Numerical Methods question:

4. (a) Consider the following points: (-2, 3). (-1,5), (0,2). (1, 4), so that To = -2, x1 = -1, x2 = 0, x3 = 1 and yo = 3, y1 = 5, y2 = 2, ya = 4. Find the coefficients do, a1, a2 and as for the Newton's interpolation formula (also known as the divided difference formula) so that the interpolating polynomial passing through the given points can be written as Py(x) = do + ai(r - ro) + a2(r - ro)(x - 1) + as(r - ro)(x - my)(1 - x2). [7] (b) Consider the following points generated from the function f(r) = 1 - Ar + 2: (0, 2), (1, -1), (4,2). Find the coefficients ap, a1, bo and b, of the linear spline function L(x) = bo(1) = do. + bo. I E [0, 1). I E [1, 4], connecting these points. Calculate the maximum error (E(x)| = (f(x) - L(x)| for T E [0, 4]. [8] [Hint: consider the functions f(x) - lo(x) and f(x) - (x) separately.]