The table below lists the closing Dow Jones Industrial Avera

The table below lists the closing Dow Jones Industrial Averages at the first day the market is open for the months from March 2013 to February 2014.

03/13 04/13 05/13 06/13 07/13 08/13 09/13 10/13 11/13 12/13 01/14 02/14

14090 14573 14701 15254 14975 15628 14834 15193 15616 16009 16441 15373

(a) Compute by hand the order 4 trigonometric polynomial interpolating the values for the first four months in the table (time unit=1 month, so the interval is [0,4]).

Solution

P(x) = x^4 + bx^3 + cx^2 + dx + e

P(0) = e = 14090

P(1) = 1 + b + c + d + e = 14573

b + c + d = 482

P(2) = 16 + 8b + 4c + 2d + e = 14701

8b + 4c + 2d = 595

P(3) = 81 + 27b + 9c + 3d + e = 15254

27b + 9c + 3d = 1083

b = 124, c = -556.5, d = 914.5

Hence the polynomial will be

P(x) = x^4 + 124x^3 -556.5x^2 + 914.5x + 14090