1. “Teach A Level Maths” Vol. 2: A2 Core Modules
18: Iteration Diagrams and Convergence
© Christine Crisp

2. Module C3
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3. In the previous presentation, we found the approximate solution to
Using the formula with
we got
The solution to the equation we called a.
We can illustrate the iteration process on a diagram.

4. For we draw
and
So, a is the x-coordinate of the point of intersection.
( to 6 d.p. )
We’ll zoom in on the graph to enlarge the part near the intersection.

5. We started the iteration with
and we substituted to get .
On the diagram we draw from to the curve . . .

6. y-value, We started the iteration with and we substituted to get .
On the diagram we draw from to the curve . . . x
which gives a
y-value,

7. y-value, We started the iteration with and we substituted to get .
On the diagram we draw from to the curve . . . x x
which gives a
y-value,
By drawing across to y = x this y-value gives the point where x This is .

8. In the iteration, is now substituted into to give .
On the diagram we draw from
to the curve . . . x x
giving the next y-value. x
The next line converts this y-value to and so on.

9. In the iteration, is now substituted into to give .
On the diagram we draw from
to the curve . . . x x
giving the next y-value. x
The next line converts this y-value to and so on.
The lines are drawn to the curve and y = x alternately, starting by joining to the curve.

10. The diagram we’ve drawn illustrates a convergent, oscillating sequence.
This is called a cobweb diagram.

11. Looking at the original graph, we have

12. SUMMARY
To draw a diagram illustrating iteration:
Draw and on the same axes.
Mark on the x-axis and draw a line parallel to the y-axis from to ( the curve ).
Continue the cobweb line, going parallel to the x-axis to meet
Repeat
Continue the cobweb line, going parallel to the y-axis to meet

13. Iteration does not always give an oscillating sequence.
We can also draw a diagram for sequences which iterate directly towards the solution.
I am going to use an example from the previous presentation: (a rearrangement of )
If you have a graphical calculator, before you look at my solution, you might like to have a go. You’ll need to zoom in to see the graphs very close to the solution, for example, from to
Copy this part of the graphs quite carefully onto paper, mark at and off you go.
( Remember you don’t need the values of etc. )

14. Before we zoom in, the graphs look like this.
The solution has 2 roots.
We will find the one nearest to the origin.

15. So, to solve using with
This is called a staircase diagram.

16. Exercise
Copy the following graphs and sketch the diagrams which illustrate the convergence of the iterative process, showing
(a)
(b)
Write on the names of the diagrams.

17. Solution: (a)
A cobweb diagram.

18. Solution: (b)
A staircase diagram.

19. In the next example we’ll look at an equation which has 2 roots and the iteration produces a surprising result.
The equation is
We’ll try the simplest iterative formula first :
If you have Autograph or another graph plotter you may like to try to find both roots before you see my solution.

20. Let’s try with close to the positive root.

21. The sequence diverges rapidly.
Let’s try with close to the positive root.
The staircase moves away from the root.
The sequence diverges rapidly.
Using the iterative formula,

22. Suppose we try a value for on the left of the root.

23. Suppose we try a value for on the left of the root.

24. Suppose we try a value for on the left of the root.

25. Suppose we try a value for on the left of the root.

26. Suppose we try a value for on the left of the root.
The sequence now converges but to the other root !

27. It is possible to use our iterative method to find in the previous example but not with the arrangement we had.
The equation was
The rearrangement we used was so the formula was
With this gives the negative root:
We can also arrange the equation as follows:
Change to log form:
giving
With ,

28. We will now look at why some iterative formulae give sequences that converge whilst others don’t and others converge or diverge depending on the starting value.
Collecting the diagrams together gives us a clue.
I’ve included the 4th type of diagram that we haven’t yet met: a diverging cobweb.
See if you can spot the important difference once you can see the 4 diagrams

29. The gradients of for the converging sequences are shallow
Cobweb: converging
Staircase: converging
The gradients of for the converging sequences are shallow
Cobweb: diverging
Staircase: diverging

30. It can be shown that gives a convergent sequence if the gradient of . . .
is between -1 and +1 at the root.
We write
or,
Unfortunately since we are trying to find we don’t know its value and can’t substitute it !
In practice, to test for convergence we use a value close to the root.
The closer is to zero, the faster will be the convergence.

31. e.g. By using calculus, determine which of the following arrangements of the equation
will give convergence to a root near x = 3 and which will not.
Solution:
The sequence will converge.

32. e.g. By using calculus, determine which of the following arrangements of the equation
will give convergence to a root near x = 3 and which will not.
Solution:
The sequence will not converge.

33. We can now see why we had the strange result when we tried to solve with
At , the gradient is less than 1 ( so the curve here is shallower than y = x ): the iteration converges.
At , the gradient is greater than 1 ( so the curve here is steeper than y = x ): the iteration diverges.

34. We can now see why we had the strange result when we tried to solve with
Both iterations started close to but the one that converged to started on the left of .

35. SUMMARY
To show that a formula of the type
will give a convergent sequence,
find
show that , where x is close to the solution.

37. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.
For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

38. SUMMARY
To draw a diagram illustrating iteration:
Draw and on the same axes.
Mark on the x-axis and draw a line parallel to the y-axis from to ( the curve ).
Continue the line, going parallel to the x-axis to meet
Continue the line, going parallel to the y-axis to meet
Repeat
4 different diagrams are possible.

39. The gradients of for the converging sequences are between -1 and +1
Staircase: diverging
Cobweb: converging
Staircase: converging
Cobweb: diverging