In the
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
field of
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
, interpolation is a type of
estimation
Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is d ...
, a method of constructing (finding) new
data points
In statistics, a unit of observation is the unit described by the data that one analyzes. A study may treat groups as a unit of observation with a country as the unit of analysis, drawing conclusions on group characteristics from data collected a ...
based on the range of a
discrete set
In mathematics, a point (topology), point is called an isolated point of a subset (in a topological space ) if is an element of and there exists a Neighborhood (mathematics), neighborhood of that does not contain any other points of . This i ...
of known data points.
In
engineering
Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
and
science
Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...
, one often has a number of data points, obtained by
sampling or
experimentation
An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs whe ...
, which represent the values of a function for a limited number of values of the
independent variable
A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical function ...
. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable.
A closely related problem is the
approximation
An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ...
of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gain in simplicity may outweigh the loss from interpolation error and give better performance in calculation process.
Example
This table gives some values of an unknown function
.
Interpolation provides a means of estimating the function at intermediate points, such as
We describe some
methods of interpolation, differing in such properties as: accuracy, cost, number of data points needed, and
smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain.
A function of class C^k is a function of smoothness at least ; t ...
of the resulting
interpolant
In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.
In engineering and science, one often has a ...
function.
Piecewise constant interpolation

The simplest interpolation method is to locate the nearest data value, and assign the same value. In simple problems, this method is unlikely to be used, as
linear
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a '' function'' (or '' mapping'');
* linearity of a '' polynomial''.
An example of a linear function is the function defined by f(x) ...
interpolation (see below) is almost as easy, but in higher-dimensional
multivariate interpolation
In numerical analysis, multivariate interpolation or multidimensional interpolation is interpolation on ''multivariate functions'', having more than one variable or defined over a multi-dimensional domain. A common special case is bivariate inter ...
, this could be a favourable choice for its speed and simplicity.
Linear interpolation

One of the simplest methods is linear interpolation (sometimes known as lerp). Consider the above example of estimating ''f''(2.5). Since 2.5 is midway between 2 and 3, it is reasonable to take ''f''(2.5) midway between ''f''(2) = 0.9093 and ''f''(3) = 0.1411, which yields 0.5252.
Generally, linear interpolation takes two data points, say (''x''
''a'',''y''
''a'') and (''x''
''b'',''y''
''b''), and the interpolant is given by:
:
:
:
This previous equation states that the slope of the new line between
and
is the same as the slope of the line between
and
Linear interpolation is quick and easy, but it is not very precise. Another disadvantage is that the interpolant is not
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
at the point ''x''
''k''.
The following error estimate shows that linear interpolation is not very precise. Denote the function which we want to interpolate by ''g'', and suppose that ''x'' lies between ''x''
''a'' and ''x''
''b'' and that ''g'' is twice continuously differentiable. Then the linear interpolation error is
:
In words, the error is proportional to the square of the distance between the data points. The error in some other methods, including
polynomial interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points in the dataset.
Given a set of data points (x_0,y_0), \ldots, (x_n,y_n), with no ...
and spline interpolation (described below), is proportional to higher powers of the distance between the data points. These methods also produce smoother interpolants.
Polynomial interpolation

Polynomial interpolation is a generalization of linear interpolation. Note that the linear interpolant is a
linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For di ...
. We now replace this interpolant with a
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
of higher
degree.
Consider again the problem given above. The following sixth degree polynomial goes through all the seven points:
:
Substituting ''x'' = 2.5, we find that ''f''(2.5) = ~0.59678.
Generally, if we have ''n'' data points, there is exactly one polynomial of degree at most ''n''−1 going through all the data points. The interpolation error is proportional to the distance between the data points to the power ''n''. Furthermore, the interpolant is a polynomial and thus infinitely differentiable. So, we see that polynomial interpolation overcomes most of the problems of linear interpolation.
However, polynomial interpolation also has some disadvantages. Calculating the interpolating polynomial is computationally expensive (see
computational complexity
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations ...
) compared to linear interpolation. Furthermore, polynomial interpolation may exhibit oscillatory artifacts, especially at the end points (see
Runge's phenomenon
In the mathematical field of numerical analysis, Runge's phenomenon () is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation ...
