MAIN POINTS
Table 2.1 makes two weaknesses of the bisection method apparent— it's not guaranteed that every recursion will produce a better answer, and it can take many recursions to produce an acceptably accurate answer (convergence is slow). Theorem 2.1 gives a formula for calculating approximation error. This is calculated by noting that the window shrinks by powers of two. The text also notes that the actual error can be much smaller than the estimated error. Example 2 shows how to start with the desired accuracy, and use logarithms to calculate the maximum number of iterations needer to attain that accuracy.
CHALLENGES
The very end of the reading confused me. I do not completely understand why they change the equation like this to factor in computer rounding errors. I also do not understand the use of the signum function— and how they prevent over/underflow errors. I understood the theorem, but it would be nice to work through an example.
REFLECTION
There's very little material for reflection here. I suppose that the ending part about considering computer rounding errors is important, particularly scientifically. It would be something that is easy to forget— possibly with bad consequences.
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