MAIN POINTS
The reading begins with Thm 3.3, which defines the error function in much the same way that it was defined for Taylor polynomials. Xi(x) is within the bounds (a,b). The proof for this takes up a page and uses Rolle's Theorem. Ex2 I don't quite understand. Ex3 I understand more— they're comparing the error when using higher and higher degree polynomials, and the steps theyve explained already. The text also explains that one often has to calculate successive degree polynomials to find the one that reaches the correct accuracy.
CHALLENGES
I don't understand what they're doing in Example 2... so they're taking a table of values of e^x, and interpolating between them... but using a Lagrange polynomial of degree 1— does this mean they select two points from the function and make a polynomial out of them? What is the Bessel function?
REFLECTION
Lagrange strike me as being useful in the computational domain because computer data deals with discrete data points (in sound, pictures, video) and any time we want to try and think of these as continuous data, we have to make a transformation. The error terms here seem sort of difficult to calculate exactly, so that could be a problem in more open-ended applications of Lagrange polynomials. However, in A/V applications, it's probbaly possible to map out the domain of the problem really well to predict what sort of errors could be encountered.
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