).
Polynomial interpolation can estimate local maxima and minima that are outside the range of the samples, unlike linear interpolation. For example, the interpolant above has a local maximum at ''x'' ≈ 1.566, ''f''(''x'') ≈ 1.003 and a local minimum at ''x'' ≈ 4.708, ''f''(''x'') ≈ −1.003. However, these maxima and minima may exceed the theoretical range of the function; for example, a function that is always positive may have an interpolant with negative values, and whose inverse therefore contains false
vertical asymptotes.
More generally, the shape of the resulting curve, especially for very high or low values of the independent variable, may be contrary to commonsense; that is, to what is known about the experimental system which has generated the data points. These disadvantages can be reduced by using spline interpolation or restricting attention to
Chebyshev polynomials
The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions:
...
.
Spline interpolation

Linear interpolation uses a linear function for each of intervals
''k'',''x''''k+1''">'x''''k'',''x''''k+1'' Spline interpolation uses low-degree polynomials in each of the intervals, and chooses the polynomial pieces such that they fit smoothly together. The resulting function is called a spline.
For instance, the
natural cubic spline
In the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. That is, instead of fitting a single, high-degree polynomial to all ...
is
piecewise
In mathematics, a piecewise function (also called a piecewise-defined function, a hybrid function, or a function defined by cases) is a function whose domain is partitioned into several intervals ("subdomains") on which the function may be ...
cubic and twice continuously differentiable. Furthermore, its second derivative is zero at the end points. The natural cubic spline interpolating the points in the table above is given by
:
In this case we get ''f''(2.5) = 0.5972.
Like polynomial interpolation, spline interpolation incurs a smaller error than linear interpolation, while the interpolant is smoother and easier to evaluate than the high-degree polynomials used in polynomial interpolation. However, the global nature of the basis functions leads to ill-conditioning. This is completely mitigated by using splines of compact support, such as are implemented in Boost.Math and discussed in Kress.
Mimetic interpolation
Depending on the underlying discretisation of fields, different interpolants may be required. In contrast to other interpolation methods, which estimate functions on target points, mimetic interpolation evaluates the integral of fields on target lines, areas or volumes, depending on the type of field (scalar, vector, pseudo-vector or pseudo-scalar).
A key feature of mimetic interpolation is that
vector calculus identities
The following are important identities involving derivatives and integrals in vector calculus.
Operator notation
Gradient
For a function f(x, y, z) in three-dimensional Cartesian coordinate variables, the gradient is the vector field:
:
...
are satisfied, including
Stokes' theorem
Stokes' theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls, or simply the curl theorem, is a theorem in vector calculus on \R^3. Given a vector field, the theorem relates th ...
and the
divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem relating the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the volume ...
. As a result, mimetic interpolation conserves line, area and volume integrals. Conservation of line integrals might be desirable when interpolating the
electric field
An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
, for instance, since the line integral gives the
electric potential
Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physic ...
difference at the endpoints of the integration path. Mimetic interpolation ensures that the error of estimating the line integral of an electric field is the same as the error obtained by interpolating the potential at the end points of the integration path, regardless of the length of the integration path.
Linear
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a '' function'' (or '' mapping'');
* linearity of a '' polynomial''.
An example of a linear function is the function defined by f(x) ...
,
bilinear and
trilinear interpolation
Trilinear interpolation is a method of multivariate interpolation on a Three dimensional space, 3-dimensional regular grid. It approximates the value of a function at an intermediate point (x, y, z) within the local axial rectangular prism (geo ...
are also considered mimetic, even if it is the field values that are conserved (not the integral of the field). Apart from linear interpolation, area weighted interpolation can be considered one of the first mimetic interpolation methods to have been developed.
