# Chapter 5 - Roots of Nonlinear Equations

### Assignment, ETC.

### Assignment: Due Thursday, March 21, 2002

Do computer problem 5.2 on page 250. For part (a) be sure to give
a thorough analysis of the convergence properties. Determine whether
or not fixed point iteration converges; if it does converge what
the rate is; if the rate is linear, the number of iterations it
would take to gain a digit of accuracy. Show all work.
For part (b) I recommend
modifying Newton1.java (one of the programs you ran in the lab before break).
Turn in the
output from your programs for each different g together with a
written description of how the actual convergence compared to
that predicted from part (a). Run the program for each g with
at least two different starting points and note any differences in
behavior.

### 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