Mathematical Investigations IV
Name
Mathematical Investigations IV
Iteration Forever
FIXED POINTS
In each of the problems on sheet 5, we looked for fixed points of a sequence of the form:
if n  1
 k
xn  
 f  xn1  if n  1
To find this fixed point, we looked for an input such that the output was equal to the input. In
other words, we solved xn  xn1 , or f  xn1   xn1 . Replacing xn 1 with x , this simplifies to
x  f ( x)
(1)
Confirm this by going back to sheet 5 and use equation (1) to find the fixed point for each
problem.
Prob. 2-4. Solve:
Prob. 5.
Prob. 6.
x = 3x – 1
It is not always reasonable, or even possible, to solve those equations in order to find a fixed
point. In these cases, the iterative technique is a useful solution method.
Let's consider more equations to solve and explore how the calculator can assist our work.
1.
Consider the equation x3  x  1  0 . Creatively rewrite it in the form of equation (1):
x3  x  1  0
 x 2  1  x  1
x
1
x 1
2
which is of the form x   ( x) , where  ( x ) 
1
.
x 1
2
Fill in the table with decimal approximations. Let the seed be 1 and xn 
n
1
xn
1
2
3
4
5
1
.
( xn 1 ) 2  1
6
If n increases without limit, what is the value of the fixed point?
Seq & Ser. 6.1
Rev. F07
Mathematical Investigations IV
Name
You should question why we chose to solve for x in such a convoluted way. Why not just
let x  1  x3 ? Try this and see what happens.
Let the seed be 1 and xn  1  ( xn 1 )3 .
n
1
xn
1
2
3
4
5
6
What is the value of the fixed point? Would a different seed help?
You now have two ways in which to find the fixed point of a sequence, if it exists:

Repeatedly using the ENTER button on your calculator.

Solving equation (1), x   ( x) .
Fixed Points and Attracting Fixed Points
A fixed point is any point (x0, x0) such that  ( x0 )  x0 .
An attracting fixed point is any point (x0, x0) such that there exists a sequence x1, x2, x3, ...
(with x1 ≠ x0) that approaches x0 when xn  f ( xn1 ) and n increases without limit.
An attracting fixed point (sometimes called a convergent point) is always a fixed point, but the
converse is not necessarily true. For example: If f ( x)  x , then the fixed points are
x = 0 and x = 1, but only x = 1 is a attracting point. If you start with any seed other than x = 0, the
sequence xn   ( xn 1 ) will converge to 1. There is no seed other than x = 0 that will produce a
sequence xn   ( xn 1 ) that will converge to 0.
Since there is only one seed that can be used to "locate" a fixed point that is not an attracting
point, such fixed points may be difficult to determine using any method other than solving the
equation x   ( x) .
Seq & Ser. 6.2
Rev. F07
Mathematical Investigations IV
Name
Find out if each of the following sequences converges (or does not converge), using one or more
of the above techniques.
2.
Let xn  5ln( xn1 )
n
1
xn
2
2
3
4
5
6
[Question: Why can't we use c = 1 as the seed?]
To what fixed point does this sequence converge?
In this case, for our equation x = ƒ(x), what is ƒ(x)?
Are there any fixed points that are not convergent points? How do you know? If so, find
them.
Can you think of other ways to solve the equation x = 5 ln(x)?
3.
Let xn  cos(xn1 )
n
1
xn
1
[Be sure your calculator is in radian mode.]
2
3
4
5
6
What is the fixed point?
In this case, what is  ( x) ?
Seq & Ser. 6.3
Rev. F07
Mathematical Investigations IV
Name
4.
Now let xn  4cos( xn1 ) .
[Be sure your calculator is still in radian mode.]
Experiment with some different seed values. (Two seeds are given.)
n
1
2
xn
–1
xn
1
3
4
Does this
seem to
converge?
5
xn
xn
a.
What do you notice so far about
convergent points?
c.
How many fixed points might
you expect?
b.
Sketch the graphs of y = x and
y  4 cos( x) .
How many convergent points did
you find?
5.
1
Let xn   
2
xn1
n
1
xn
1
2
What is the convergent point?
3
4
5
6
Is there a fixed point that is not a convergent
point? Justify your answer.
Seq & Ser. 6.4
Rev. F07