Interpolation

:This article is about interpolation in mathematics. See also interpolation (music) and interpolation (manuscripts).

Polynomial interpolation

Main article: Polynomial interpolation

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Polynomial interpolation is a generalization of linear interpolation. Note that the linear interpolant is a linear function. We now replace this interpolant by a polynomial of higher degree.

Related Topics:
Linear function - Polynomial - Degree

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Consider again the problem given above. The following sixth degree polynomial goes through all the seven points:

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: f(x) = -0.0001521 x^6 - 0.003130 x^5 + 0.07321 x^4 - 0.3577 x^3 + 0.2255 x^2 + 0.9038 x.

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Substituting x = 2.5, we find that f(2.5) = 0.5965.

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Generally, if we have n data points, there is exactly one polynomial of degree n−1 going through all the data points. The interpolation error is proportional to the distance between the data points to the power n. Furthermore, the interpolant is a polynomial and thus infinitely differentiable. So, we see that polynomial interpolation solves all the problems of linear interpolation.

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However, polynomial interpolation also has some disadvantages. Calculating the interpolating polynomial is very 'expensive' (in relative terms of computer calculation time). Furthermore, polynomial interpolation may not be so exact after all, especially at the end points (see Runge's phenomenon). These disadvantages can be avoided by using spline interpolation.

Related Topics:
Computer - Runge's phenomenon

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