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 :
Let’s try with close to the positive root.
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,
Suppose we try a value for on the left of the root.
Suppose we try a value for on the left of the root.
Suppose we try a value for on the left of the root.
Suppose we try a value for on the left of the root.
Suppose we try a value for on the left of the root. The sequence now converges but to the other root !
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 ,
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
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
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.
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.
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.
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.
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 .
SUMMARY To show that a formula of the type will give a convergent sequence, find show that , where x is close to the solution.