This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# Interpolation

 Interpolationconstructs an interpolation of the function values , assumed to correspond to x values 1, 2, ... . Interpolationconstructs an interpolation of the function values corresponding to x values . Interpolationconstructs an interpolation of multidimensional data. Interpolationconstructs an interpolation that reproduces derivatives as well as function values. Interpolationfind an interpolation of data at the point x.
• The interpolating function returned by Interpolation[data] is set up so as to agree with data at every point explicitly specified in data.
• The function values can be real or complex numbers, or arbitrary symbolic expressions.
• The can be lists or arrays of any dimension.
• The function arguments , , etc. must be real numbers.
• Different elements in the data can have different numbers of derivatives specified.
• For multidimensional data, the n derivative can be given as a tensor with a structure corresponding to D.
• Partial derivatives not specified explicitly can be given as Automatic.
• Interpolation works by fitting polynomial curves between successive data points.
• Interpolation allows any derivative to be given as Automatic, in which case it will attempt to fill in the necessary information from other derivatives or function values.
• Interpolation supports a Method option. Possible settings include for spline interpolation and for Hermite interpolation.
Construct an approximate function that interpolates the data:
Apply the function to find interpolated values:
Plot the interpolation function:
Compare with the original data:
Find the interpolated value immediately:
Construct an approximate function that interpolates the data:
 Out[1]=
Apply the function to find interpolated values:
 Out[2]=
Plot the interpolation function:
 Out[3]=
Compare with the original data:
 Out[4]=

Find the interpolated value immediately:
 Out[1]=
 Scope   (3)
Interpolate between points at arbitrary values:
Create data with Table:
Form the interpolation:
Plot the interpolated function:
Create a list of multidimensional data:
Create an approximate interpolating function:
Plot the interpolating function:
Create data that includes derivatives at each point:
Construct an interpolation:
Plot the interpolation:
Create 2D data that includes a gradient vector at each point:
Compare with data that does not include gradients:
Also include tensors of second derivatives:
 Options   (5)
Make a zeroth-order interpolation:
Make a linear interpolation:
Compare splines with piecewise Hermite interpolation for random data:
The curves appear close, but the spline has a continuous derivative:
Make an interpolating function that repeats periodically:
 Applications   (2)
Interpolate random data:
Find a continuous interpolation of the GCD function:
The interpolating function always goes through the data points:
Find the integral of an interpolating function:
Plot the interpolating function and its integral:
Extrapolation is attempted to go beyond the original data:
With the default choice of order, at least 4 points are needed in each dimension:
With a lower order, fewer points are needed:
The interpolation function will always be continuous, but may not be differentiable:
Interpolate the sequence of primes: