Interpolation
Usage Message: Interpolation[data] constructs an InterpolatingFunction object which represents an approximate function that interpolates the data. The data can have the forms {{x1, f1}, {x2, f2}, ... } or {f1, f2, ... }, where in the second case, the xi are taken to have values 1, 2, ... . Attributes[Interpolation] = {Protected} Options: InterpolationOrder
-> 3 Related Symbols Fit
, FunctionInterpolation
, InterpolatingFunction
, InterpolatingPolynomial
, ListInterpolation
Notes: Interpolation[data]
returns an InterpolatingFunction
expression which is used for interpolating between the points specified in data
. Interpolation
is based on divided difference interpolation, a standard method in which the interpolating polynomial in each interval is chosen to match the function values (and, if specified, the derivatives) at points adjacent to the interval. A good discussion of divided difference interpolation in one dimention can be found in Robert D. Skeel and Jerry B. Keiper, "Elementary Numerical Computing with Mathematica", McGraw-Hill, Inc. (1993), chapter 5. For multi-dimensional interpolation, the one-dimensional method is applied recursively, interpolating along lines surrounding the desired point, and then interpolating perpendicular to those lines. Although it is not used by Interpolation
, another common interpolation method is spline interpolation, in which the interpolating polynomial in each interval is chosen not only to match the data points adjacent to the interval, but also to match the derivatives of the interpolating polynomials in adjacent intervals. Spline interpolation is computationally more difficult, and is no more accurate than divided difference interpolation, but it does lead to smoother-looking curves, a characteristic which can be particularly important in graphics applications. It is likely that functions for spline interpolation will be added for some future version of Mathematica. In the meantime, if you need spline interpolation, we recommend that you look on MathSource for spline interpolation programs written by other users, or consider writing a short spline interpolation function of your own. The NumericalMath`SplineFit` and Graphics`Spline` packages are intended for spline interpolation of curves rather than functions. These packages are analogous to Interpolation
only in special cases, such as when the values of the independent coordinates are consecutive integers starting with zero.
Additional Online Documentation:
Mathematica 3.0
http://documents.wolfram.com/v3/RefGuide/Interpolation.html
Mathematica 4.0
http://documents.wolfram.com/v4/RefGuide/Interpolation.html
Questions or comments? Send email to support@wolfram.com.
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