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Interpolation
Interpolation[
data
] constructs an InterpolatingFunction object which represents an approximate function that interpolates the data.
The data can have the forms 
,
,
,
, ...  or 
,
, ...  , where in the second case, the  are taken to have values 1, 2, ... .
Data can be given in the form 
,
,
,
, ... 
, ...  to specify derivatives as well as values of the function at the points  . You can specify different numbers of derivatives at different points.
Function values and derivatives may be real or complex numbers, or arbitrary symbolic expressions. The  must be real numbers.
Multidimensional data can be given in the form 
,
, ... ,
, ...  . Derivatives in this case can be given by replacing  and so on by 
,
,
, ... 
 .
Interpolation works by fitting polynomial curves between successive data points.
The degree of the polynomial curves is specified by the option InterpolationOrder.
The default setting is InterpolationOrder
->
3.
You can do linear interpolation by using the setting InterpolationOrder
->
1.
Interpolation[
data
] generates an InterpolatingFunction object which returns values with the same precision as those in data.
See the Mathematica book: Section 3.8.2.
See also: ListInterpolation, FunctionInterpolation, InterpolatingPolynomial, Fit.
Related packages: NumericalMath`SplineFit`, NumericalMath`PolynomialFit`, NumericalMath`Approximations`, DiscreteMath`ComputationalGeometry`.
Further Examples
Approximating Sqrt
Here is a table of values of the square root function at the points  .
In[1]:= 
Out[1]= 
This constructs an approximate function that represents these 11 values on the domain  .
In[2]:= 
Out[2]= 
The values of the function match the data at the given points.
In[3]:= 
Out[3]= 
The function also gives a fair approximation to the square root function at other points between  and  .
In[4]:= 
Out[4]= 
A plot of the difference between the two functions shows that the approximation is better at some points than at others.
Evaluate the cell to see the graphic.
In[5]:= 
In[6]:= 
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