Consider the points cos k pi 2n sin k pi 2n where k 0 1

Consider the points (cos k pi / 2^n, sin k pi / 2^n), where k = 0, 1, ..., 2^n. These points are lying on the circumference of the upper part of a unit circle. Let {S_i}2^n_i = 1 be the set of natural cubic splines interpolate these points for any given n. Let A_i be the area under the spline S_i. A_i can be calculated by using Simpson\'s composite rule. The error E_n is estimated by E_n = |summation_ i = i^2^n A_i - pi/2|. Calculate {E_n} for n = 1, 2, ... 8 and plot E_n vs. 2^n in log log and semilogy axes separately. Write a report to present your numerical methods, data, figure, comparisons and conclusions.

Solution