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Answer :
Hermite interpolation uses not only function values but also their derivatives to construct the interpolating polynomial, while ordinary interpolation only uses function values.
A cubic spline interpolant is a piecewise polynomial function consisting of several cubic polynomials, while a Hermite cubic interpolant is a single polynomial of degree 3 that interpolates function and their derivatives.
Hermite interpolation is a type of interpolation that not only matches the function values but also its derivatives at the given data points. This means that Hermite interpolation provides a more accurate and smooth approximation of the underlying function than ordinary interpolation, which only matches the function values at the given points.
Hermite interpolation is particularly useful when the function being approximated is expected to be smooth and differentiable.
A cubic spline interpolant, on the other hand, is a piecewise function that consists of multiple cubic polynomials, each defined over a subinterval of the data points.
The cubic spline interpolant is designed to be smoother than the Hermite cubic interpolant. In contrast to the Hermite cubic interpolant, the cubic spline interpolant matches the function values at the data points and also ensures that the first and second derivatives are continuous across the subintervals.
This results in a smoother and more natural-looking approximation of the function. Overall, both Hermite interpolation and cubic spline interpolation provide powerful tools for approximating functions with a high degree of accuracy and smoothness.
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