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  1. 1. From Wikipedia, the free encyclopedia InterpolationInterpolationIn the mathematical subfield of numericalanalysis, interpolation is a method of con-structing new data points within the range ofa discrete set of known data points. In engineering and science one often has anumber of data points, as obtained bysampling or experimentation, and tries toconstruct a function which closely fits thosedata points. This is called curve fitting or re-gression analysis. Interpolation is a specificcase of curve fitting, in which the functionmust go exactly through the data points. A different problem which is closely re-lated to interpolation is the approximation ofa complicated function by a simple function.Suppose we know the function but it is toocomplex to evaluate efficiently. Then wecould pick a few known data points from thecomplicated function, creating a lookup An interpolation of a finite set of points on antable, and try to interpolate those data points epitrochoid. Points through which curve isto construct a simpler function. Of course, splined are red; the blue curve connectingwhen using the simple function to calculate them is interpolation.new data points we usually do not receive thesame result as when using the original func- This will give the same result as a lineartion, but depending on the problem domain function evaluated at the midpoint.and the interpolation method used the gain in Given a sequence of n distinct numbers xksimplicity might offset the error. called nodes and for each xk a second num- It should be mentioned that there is anoth- ber yk, we are looking for a function f so thater very different kind of interpolation inmathematics, namely the "interpolation of op-erators". The classical results about interpol- A pair xk,yk is called a data point and f isation of operators are the Riesz-Thorin theor- called an interpolant for the data points.em and the Marcinkiewicz theorem. There When the numbers yk are given by aalso are many other subsequent results. known function f, we sometimes write fk.Definition ExampleFrom inter meaning between and pole, the For example, suppose we have a table likepoints or nodes. Any means of calculating a this, which gives some values of an unknownnew point between two or more existing data function f.points is interpolation. x f(x) There are many methods for doing this,many of which involve fitting some sort of 0 0function to the data and evaluating that func- 1 0 . 8415tion at the desired point. This does not ex- 2 0 . 9093clude other means such as statistical meth- 3 0 . 1411ods of calculating interpolated data. 4 −0 . 7568 One of the simplest forms of interpolation 5 −0 . 9589is to take the arithmetic mean of the value of 6 −0 . 2794two adjacent points to find the mid point. 1
  2. 2. From Wikipedia, the free encyclopedia Interpolation but in higher dimensions, in multivariate in- terpolation, this can be a favourable choice for its speed and simplicity. Linear interpolationPlot of the data points as given in the table.Interpolation provides a means of estimatingthe function at intermediate points, such asx = 2.5. There are many different interpolation Plot of the data with linear interpolationmethods, some of which are described below. superimposedSome of the concerns to take into accountwhen choosing an appropriate algorithm are: One of the simplest methods is linear inter-How accurate is the method? How expensive polation (sometimes known as lerp). Consideris it? How smooth is the interpolant? How the above example of determining f(2.5).many data points are needed? Since 2.5 is midway between 2 and 3, it is reasonable to take f(2.5) midway betweenPiecewise constant f(2) = 0.9093 and f(3) = 0.1411, which yields 0.5252.interpolation Generally, linear interpolation takes two data points, say (xa,ya) and (xb,yb), and the interpolant is given by: at the point (x,y) Linear interpolation is quick and easy, but it is not very precise. Another disadvantage is that the interpolant is not differentiable at the point xk. The following error estimate shows that linear interpolation is not very precise. De- note the function which we want to interpol- ate by g, and suppose that x lies between xaPiecewise constant interpolation, or nearest- and xb and that g is twice continuously differ-neighbor interpolation. entiable. Then the linear interpolation error isFor more details on this topic, see Nearest-neighbor interpolation.The simplest interpolation method is to loc-ate the nearest data value, and assign the In words, the error is proportional to thesame value. In one dimension, there are sel- square of the distance between the datadom good reasons to choose this one over lin- points. The error of some other methods, in-ear interpolation, which is almost as cheap, cluding polynomial interpolation and spline 2
