(a) suppose we have n +1 data points (x , yo). (x, y). , (x , yn). the points are equally spaced, with point spacing h , so that hx1 for j0,1,,n-1. we want to fit a cubic spline s(x) through these points; this consists of n separate cubics, between each sets of points, and joined head-to-tail. show that, in the interval x x, xj,], the cubic spline takes the form s(x) %3d 씨 (x-z) m 1 h h mm-2)m) here, m s"(x) are the second derivatives ("moments") of the spline at each of the knot points x x [10 points (b) the moments mi are determined by making the first derivative of s(x) continuous at all the internal knot points x,,n-1. for the "natural" spline, which takes mo 0 and mn = 0, show that this yields the tri- diagonal matrix system of equations m-1 + 4m,+ m6.1-2y + y-1 h2 for j = ,n -1, and write as a matrix equation. [7 points] (c) explain why the matrix equation in part (b) could be solved successfully by an iterative method like the jacobi method.