this is my first question here and I hope to find the answer because I already tried to find it in some books and videos (even in french) but I didn't find the answer.
My question is: Where did Hermite interpolation come from?
I know the formula:
$$P_{2n+1}(x)=\sum_{i=0}^{n}{f(x_{i})\left[1-2(x-x_{i})L'_{i}(x)\right]L^2_{i}(x)}+\sum_{i=0}^{n}{f'(x_{i})(x-x_{i})L^2_{i}(x)}$$
But how did they find it?
Someone told me it was linear algebra that helps to find the formula but I don't understand how.
N.B. :Sorry for any language faults, I don't speak English.