  3. 3. From Wikipedia, the free encyclopedia Interpolationinterpolation (described below), is propor- Runge’s phenomenon). These disadvantagestional to higher powers of the distance can be avoided by using spline interpolation.between the data points. These methods alsoproduce smoother interpolants. Spline interpolationPolynomial interpolation Plot of the data with Spline interpolation appliedPlot of the data with polynomial interpolationapplied Remember that linear interpolation uses a linear function for each of intervals [xk,xk+1].Polynomial interpolation is a generalization Spline interpolation uses low-degree polyno-of linear interpolation. Note that the linear mials in each of the intervals, and choosesinterpolant is a linear function. We now re- the polynomial pieces such that they fitplace this interpolant by a polynomial of smoothly together. The resulting function ishigher degree. called a spline. Consider again the problem given above. For instance, the natural cubic spline isThe following sixth degree polynomial goes piecewise cubic and twice continuously dif-through all the seven points: ferentiable. Furthermore, its second derivat- f(x) = − 0.0001521x6 − 0.003130x5 + ive is zero at the end points. The natural cu- 0.07321x4 − 0.3577x3 + 0.2255x2 + bic spline interpolating the points in the table 0.9038x. above is given bySubstituting x = 2.5, we find that f(2.5) =0.5965. Generally, if we have n data points, thereis exactly one polynomial of degree at mostn−1 going through all the data points. The in- In this case we get f(2.5)=0.5972.terpolation error is proportional to the dis- Like polynomial interpolation, spline inter-tance between the data points to the power polation incurs a smaller error than linear in-n. Furthermore, the interpolant is a polyno- terpolation and the interpolant is smoother.mial and thus infinitely differentiable. So, we However, the interpolant is easier to evaluatesee that polynomial interpolation solves all than the high-degree polynomials used inthe problems of linear interpolation. polynomial interpolation. It also does not suf- However, polynomial interpolation also fer from Runge’s phenomenon.has some disadvantages. Calculating the in-terpolating polynomial is computationally ex-pensive (see computational complexity) com- Interpolation via Gaussi-pared to linear interpolation. Furthermore,polynomial interpolation may not be so exact an processesafter all, especially at the end points (see Gaussian process is a powerful non-linear in- terpolation tool. Many popular interpolation tools are actually equivalent to particular 3
  4. 4. From Wikipedia, the free encyclopedia InterpolationGaussian processes. Gaussian processes can In curve fitting problems, the constraintbe used not only for fitting an interpolant that the interpolant has to go exactly throughthat passes exactly through the given data the data points is relaxed. It is only requiredpoints but also for regression, i.e. for fitting a to approach the data points as closely as pos-curve through noisy data. In the geostatistics sible. This requires parameterizing the poten-community Gaussian process regression is tial interpolants and having some way ofalso known as Kriging. measuring the error. In the simplest case this leads to least squares approximation.Other forms of Approximation theory studies how to find the best approximation to a given function byinterpolation another function from some predetermined class, and how good this approximation is.Other forms of interpolation can be construc- This clearly yields a bound on how well theted by picking a different class of inter- interpolant can approximate the unknownpolants. For instance, rational interpolation is function.interpolation by rational functions, and tri-gonometric interpolation is interpolation bytrigonometric polynomials. The discrete ReferencesFourier transform is a special case of trigono- • David Kidner, Mark Dorey and Derekmetric interpolation. Another possibility is to Smith (1999). What’s the point?use wavelets. Interpolation and extrapolation with a The Whittaker–Shannon interpolation for- regular grid DEM. IV Internationalmula can be used if the number of data Conference on GeoComputation,points is infinite. Fredericksburg, VA, USA. Multivariate interpolation is the interpola- • Kincaid, David; Ward Cheney (2002).tion of functions of more than one variable. Numerical Analysis (3rd edition). Brooks/Methods include bilinear interpolation and Cole. ISBN 0-534-38905-8. Chapter 6.bicubic interpolation in two dimensions, and • Schatzman, Michelle (2002). Numericaltrilinear interpolation in three dimensions. Analysis: A Mathematical Introduction. Sometimes, we know not only the value of Clarendon Press, Oxford. ISBNthe function that we want to interpolate, at 0-19-850279-6. Chapters 4 and 6.some points, but also its derivative. Thisleads to Hermite interpolation problems. External linksRelated concepts • DotPlacer applet : Applet showing various interpolation methods, with movableThe term extrapolation is used if we want to pointsfind data points outside the range of knowndata points.Retrieved from "http://en.wikipedia.org/wiki/Interpolation"Categories: InterpolationThis page was last modified on 14 May 2009, at 22:02 (UTC). All text is available under theterms of the GNU Free Documentation License. (See Copyrights for details.) Wikipedia® is aregistered trademark of the Wikimedia Foundation, Inc., a U.S. registered 501(c)(3) tax-deductible nonprofit charity. Privacy policy About Wikipedia Disclaimers 4