Fixed-Point Iteration
Summary
Suppose that you want to solve the equation
The value of x that satisfies this equation is called a fixed point
for the function 
, because it is a point such that g(x)=x -
the image is the same as the argument.
One way to go about finding the fixed point would be to rewrite the equation as
and to use bisection to find a root of f (in fact, the unique root, as one can see from the graph of f). Here it is in lisp:
Fixed-point iteration is based on a couple of results from calculus: the IVT, and the MVT, as follows:
Theorem 2.2:
and 
for all 
, then g has a
fixed point in [a,b]. 
then the fixed point in [a,b] is unique.
So we know that there's a fixed point on an interval [a,b], and may even know that it's unique. What now?
Now we assume that, perhaps, if we start with a value 
that's close to the
real fixed point p, that by simply computing 
(which is 
) we'll actually get closer to p.
Let's look at the ``cobweb diagram'' of this situation.
Here it is in lisp:
Under what circumstances will that happen? In what circumstances would the same ``cobwebbing'' procedure fail? Here it doesn't work too well: convergence is slow. Why?
Here it just plain fails....
Well, in some circumstances, it's guaranteed to work:
Theorem 2.3: Fixed-Point Theorem Let be such that , for all x in [a,b]. Suppose, in addition, that g' exists on
(a,b), and that a constant 0<k<1 exists with
for all 
. Then for any number 
, the sequence
, converges to the unique fixed point p in [a,b].
Proof: MVT applied to 
.
Corollary 2.4: If g satisfies the hypotheses of Theorem 2.3, then
bounds for the error involved in using 
to approximate p are given by
and
for all .
Proof: by use of various inequalities.
There may be lots of ways to create a fixed-point function, and some of them are better than others.
Example: consider exercises 1 and 2.