Chapter 5 - Roots of Nonlinear Equations

Assignment, Topics, and Practice Exercises

Assignment: Due Friday, April 9, 2004 by 4:00 pm

1. Finish the Root Finding Exploration (handout from Tuesday's class)

2. Do computer problem 5.7 on page 250 as follows:
   1. Write a program to do Newton's method (just modify one of the programs from the Exploration - that is the easiest thing to do).
   2. Run the program with the starting point given in the problem. Answer the problem's question of what happens AND using a graph of the function (you can easily get Mathematica to draw a graph for you if you want) illustrate what is going on.
   3. For part (b), use your program to answer the question. In particular, try several different starting points and determine the convergence properties (write them down). Write your observations about the roots (pathological?) and the behavior of Newton and relate it to the graph.

3. Do computer problem 5.8 on page 250 as follows:
   1. Do what the problem asks you to do. {NOTE: In Java, the arccosine function is Math.acos(x), the exponential function is Math.exp(x), the log function is Math.log(x), and the cosine function is Math.cos(x))
   2. The problem tells you to use 3 as the starting point for the iteration schemes. Try other starting values -- a wide variety in the range of 1 to 100. Comment on the convergence properties for the different strategies. Use a graph of the function to help explain what is going.

4. Modify the two Secant programs from the Root Exploration Lab Exercise to do the False Position method (a safeguarded variation of secant). Run your programs (for the two functions in the lab exercise) and compare the convergence properties (for example, how do the convergence rates seem to compare? are there starting points for which one method converges but the other doesn't?).

HAND IN: Turn in the output from your programs together with your written analyses (and graphs, where appropriate). Email a tar file of the programs to ingram@roanoke.edu.

Topics from the Chapter

• Sections 5.1 and 5.2 -- What is a non-linear equation? A system of non-linear equations? Existence and uniqueness of solutions: Anything can happen -- no nice theoretical results as there are for linear equations. Definition of multiple root
• Section 5.3 -- Sensitivity and Conditioning. How is root finding related to evaluation of a function? Why do multiple roots cause problems?
• Section 5.4 -- Convergence Rates and Stopping Criteria. What is the definition of convergence rate? What does the r in the definition tell you? What does the C tell you? How can you look at output from an iterative method and guess the convergence rate of the method? What are some typical stopping criteria for iterative algorithms?
• Bisection Method, Newton's Method, and Secant Method -- Section 5.5.1, 5.5.3, and 5.5.4 Know each algorithm, its convergence rate, its pros and cons, and its "geometry" (be able to illustrate what each method does with pictures)
• Fixed Point Iteration (Section 5.5.2): Know the algorithm, the relationship to root finding, the relationship to Newton (or vice versa), the convergence properties. Be able to find fixed point schemes to find roots of a function and determine whether or not the fixed point scheme will converge (and at what rate).
• Other methods (Section 5.5.5 and 5.5.7): Understand the idea of interpolation by functions more complex than the linear ones used in Newton and secant; understand why inverse interpolation is more reasonable than interpolation; Know what is meant by a safeguarded method.

Other Practice/Review Exercises

• Review Questions: pages 246-248: #5.1-5.5, 5.9 - 5.20, 5.23, 5.24, 5.27 - 5.29 - 5.31
• Exercises: Pages 248 - 250: #5.1 - 5.4, 5.6, 5.11
• Computer Problem: Page 250, #5.2 (analysis of convergence rate)