For a given set of discrete data points (x i , f i ) (i 0, 1, 2, , n), show that the coefficients a k (k 0, 1, , n) fitted to the polynomial are given by the solution of the equations written in matrix form as The following data is taken from the tables of the Airy function f(x) Ai(x) Estimate from the polynomial approximation the values of f(2 0) and f (3 5)

Question: For a given set of discrete data points (x i , f i ) (i = 0, 1, 2, . . . , n), show

For a given set of discrete data points (x_i, f_i) (i = 0, 1, 2, . . . , n), show that the coefficients a_k (k = 0, 1, . . . , n) fitted to the polynomial

11 y(x) = ant k=0

are given by the solution of the equations written in matrix form as

where Aa = f A = xo xo 1xx = : [1 X x a [a, a f = [fo fi ... a,] fil XO

The following data is taken from the tables of the Airy function f(x) = Ai(–x):

x 1 1.5 2.3 3.0 3.9 f(x) 0.535 56 0.464 26 0.02670 -0.378 81 -0.147 42

Estimate from the polynomial approximation the values of f(2.0) and f (3.5).

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