MATH 163, Spring 2001 Due Date: Name(s):
Honors Project 1a: The Iteration Method
Objective
In addition to the Bisection Method and Newton's Method, an approximate solution to the equation
f(x) = 0 can be found with the Iteration Method. The Iteration Method often arises in applications such as
economics in the form of "cobweb cycles". In this project we discuss the Iteration Method.
Narrative
To find an approximate solution to the equation f(x) = 0 by the Iteration Method, look for a value of x
for which g(x) = x for an appropriate function g(x); an x for which g(x) = x is called a fixed point of g. For
example, to find an approximate solution to the equation x2
- 2 = 0 we might we choose an arbitrary value
for x0  [1, 1.99] and repeatedly compute xn using the formula
xn = g(xn-1), n = 1, 2, 3, . . .
where g(x) = -1
2 (x2
- 2) + x. The reason the iteration method works is that
g(x) = -
1
2
(x2
- 2) + x = x if and only if x2
- 2 = 0.
The reason we didn't simply let g(x) = x2
- 2 + x is that with this choice of g, the Iteration Method does
not converge.
One strength of the Iteration Method is that after the appropriate function g and initial value x0 are
identified, it is fairly easy to implement. One weakness of it is that given f you have to come up with g, and
this is not always easy.
Task
a) Type the command lines below into Maple in the order in which they are listed. These command lines
illustrate how the iteration method as described in the Narrative converges. (Note again how little effort it
takes to implement this method!)
> # Honors Project 1a: The Iteration Method
> restart:
> # Task a
> g := x -> -(x^2-2)/2+x;
> g(1.5);
> for i from 1 to 20 do g(%) od;
b) Continue by typing the following command lines into Maple. These command lines illustrate how the
iteration method as described in the Narrative diverges.
> # Task b
> g := x -> x^2-2+x;
> g(1.5);
> for i from 1 to 20 do g(%) od;
At this time make a hard-copy of your typed input and Maple's responses. Then, ...
The condition that determines whether the iteration method converges is |g (x)| < 1 for all x in some
closed interval I containing the root of the equation g(x) = 0.
c) Show by hand that if g(x) = -1
2 (x2
- 2) + x then |g (x)| = | - x + 1| < 1 for all x  [1, 1.99].
d) Show by hand that if g(x) = (x2
- 2) + x then |g (x)| = |2x + 1| < 1 for any x  [1, 1.99].
Comments
1. As indicated above, one of the strengths of the iteration method is that after "set-up", it is very easy to
program. Since "set-up" includes finding a function g satisfying the convergence condition, and since it
is not always "obvious" how to do this, getting through the "set-up" can be a weakness of the iteration
method.
2. The iteration method can be extended to multiple variables.
3. The use of the iteration method in applications is one of the factors that motivates the study of "fixed
point theory" in mathematics.