So... Newton's Method
is great, but does it ever not work? Think of some of the elementary functions such as 1/x, x, x^2, and
x^3. So now that we are imaging these functions, start thinking about how Newton's
Method works. We want to find the root of an equation. Then using an initial
guess, we iterate the equation
Using the function y=1/x, what does Newton's Method do? Well, the root will be found at infinity and at different infinities in fact, positive infinity from the positive side and negative infinity from the negative direction. But a problem arises from this. We would keep iterating forever! Why does this occur? Mainly because Newton's Method has problems with roots that go off to infinity.
What is another way that we can break Newton's Method you might ask? Well, lets look at the function y=x^2. As mentioned earlier, the TI-83 has a sign change error with this function. But we also know that Newton's method is not used in the TI-83, so would it have the same problem? After a little bit of calculations, and I recommend that you only spend a little bit of time on this, you will realize that the method could go on forever, never hitting that zero. Newton's Method is dependent on the derivative to find the root. If the function is either positive or negative before and after the root, what does that tell us about the root? That root in fact has a derivative of 0. Looking back at the equation for Newton's Method, you would see that the closer you got to the root, the closer you would get to dividing by a zero. Dividing by a zero is never good.
Speaking of dividing by a zero, what would happen if you chose a guess point that had a derivative of zero? When you draw the tangent line at a point with a slope of zero, that line will never hit the x-axis, unless of course that point is a root, in which case you have already found your root. This also tells us that there are points that can be chosen but will never lead us to roots. Some of these points are the mentioned points with derivatives of zero, but there are also a few troublesome points that look like decant points to pick.
Because the method uses tangent lines, there is the slight possibility that a point A will lead you to a point B. That point B then leads you to point C. Point C leads you to point D, but point D leads you back to point A! If we are hand calculating the root, we will notice that we have gotten to a point we have already seen before. But if a computer is doing it for us, it does not know that it is in this loop, unless we put in a safe guard to have it watch out for repeated points. If the loop happens to be over a million points in size, it could prove to be difficult to save every point to make sure that it does show up again. When using Newton's Method in the complex plane, these looping points cause some interesting effects to happen.
To summarize, we can break Newton's Method by
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