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Hermite Interpolation extended to second derivative

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  1. Apr 12, 2015 #1
    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. jcsd
  3. Apr 12, 2015 #2
    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
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