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Find an interpolating function for data—Wolfram Documentation

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• See Also
  • ListInterpolation
  • FunctionInterpolation
  • InterpolatingPolynomial
  • Fit
  • FindFit
  • Quantile
  • Nearest
  • InterpolatingFunction
  • Piecewise
  • BSplineFunction
  • BezierFunction
  • ListLinePlot
  • ListPlot3D
  • FindSequenceFunction
  • SequencePredict
• Related Guides
  • Curve Fitting & Approximate Functions
  • Splines
  • Time Series Processing
  • Supervised Machine Learning
  • Numerical Data
  • Creating & Importing Signals
• Tech Notes
  • Approximate Functions and Interpolation

Interpolation[{f₁,f₂,…}]

constructs an interpolation of the function values f_i, assumed to correspond to x values 1, 2, … .

Interpolation[{{x₁,f₁},{x₂,f₂},…}]

constructs an interpolation of the function values f_i corresponding to x values x_i.

Interpolation[{{{x₁,y₁,…},f₁},{{x₂,y₂,…},f₂},…}]

constructs an interpolation of multidimensional data.

Interpolation[{{{x₁,…},f₁,df₁,…},…}]

constructs an interpolation that reproduces derivatives as well as function values.

Interpolation[data,x]

find an interpolation of data at the point x.

• Interpolation returns an InterpolatingFunction object, which can be used like any other pure function.
• 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 f_i can be real or complex numbers, or arbitrary symbolic expressions.
• The f_i can be lists or arrays of any dimension.
• The function arguments x_i, y_i, 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[f,{{x,y,…},n}].
• Partial derivatives not specified explicitly can be given as Automatic.
• 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 that returns values with the same precision as those in data.
• 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.
• The following options can be given:
• Interpolation supports a Method option. Possible settings include "Spline" for spline interpolation and "Hermite" 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:

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:

Form an interpolation from data given as a TimeSeries:

Plot the interpolated 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:

Create a vector-valued InterpolatingFunction of one variable from unstructured vector-valued data:

The value is a vector:

Plot will show both components:

Unstructured data can be used in the one-variable case.

Create a vector-valued InterpolatingFunction of two variables from structured vector-valued data:

The value is a vector:

Plot3D will show all three components:

A single component may be plotted using Part:

Derivatives may also be included for Hermite interpolation:

Make a zeroth‐order interpolation:

Make a linear interpolation:

Make a quadratic 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:

Interpolate random data:

Find a continuous interpolation of the GCD function:

Interpolate through a set of points with a spline:

Create a spline interpolation:

Visualize the interpolation and the set of points:

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:

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