So the reason that I (Marcus) got interested in this project had to due with the TI-83 and 89 calculators. It always bugged me that the 89 would give wrong answers for the same question that an 83 would give a correct answer for. After all, the 89 is the more advanced machine – should it not be more accurate? When I read the problem in the book, it sounded like Newton's method might be what the calculators used. I had never heard of root finding methods, and so I assumed that there was only way to do it (especially if Newton was the creator).
Here's one way to get the error that I am talking about. The function y=x^2 has only one real root, at x=0. Yet the calculator (for reasons you'll see in the next section) has grave issues finding this root. Using a TI-89,
You might have to wait 30 seconds or more before an answer appears. When it does, the answer will not be 0 (though it will be a very small number). You can get the same kind of close-but-wrong answers with roots in general for poor choices of initial bounds. The calculator will become both slow to converge and return anwers out of tolerance when you ask for the root of a function that doesn't change sign at the root.
The TI-83 fails, but in a different way than the TI-89. Let's again find the root for y=x^2:
After learning how the Newton method works, it became pretty obvious that the calculators were using some butchered form of this method or using something else entirely. I decided to e-mail Texas Instruments and ask if they would be willing to answer my questions on their machines discrepancies. I got an email back that day saying I could get my help without any trouble; I would just need to be more specific in the question. I called TI and talked to one of their personal describing the problem. He was able to reproduce the error and decided to talk to their programmers. A couple hours later, I got a phone call with my answer.
The reason that the 89 gives an approximate wrong answer is due to a rounding error. It has can hold a number up to 14 decimal places long in its memory and up to 16 decimal places long in its computations. When the calculator is using its root operation, it switches back and forth between the two, loosing some data as it goes from back and forth from 14 and 16 decimal places. The 83 uses a secant method for finding the root that has too go from a positive value to a negative value or visa versa. If the function does not do this (in the case of y=x^2 where the function just "kisses" the x-axis), then the root finder suffers a "no sign change" error.
The reason that the 89 that does suffer from this is that the root finder is more complex and can handle the "kissing" roots. The downside to this more advance root finder is that rounding errors can occur. The 83 cannot handle kissing roots but it does not have the rounding problem.
An irony about the TI-89 that the man helping me discovered is that if you use the root function in the regular window of the calculator, it will give you the correct answer. The error only occurs when you ask for the root in the graphing window, using the zero command. It makes me ask the question, "Why does it not use the root function that works with the equation?"
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