Example 2
In the first two
frames, we see the standard process moving from x0 to
x1.
An interesting event occurs when the process moves from x1 to
x2. We see that the tangent line at x1
actually passes
through the x-axis at x0, so that x2 is,
in fact,
x0!
We have returned to our point of origin. This event forces the
process
to repeat itself forever. For example, notice that in finding x3,
since we start at x0 again, we must end up at x1
again. x3 is, in fact, x1! This situation is
called cycling,
and the pair of x-values x0 and x1 is
called a two-cycle for Newton's Method.
Copyright c 1996 by Steve Hetzler
and Bob Tardiff Department of Mathematics
and Computer Science, Salisbury State University, Salisbury, MD 21801.