Interpolation
✖
Interpolation
constructs an interpolation of the function values fi, assumed to correspond to x values 1, 2, … .
constructs an interpolation of the function values fi corresponding to x values xi.
constructs an interpolation of multidimensional data.
constructs an interpolation that reproduces derivatives as well as function values.
Details and Options


- 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 fi can be real or complex numbers, or arbitrary symbolic expressions.
- The fi can be lists or arrays of any dimension.
- The function arguments xi, yi, 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:
-
InterpolationOrder Automatic set the order of the interpolation Method Automatic set a specifc method to use PeriodicInterpolation Automatic specify if an interpolation is periodic - Interpolation supports a Method option. Possible settings include "Spline" for spline interpolation and "Hermite" for Hermite interpolation.
Examples
open allclose allBasic Examples (2)Summary of the most common use cases
Construct an approximate function that interpolates the data:

https://wolfram.com/xid/0ldf0pe-g6ewuf

Apply the function to find interpolated values:

https://wolfram.com/xid/0ldf0pe-cocphs

Plot the interpolation function:

https://wolfram.com/xid/0ldf0pe-numkhs

Compare with the original data:

https://wolfram.com/xid/0ldf0pe-d2eg57

Find the interpolated value immediately:

https://wolfram.com/xid/0ldf0pe-bdisut

Scope (4)Survey of the scope of standard use cases
Interpolate between points at arbitrary values:

https://wolfram.com/xid/0ldf0pe-crnp6r


https://wolfram.com/xid/0ldf0pe-b9sr46

Create data with Table:

https://wolfram.com/xid/0ldf0pe-j62uy


https://wolfram.com/xid/0ldf0pe-b92rsi

Plot the interpolated function:

https://wolfram.com/xid/0ldf0pe-c8dhnl

Create a list of multidimensional data:

https://wolfram.com/xid/0ldf0pe-hvle88

Create an approximate interpolating function:

https://wolfram.com/xid/0ldf0pe-da0d3v

Plot the interpolating function:

https://wolfram.com/xid/0ldf0pe-mjvwvb

Form an interpolation from data given as a TimeSeries:

https://wolfram.com/xid/0ldf0pe-nt9st8

https://wolfram.com/xid/0ldf0pe-zjhkoc


https://wolfram.com/xid/0ldf0pe-vt3l34

Plot the interpolated function:

https://wolfram.com/xid/0ldf0pe-u9bek3

Generalizations & Extensions (5)Generalized and extended use cases
Create data that includes derivatives at each point:

https://wolfram.com/xid/0ldf0pe-brgn5x


https://wolfram.com/xid/0ldf0pe-g84l4e


https://wolfram.com/xid/0ldf0pe-gdsn88

Create 2D data that includes a gradient vector at each point:

https://wolfram.com/xid/0ldf0pe-cg43a7


https://wolfram.com/xid/0ldf0pe-c1wxyx


https://wolfram.com/xid/0ldf0pe-ggs829

Compare with data that does not include gradients:

https://wolfram.com/xid/0ldf0pe-hlt3lq


https://wolfram.com/xid/0ldf0pe-xi4c7


https://wolfram.com/xid/0ldf0pe-g2sktj

Also include tensors of second derivatives:

https://wolfram.com/xid/0ldf0pe-52u9h


https://wolfram.com/xid/0ldf0pe-bs8p08


https://wolfram.com/xid/0ldf0pe-jipi7s

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

https://wolfram.com/xid/0ldf0pe-cqv76e

https://wolfram.com/xid/0ldf0pe-cpxztm


https://wolfram.com/xid/0ldf0pe-ew8f7e

Plot will show both components:

https://wolfram.com/xid/0ldf0pe-ci4trg

Unstructured data can be used in the one-variable case.
Create a vector-valued InterpolatingFunction of two variables from structured vector-valued data:

https://wolfram.com/xid/0ldf0pe-eygcp5

https://wolfram.com/xid/0ldf0pe-d370pg


https://wolfram.com/xid/0ldf0pe-cdjaf4

Plot3D will show all three components:

