While trying to discover why the TI-89 has the above mentioned problems, we did some internet research to try to find out about other root finding methods. We came across a very useful page:
http://www.efunda.com/math/num_rootfinding/num_rootfinding.cfm,which described not only Newton's Method, but also the Bisection, Secant, and False Position Method's. It was really interesting to see the other methods, and as mentioned above, to realize that other methods are used to find the roots.
The first method on the webpage is the Bisection Method. It is a very slow and tedious method that works on the assumption that the function changes signs at the root. We have seen what this assumption can lead to with function like y=x^2. The next method is the Secant Method, which is an extension of the Bisection method, but is not nearly as slow and tedious as the Bisection method. It works on the assumption that a function is approximately linear near the root. It can screw up if the root is not smooth at that root because of this assumption.
An improvement on the Secant Method is the False-Position Method. It improves on the secant method by replacing values where the secant method does not; thereby requiring more calculations but also allowing it to find more difficult roots. As you can tell, the root finding methods lead into each other, Bisection to Secant to False-Position Method; each requires an interval containing at least one root. (Specifically, they require an interval over which the function changes sign). The Newton Method, (on the above page referred to as the Newton-Raphson Method), works on a completely different basis: it needs only an intial guess and the derivative at that point.
Computers and Calculators, for the most part, use one of the interval-based methods. This is because evaluating the derivative at a point is computationally expensive.
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