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Answer :
it does not use the additional information provided by the second derivative to ensure continuity between the interpolating polynomials, which can result in a less smooth interpolant than a cubic spline.
Hermite interpolation and ordinary interpolation are both methods used to find an approximation of a function based on a given set of data points. However, they differ in the way they approach the problem.
In ordinary interpolation, a unique polynomial of degree n-1 (where n is the number of data points) is constructed that passes through all the given data points. The polynomial is then used to approximate the function between the data points. The drawback of ordinary interpolation is that it can result in a very oscillatory or wiggly interpolant, especially when the data points are unequally spaced.
On the other hand, Hermite interpolation constructs a polynomial of degree 2n-1 that not only passes through all the given data points but also includes the values of the first n-1 derivatives at each data point. This additional information allows Hermite interpolation to produce a smoother interpolant that more accurately represents the behavior of the function.
A cubic spline interpolant is a type of piecewise interpolation where a polynomial of degree 3 is used to approximate the function between each adjacent pair of data points. The splines are connected at the data points such that the function and its first and second derivatives are continuous across the entire range of data points.
In contrast, a Hermite cubic interpolant constructs a single polynomial of degree 3 that passes through all the given data points and includes the values of the first derivative at each data point. Therefore, it does not use the additional information provided by the second derivative to ensure continuity between the interpolating polynomials, which can result in a less smooth interpolant than a cubic spline.
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Final answer:
Hermite interpolation differs from ordinary interpolation by incorporating derivative information for more accuracy. And cubic spline interpolant differs from a Hermite cubic interpolant in that while both interpolate using a cubic polynomial, the latter uses derivative information to provide more accurate estimates.
Explanation:
In mathematics, both Hermite interpolation and ordinary interpolation are methods used to estimate unknown values between two known values. However, Hermite interpolation uses not just the function values but also their derivatives to create a more accurate estimate.
Ordinary interpolation, using for example Lagrange polynomials, fits the function through the given points. On the other hand, Hermite interpolation creates a polynomial that not only goes through the given points, but also has the same derivative at these points. This makes Hermite interpolation more accurate, especially when the function estimated has higher order derivatives.
Similarly, a cubic spline interpolant and a Hermite cubic interpolant are two different estimation techniques. Both fit a cubic polynomial between pairs of points, but a Hermite cubic interpolant, similar to Hermite interpolation, uses derivative information at the points to improve accuracy. A cubic spline interpolant ensures that the resulting piecewise polynomial function is smooth and continuous throughout the entire interval.
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