Consider 2n+1 regularly spaced interpolation nodes x-n,x-n+1,,,,x-1,,,,,xn with xk=xo+kh,k= -n,-n 1,,,;-1,0,1,….,n-1.
(a) Derive a formula for the interpolation polynomial of order 2n, using Newton’s method and adding points in the order x0,x_1,x_2,x2,….,x-n,xn.
(b) Derive a another formula, adding points in the order x0,x1,x2,x-2\,….,n,xn-n
(c) Take the average of two formulas and show that it is equivalent to Stirling’s interpolation formula
P(x)= f(xo) + s ^f(x-1) +^ (x0) + s2/2 ^2 f (x-1) + S(
Where s=(x-xo) /h,and ^ ^f(x)=f(x+h)-f(x)
Check each of three formula’s using n=2=h1, and 1,and f9x)=x^2