Functional interpolation
The
Theory of Functional Connections (TFC) is a mathematical framework specifically developed fo
functional interpolation Given any interpolant that satisfies a set of constraints, TFC derives a functional that represents the entire family of interpolants satisfying those constraints, including those that are discontinuous or partially defined. These functionals identify the subspace of functions where the solution to a constrained optimization problem resides. Consequently, TFC transforms constrained optimization problems into equivalent unconstrained formulations. This transformation has proven highly effective in the solution of
differential equations. TFC achieves this by constructing a constrained functional (a function of a free function), that inherently satisfies given constraints regardless of the expression of the free function. This simplifies solving various types of equations and significantly improves the efficiency and accuracy of methods like
Physics-Informed Neural Networks (PINNs). TFC offers advantages over traditional methods like
Lagrange multipliers
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...
and
spectral method
Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain " basis funct ...
s by directly addressing constraints analytically and avoiding iterative procedures, although it cannot currently handle inequality constraints.
Function approximation
Interpolation is a common way to approximate functions. Given a function
with a set of points
, b
The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> one can form a function
such that
for
(that is, that
interpolates
at these points). In general, an interpolant need not be a good approximation, but there are well known and often reasonable conditions where it will. For example, if
(four times continuously differentiable) then
cubic spline interpolation has an error bound given by
where
and
is a constant.
Via Gaussian processes
Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
is a powerful non-linear interpolation tool. Many popular interpolation tools are actually equivalent to particular Gaussian processes. Gaussian processes can be used not only for fitting an interpolant that passes exactly through the given data points but also for regression; that is, for fitting a curve through noisy data. In the geostatistics community Gaussian process regression is also known as
Kriging
In statistics, originally in geostatistics, kriging or Kriging (), also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging g ...
.
Inverse Distance Weighting
Inverse Distance Weighting
Inverse distance weighting (IDW) is a type of Deterministic algorithm, deterministic method for multivariate interpolation with a known homogeneously scattered set of points. The assigned values to unknown points are calculated with a Weighted m ...
(IDW) is a spatial interpolation method that estimates values based on nearby data points, with closer points having more influence. It uses an inverse power law for weighting, where higher power values emphasize local effects, while lower values create a smoother surface. IDW is widely used in
GIS,
meteorology
Meteorology is the scientific study of the Earth's atmosphere and short-term atmospheric phenomena (i.e. weather), with a focus on weather forecasting. It has applications in the military, aviation, energy production, transport, agricultur ...
, and environmental modeling for its simplicity but may produce artifacts in clustered or uneven data.
Other forms
Other forms of interpolation can be constructed by picking a different class of interpolants. For instance, rational interpolation is interpolation by
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s using
Padé approximant
In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is ap ...
, and
trigonometric interpolation is interpolation by
trigonometric polynomial
In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The c ...
s using
Fourier series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
. Another possibility is to use
wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the n ...
s.
The
Whittaker–Shannon interpolation formula can be used if the number of data points is infinite or if the function to be interpolated has compact support.
Sometimes, we know not only the value of the function that we want to interpolate, at some points, but also its derivative. This leads to
Hermite interpolation
In numerical analysis, Hermite interpolation, named after Charles Hermite, is a method of polynomial interpolation, which generalizes Lagrange interpolation. Lagrange interpolation allows computing a polynomial of degree less than that takes th ...
problems.
When each data point is itself a function, it can be useful to see the interpolation problem as a partial
advection
In the fields of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is a ...
problem between each data point. This idea leads to the
displacement interpolation problem used in
transportation theory.
In higher dimensions
Multivariate interpolation is the interpolation of functions of more than one variable.
Methods include
nearest-neighbor interpolation
Nearest-neighbor interpolation (also known as proximal interpolation or, in some contexts, point sampling) is a simple method of multivariate interpolation in one or more dimensions.
Interpolation is the problem of approximating the value of a ...
,
bilinear interpolation
In mathematics, bilinear interpolation is a method for interpolating functions of two variables (e.g., ''x'' and ''y'') using repeated linear interpolation. It is usually applied to functions sampled on a 2D rectilinear grid, though it can be ge ...
and
bicubic interpolation
In mathematics, bicubic interpolation is an extension of cubic spline interpolation (a method of applying cubic interpolation to a data set) for interpolating data points on a two-dimensional regular grid. The interpolated surface (meaning the k ...
in two dimensions, and
trilinear interpolation
Trilinear interpolation is a method of multivariate interpolation on a Three dimensional space, 3-dimensional regular grid. It approximates the value of a function at an intermediate point (x, y, z) within the local axial rectangular prism (geo ...
in three dimensions.
