_TZ_6393-periastron.dviMATH 3221 ADVANCED LINEAR ALGEBRADAILY ASSIGNMENT MARCH 6, 202328. Okay, after working this one out for myself, it’s too neat not to give to you.,Use the techniques we...

Linear Algebra


_TZ_6393-periastron.dvi MATH 3221 ADVANCED LINEAR ALGEBRA DAILY ASSIGNMENT MARCH 6, 2023 28. Okay, after working this one out for myself, it’s too neat not to give to you., Use the techniques we discussed in class to find the polynomial of degree 3 that best approximates sin πx on the interval [−1, 1], in the sense of minimizing the integral ∫ 1 −1 ( sin πx− p(x) )2 dx. Suggestion: Start from the orthogonal basis we computed in the previous problem (which I’ve given you below). Use a computer to integrate the functions in the basis to normal- ize them, and then use the techniques we discussed in class to compute the orthogonal projection p — again, using a computer for the integration. Also, remember to make use of odd function properties. Plot sin πx and p(x) on the same graph, together with the degree-3 Taylor polynomial of sin πx at 0. This should illustrate the discussion in class about approximation at one point versus over the interval. If you really want to see a nice illustration of this, compute the fifth degree approximation, and plot it together with sin πx. (This is not actually much more work.) How good is the approximation? Then add the degree-5 Taylor approximation to the picture. Non-normalized orthogonal basis for P5 (which up to normalizing constants is the an- swer to last day’s problem, at least the first 4 elements): { 1, x, (3x2 − 1), (5x3 − 3x), (35x4 − 30x2 + 3), (63x5 − 70x3 + 15x) } . I’ve given you up to the fifth degree in case you want to do the optional fifth-order ap- proximation, but the basic question only requires up to degree 3. Okay, I don’t know how much work that’s going to be, so I’ll let you off the hook for another problem. That’s it for today!