Do I interpolate a polynomial using derivatives instead of divided differences correctly?

Question

I need to construct a 6 degree polynomial given some data points and their derivatives:

Say if I have points their corresponding values and derivatives: $$p(x_0)=p_0, p(x_1)=p_1, p'(x_1)=p'_1, p(x_2)=p_2, p'(x_2)=p'_2, p''(x_2)=p''_2, p(x_3)=p_3$$

So using the fact that $$p[x_0,...x_n]=\frac{p^{(n)}(x_n)}{n!}$$

We get:

Newtons polynomial: $p_1+p'_1(x-x_0)+p''_2(x-x_0)(x-x_1)$ And then I am stuck. How can I find the divided difference of $p[x_0...x_3]$? And how will I get the 6th degree polynomial if the max degree will be 3?

Answer 1

You're looking for Hermite interpolation.

The general idea is that, when you're constructing your divided difference table, whenever you have to fix the first $k$ derivatives of point, you repeat that point $k+1$ times in the leftmost column of the table and replace succeeding ambiguous divided differences by derivatives.