# Hermite Interpolation extended to second derivative

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1. Apr 12, 2015

### Haydo

SOLVED

1. The problem statement, all variables and given/known data

Find polynomial of least degree satisfying:
p(1)=-1, p'(1)=2, p''(1)=0, p(2)=1, p'(2)=-2

2. Relevant equations
In general, a Hermite Polynomial is defined by the following:
∑[f(xi)*hi(x)+f'(xi)*h2i(x)]
where:
hi(xj)=1 if i=j and 0 otherwise. Similarly with h'2. h'i(x)=0 and h2(x)=0. i.e., they are zero if they are integrated or derived.

Here is a page from wolfram with general information: http://mathworld.wolfram.com/HermitesInterpolatingPolynomial.html

3. The attempt at a solution
First, I recognized x0=1 and x1=2. I tried to create some third coefficient term in order to satisfy p''(1)=0, but that seems to mean that I would have to make hi(xj) be zero for the second derivative, and I have no idea how to do this. I tried just setting this new h (call it h3) equal to (x)(x-2) when derived twice and integrating (so that h3''=1 for x1 and zero otherwise) but that was a total flop.

I'm starting to think that, despite the question's section (it was in the section regarding Hermite Interpolating Polynomials), there is a better way to approach it. Any help would be greatly appreciated.

Last edited: Apr 13, 2015
2. Apr 12, 2015

### Haydo

For anyone reading this, I believe I have to use Newton's Formalism for the Hermite Interpolating polynomial, instead of Lagrange's.

EDIT- Yep, this is how to solve it. Answer is:
-1+2(x-1)-4(x-1)^3(x-2)

Here is a page describing Newton's Form for the Hermite Interpolant

Last edited: Apr 13, 2015