Further Reading

Here are some resources we found useful.

• Course Text: Mathematics for High School Teachers: An Advanced Perspective by Peressini, Usiskin, Marchisotto and Stanley -- the textbook for our course; this project is one of the ones from chapter 4.
• Newton's Method Basics:
  • MathWorld: Newton's Method
  • Wikipedia: Newton's Method
  • Interactive Newton's Method Iterating Applet
• Calculators:
  • We'd like to gratefully acknowledge the assistance of the TI calculators helpdesk. Marcus worked with them over several days to understand the perplexities of finding roots on the calculator (and how the calculator chooses to fail in the degenerate case). They were very friendly and forthcoming.
  • EFunda: Root Finding
• Newton's Method Fractals:
  • Nonlinear Dynamics and Chaos: With Applications by Steven H. Strogatz. An excellent and accessible introduction to Nonlinear Dynamics and Chaos. Learn about general features of dynamical systems like those we met exploring Newton's Method as an iterative map: fixed points, basins of attraction, stability, self-similarity, patterns and chaos. Many connections to biology, engineering, and physics.
  • The Beauty of Fractals: Images of Complex Dynamical Systems by Heinz-Otto Peitgen. An old book with beautiful pictures, somewhat clear writing, and useable pseudo-code to generate fractals.
  • The Science of Fractal Images by Heinz-Otto Peitgen, Dietmar Saupe. Somewhat dense; a hit-and-miss early collection of essays.
  • An Introduction to Chaotic Dynamical Systems by Robert L. Devaney. An advanced text.
• Powerpoint Presentation: Here is a Powerpoint Presentation that we gave in our class. It covers the same ground as this webpage.
• Mathematica Notebooks: If you have Mathematica, you may be able to make use of the Mathematica notebooks we used to generate the graphs in this webpage. Warning: they've not been cleaned up, so you're on your own to get them to work for you.
  • NewtonsMethod.nb -- Two different algorithms to generate Newton's Method fractals.
  • ZeroFinding.nb -- All of the polynomial-with-zero graphs, and the map of real-initial-guesses.
  • A cleaner example of Mathematica code for exploring Newton's Method.

In closing, here is a quote from mathematician Adrien Douady:

"I must say that, in 1980, whenever I told my friends that I was just starting, with J.H. Hubbard, a study of polynomials of degree 2 in one complex variable (and more specifically those of the form z_n+1 <- z_n² + c), they would all stare at me and ask: 'Do you expect to find anything new?'"