I am currently studying numerical method. I understand that the regular way of cubic Hermite interpolation on arbitrary interval $[a,b]$ is:
$ p(u)=\left( \begin{array}{c} 2t^3-3t^3+1\\ t^3-2t^2+t\\ -2t^3+3t^2\\ t^3-t^2\\ \end{array}\right)^T \left( \begin{array}{c} p(a)\\ (b-a)p'(a)\\ p(b)\\ (b-a)p'(b)\\ \end{array}\right)$, where $t=\frac{u-a}{b-a}$
But I don't know how to approach the following example. Can anyone tell me an approach to it? Thanks!(I don't get it what it means 'minimal degree')
Given $p(0)=0,p(1)=2,p(2)=18$ and derivatives $p'(0)=1,p'(2)=33$. Use the generalized Hermite interpolation scheme to develop the interpolation conditions for a curve $p(u)$ of minimal degree that interpolates the given points and derivatives at the respective knots. Compute the generalized Hermite polynomials and the interpolating curve.
I suppose the "generalized Hermite polynomials" are the functions of $u$ that you get by substituting $t = (u-a)/(b-a)$ into $2t^3-3t^3+1$, $t^3-2t^2+t$, $-2t^3+3t^2$, $t^3-t^2$.
Then just apply the formula you wrote.
Don't worry about the "minimal degree" requirement. Your formula will give you a curve of degree 3, and this is the lowest degree that allows you to interpolate two values and two derivatives.