In the context of a problem I am working on, it would be useful to have examples of polynomials with certain features. I know given $n$ points, you can use a divided difference table to find a polynomial of degree $n+1$ going through these points. This is easy. However, I have heard you can also find polynomials with specified derivative values.
Can anyone explain how to use the divided difference table to find a polynomial with $p(0)=1$, $p(3)=1$, and $p'(2)=0$? Or something like $p(0)=1$, $p'(2)=3$, and $p''(3)=-1$? Even links to resources explaining this process would be helpful. I have not found any searching for this.
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$\begingroup$I think you are looking for osculating polynomials.
Here are some links that include nice notes and examples:
Burden and Faires has a nice section on Hermite Polynomials as well as many other texts on Numerical Analysis / Algorithms.
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