Question

4. (a) Show directly for uniform spacingh =r1-ro=r2-1 that 42 fo f\ro, I1, F2] = 2h? Afo f{rO, x1] h and where fi f(r), Afi = fi+1-fi, and A?f,= Afi+1=Afi (b) Establish Newton's forward-difference form of the interpolating polyno mial with uniform spacing P2(r) =fo+ where p2 is the polynomial of degree < 2 that interpolatesf at ro, , , and =o+sh. Here, ()--1)m+1) is a binomial coefficient.

Answer 1

F[x_0, x_1] = f(x - 1) - f(x_0)/h = delta f/h f(x_0, x_1, x_2] = f[x_1, x_2] - f [x_0, x_1]/x_2 - x_0 = f[x - 2) - f(x - 1)/h - f(x_1) - f(x_0)/h/2h = delta f_1 - delta f_0/2 h^2 = del: ta^2 f_0/2h^2 Interpolating polynomial with 3 points p_2(x) = f(x_0) + (x - x_0) f [x_0, x_1] + (x - x_0) (x - x_1) f [x_0, x_1, x_2] = f(x_0) + sh (delta f_0/h) + sh (sh - h) delta^2 f_0 /2h^2 = f(x_0) + s delta f_0 + s (s -1)/2 delta^2 f_0 = f_0 + (s_1) delta f_0 + (s_2) deta^2 f_0