They can be applied to gridded or scattered data. Mimetic interpolation generalizes to
dimensional spaces where
.
Image:Nearest2DInterpolExample.png, Nearest neighbor
Image:BilinearInterpolExample.png, Bilinear
Image:BicubicInterpolationExample.png, Bicubic
In digital signal processing
In the domain of digital signal processing, the term interpolation refers to the process of converting a sampled digital signal (such as a sampled audio signal) to that of a higher sampling rate (
Upsampling
In digital signal processing, upsampling, expansion, and interpolation are terms associated with the process of sample rate conversion, resampling in a multi-rate digital signal processing system. ''Upsampling'' can be synonymous with ''expansion'' ...
) using various digital filtering techniques (for example, convolution with a frequency-limited impulse signal). In this application there is a specific requirement that the harmonic content of the original signal be preserved without creating aliased harmonic content of the original signal above the original
Nyquist limit
In signal processing, the Nyquist frequency (or folding frequency), named after Harry Nyquist, is a characteristic of a sampler, which converts a continuous function or signal into a discrete sequence. For a given sampling rate (''samples per ...
of the signal (that is, above fs/2 of the original signal sample rate). An early and fairly elementary discussion on this subject can be found in Rabiner and Crochiere's book ''Multirate Digital Signal Processing''.
Related concepts
The term ''
extrapolation
In mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. ...
'' is used to find data points outside the range of known data points.
In
curve fitting
Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is ...
problems, the constraint that the interpolant has to go exactly through the data points is relaxed. It is only required to approach the data points as closely as possible (within some other constraints). This requires parameterizing the potential interpolants and having some way of measuring the error. In the simplest case this leads to
least squares
The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
approximation.
Approximation theory
In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characteri ...
studies how to find the best approximation to a given function by another function from some predetermined class, and how good this approximation is. This clearly yields a bound on how well the interpolant can approximate the unknown function.
Generalization
If we consider
as a variable in a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, and the function
mapping to a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
, then the problem is treated as "interpolation of operators".
[Colin Bennett, Robert C. Sharpley, ''Interpolation of Operators'', Academic Press 1988] The classical results about interpolation of operators are the
Riesz–Thorin theorem and the
Marcinkiewicz theorem. There are also many other subsequent results.
See also
*
Barycentric coordinates – for interpolating within on a triangle or tetrahedron
*
Brahmagupta's interpolation formula
*
Discretization
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numeri ...
*
Fractal interpolation
*
Imputation (statistics)
In statistics, imputation is the process of replacing missing data with substituted values. When substituting for a data point, it is known as "unit imputation"; when substituting for a component of a data point, it is known as "item imputation". ...
*
Lagrange interpolation
In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data.
Given a data set of coordinate pairs (x_j, y_j) with 0 \leq j \leq k, the x_j are called ''nodes'' ...
*
Missing data
In statistics, missing data, or missing values, occur when no data value is stored for the variable in an observation. Missing data are a common occurrence and can have a significant effect on the conclusions that can be drawn from the data.
Mi ...
*
Newton–Cotes formulas
In numerical analysis, the Newton–Cotes formulas, also called the Newton–Cotes quadrature rules or simply Newton–Cotes rules, are a group of formulas for numerical integration (also called ''quadrature'') based on evaluating the integrand a ...
*
Radial basis function interpolation
Radial basis function (RBF) interpolation is an advanced method in approximation theory for constructing Order of accuracy, high-order accurate interpolation, interpolants of unstructured data, possibly in high-dimensional spaces. The interpolant t ...
*
Simple rational approximation
*
Smoothing
In statistics and image processing, to smooth a data set is to create an approximating function that attempts to capture important patterns in the data, while leaving out noise or other fine-scale structures/rapid phenomena. In smoothing, the d ...
References
External links
* Online tools fo
linearquadraticcubic spline, an
polynomial interpolation with visualisation and
JavaScript
JavaScript (), often abbreviated as JS, is a programming language and core technology of the World Wide Web, alongside HTML and CSS. Ninety-nine percent of websites use JavaScript on the client side for webpage behavior.
Web browsers have ...
source code.
Sol Tutorials - Interpolation Tricks
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Video
Video signal