Editing Interpolation (section)

===Polynomial interpolation===
[[File:Interpolation example polynomial.svg|right|thumb|230px|Plot of the data with polynomial interpolation applied]]
{{Main|Polynomial interpolation}}
Polynomial interpolation is a generalization of linear interpolation. Note that the linear interpolant is a [[linear function]]. We now replace this interpolant with a [[polynomial]] of higher [[degree of a polynomial|degree]].

Consider again the problem given above. The following sixth degree polynomial goes through all the seven points:
:<math> 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. </math>
<!-- Coefficients are 0, 0.903803333333334, 0.22549749999997, -0.35772291666664, 0.07321458333332, -0.00313041666667, -0.00015208333333. -->
Substituting ''x'' = 2.5, we find that ''f''(2.5) = ~0.59678.

Generally, if we have ''n'' data points, there is exactly one polynomial of degree at most ''n''&minus;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 overcomes most of the problems of linear interpolation.

However, polynomial interpolation also has some disadvantages. Calculating the interpolating polynomial is computationally expensive (see [[Computational complexity theory|computational complexity]]) compared to linear interpolation. Furthermore, polynomial interpolation may exhibit oscillatory artifacts, especially at the end points (see [[Runge's phenomenon]]).

Polynomial interpolation can estimate local maxima and minima that are outside the range of the samples, unlike linear interpolation. For example, the interpolant above has a local maximum at ''x'' ≈ 1.566, ''f''(''x'') ≈ 1.003 and a local minimum at ''x'' ≈ 4.708, ''f''(''x'') ≈ −1.003. However, these maxima and minima may exceed the theoretical range of the function; for example, a function that is always positive may have an interpolant with negative values, and whose inverse therefore contains false [[division by zero|vertical asymptotes]].

More generally, the shape of the resulting curve, especially for very high or low values of the independent variable, may be contrary to commonsense; that is, to what is known about the experimental system which has generated the data points. These disadvantages can be reduced by using spline interpolation or restricting attention to [[Chebyshev polynomials]].
{{clear}}