# Regularly Spaced Interpolation Nodes

by Gilmar

**Stirling’s interpolation formula**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.

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.

Derive a another formula, adding points in the order x0,x1,x2,x-2\,….,n,xn-n

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 f(9x)=x²