https://wolfram.com/xid/0ldf0pe-k1l4tp

A single component may be plotted using Part:

https://wolfram.com/xid/0ldf0pe-m0m562

Derivatives may also be included for Hermite interpolation:

https://wolfram.com/xid/0ldf0pe-j9ymoz


https://wolfram.com/xid/0ldf0pe-dblutd

https://wolfram.com/xid/0ldf0pe-fkfg

Options (5)Common values & functionality for each option
InterpolationOrder (3)
Method (1)
Compare splines with piecewise Hermite interpolation for random data:

https://wolfram.com/xid/0ldf0pe-bex8j3

https://wolfram.com/xid/0ldf0pe-ebse7i


https://wolfram.com/xid/0ldf0pe-ctt1b

The curves appear close, but the spline has a continuous derivative:

https://wolfram.com/xid/0ldf0pe-ndcptz

Applications (3)Sample problems that can be solved with this function

https://wolfram.com/xid/0ldf0pe-gb2


https://wolfram.com/xid/0ldf0pe-u37

Find a continuous interpolation of the GCD function:

https://wolfram.com/xid/0ldf0pe-czn0vv


https://wolfram.com/xid/0ldf0pe-d8qs4x

Interpolate through a set of points with a spline:

https://wolfram.com/xid/0ldf0pe-xoulw4
Create a spline interpolation:

https://wolfram.com/xid/0ldf0pe-mykwew

Visualize the interpolation and the set of points:

https://wolfram.com/xid/0ldf0pe-ojsj4

Properties & Relations (2)Properties of the function, and connections to other functions
The interpolating function always goes through the data points:

https://wolfram.com/xid/0ldf0pe-g8s3sk

https://wolfram.com/xid/0ldf0pe-cxdbk


https://wolfram.com/xid/0ldf0pe-jfa1wu

Find the integral of an interpolating function:

https://wolfram.com/xid/0ldf0pe-e7zogj


https://wolfram.com/xid/0ldf0pe-b1wcgz

Plot the interpolating function and its integral:

https://wolfram.com/xid/0ldf0pe-ydbna


https://wolfram.com/xid/0ldf0pe-no6yz9


Possible Issues (3)Common pitfalls and unexpected behavior
Extrapolation is attempted to go beyond the original data:

https://wolfram.com/xid/0ldf0pe-bp8rj2


https://wolfram.com/xid/0ldf0pe-es7wz6


With the default choice of order, at least 4 points are needed in each dimension:

https://wolfram.com/xid/0ldf0pe-tqsi4


With a lower order, fewer points are needed:

https://wolfram.com/xid/0ldf0pe-ft54qz

The interpolation function will always be continuous, but may not be differentiable:

https://wolfram.com/xid/0ldf0pe-jhakc


https://wolfram.com/xid/0ldf0pe-f3p7u1

Wolfram Research (1991), Interpolation, Wolfram Language function, https://reference.wolfram.com/language/ref/Interpolation.html (updated 2008).
Text
Wolfram Research (1991), Interpolation, Wolfram Language function, https://reference.wolfram.com/language/ref/Interpolation.html (updated 2008).
Wolfram Research (1991), Interpolation, Wolfram Language function, https://reference.wolfram.com/language/ref/Interpolation.html (updated 2008).
CMS
Wolfram Language. 1991. "Interpolation." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2008. https://reference.wolfram.com/language/ref/Interpolation.html.
Wolfram Language. 1991. "Interpolation." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2008. https://reference.wolfram.com/language/ref/Interpolation.html.
APA
Wolfram Language. (1991). Interpolation. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Interpolation.html
Wolfram Language. (1991). Interpolation. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Interpolation.html
BibTeX
@misc{reference.wolfram_2025_interpolation, author="Wolfram Research", title="{Interpolation}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/Interpolation.html}", note=[Accessed: 25-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_interpolation, organization={Wolfram Research}, title={Interpolation}, year={2008}, url={https://reference.wolfram.com/language/ref/Interpolation.html}, note=[Accessed: 25-July-2025
]}