- 24 Variation Diminishing Matrices
- 25 Approximation by Bernstein Polynomials
- 26 Properties of Bernstein Polynomials
- 27 Bézier Curves
- 28 Cubic Interpolatory Splines
- 29 Moving Averages
- 30 Approximation of Surfaces
- 31 Properties of Triangular Patches
- 32 Convexity of Patches
- Appendix A Approximation
- Appendix B Limits and Continuity
- Appendix C Convexity
- Bibliography
- Hints and Solutions
- Index
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- Over and Over Again , pp. 109 - 114Publisher: Mathematical Association of AmericaPrint publication year: 1997