Hermite Interpolation and the Extended Neville Algorithm
Lagrange polynomials interpolate positions; Hermite polynomials interpolate positions and directions—points and vectors, function values and derivatives. Hermite interpolation is important for several reasons. Frequently in computational science and engineering we have information about tangents, curvatures, or other higher-order derivatives at various locations, and we need to generate curves and surfaces that fit this data. In geometric design, interpolating derivative data gives us more control over the shape of the curve or surface. Moreover, often we want to connect two or more curves or surfaces; to join them smoothly, we require the ability to interpolate derivatives across ...
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