Use Newton's forward-difference formula to construct interpolating polynomials of degree two, and three for the following data. Approximate the specified value using each of the polynomials.
I have to approximate $f(8.4)$ if $f(8.1)=16.94410$ , $f(8.3)=17.56492$ , $f(8.6)=18.50515$ , $f(8.7)=18.82091$
I know that the formula in this case is $$P(x_{0}+hs)=f_{0}+s\Delta f_{0}+\frac{s(s-1)}{2!}\Delta^{2}f_{0}+\cdots+\frac{s(s-1)(s-2)...(s-n+1)}{n!}\Delta^{n}f_{0}$$ where $$s=\frac{(x-x_{0})}{h}; \qquad f_0=f(x_0); \qquad\Delta^k f_i=\sum_{j=0}^k{(-1)^j \frac{k!}{j!(k-j)!}f_{i+k-j}}$$
Problem is,what is h in this case,since $x_{1}-x_{0}=0.2$ , $x_{2}-x_{1}=0.3$ and $x_{3}-x_{2}=0.1$ ?
For this problem, Newton's forward Interpolation formula can not be used.
This formula can be used only for equi-spaced arguments.
For non-equispaced arguments apply Lagrange's interpolation formula.