Re: Spline Interpolation and cubic interpolation

for 3rd order polynomial interpolation, it's usually 4 data points (and
you're interpolating between the inner two data points). there are
many different interpolation polynomials, even for a given order (such
as 3).

Lagrange interpolation will fit an Nth order polynomial through N+1
adjacent data points, and assuming that N+1 is even and we start
counting the data points from "0" to "N", the interpolation is used for
the space between data points (N-1)/2 to (N+1)/2.

Hermite polynomial interpolation will fit an Nth order polynomial only
to the two adjacent data points at indices (N-1)/2 to (N+1)/2, but will
also match as many derivatives as possible at those two adjacent data
points with the interpolation polynomials that go through those
neighbors and *their* neighbors. that means that these derivatives of
the Nth order Hermite at the (N-1)/2 point can only depend on data
points 0 to N-1 and the derivatives at the (N+1)/2 point can only
depend on data points 1 to N (and must be the same function of those N
points).

a 3rd order Hermite interpolation polynomial will match the 0th and 1st
derivative at the 1st and 2nd point. the value of the polynomial (the
"0th derivative") at x[1] *is* x[1] and the value of the 1st derivative
at x[1] is (x[2]-x[0])/2 (assuming uniform spacing). similarly, the
value of the polynomial (the "0th derivative") at x[2] is x[2] and the
value of the 1st derivative at x[2] is (x[3]-x[1])/2. that way, both
the polynomial value *and* slope matches the polynomial and slope of
the neighboring interpolations.

there are more (such as B-splines), but, for the most part, you need 4
points for 3rd order polynomial interpolation.

r b-j

.

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