Applied Linear Algebra - Fitting a rational function with a polynomial.

Between tnelr logarltnms, r : logy logy. onow that n : E" ' 1. 13.17 Fitting a rational function with a. polynomial. Let $1,...,a:11 be 11 points uniformly spaced in the interval [1,1]. (This means 93,: = 1.0 + 0.2(i 1) for i = 1,...,11.) Take ya: : (1 + sing/(1 + 535?), for i : 1,. . . ,11. Find the least squares t of polynomials of degree 0,1,...,8 to these points. Plot the tting polynomials, and the true function y = (1 + 9:)/(1 + 5332), over the interval [1.1,1.1] (say, using 100 points). Note that the interval for the plot, [11, 1.1], extends a bit outside the range of the data used to t the polynomials, [1, 1]; this gives us an idea of how well the polynomial ts can extrapolate. Exercises 283 Generate a test data set by choosing n.1,. . . ,UIO uniformly spaced over [1.1,1.1], with vi = (1 +1\") / (1 + 51.5%). Plot the RMS error of the polynomial ts found above on this test data set. On the same plot= show the RMS error of the polynomial ts on the training data set. Suggest a reasonable value for the degree of the polynomial t, based on the RMS ts on the training and test data. Remark. There is no practical reason to t a rational function with a polynomial. This exercise is only meant to illustrate the ideas of tting with different basis functions, over-t, and validation with a test data. set