# interpolation

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## interpolation

[in‚tər·pə′lā·shən]
(mathematics)
A process used to estimate an intermediate value of one (dependent) variable which is a function of a second (independent) variable when values of the dependent variable corresponding to several discrete values of the independent variable are known.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

## interpolation

see MEASURES OF CENTRAL TENDENCY.
Collins Dictionary of Sociology, 3rd ed. © HarperCollins Publishers 2000
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Interpolation

an insertion or correction in an original text made by someone other than the author.

Interpolations played a pivotal role in the texts by Roman jurists that are compiled in the Digest. They were made to eliminate contradictions in the texts as well as the statutes and attitudes that were inappropriate for the Justinian era. Various kinds of interpolations were made, including specification and substitutions of the rule of law, substitution or elimination of terms, and lexical changes. The medieval humanists were the first to discover the interpolations in the Digest.

## Interpolation

in mathematics and statistics, the process of finding values of a quantity between some of its known values. An example is finding values of the function f(x) at points x lying between the points (nodes of interpolation) x0 < x1 < … < xn by means of the known values yi = f(x1), where i = 0, 1, …, n. In the case when x lies outside the interval included between x0 and xn, the analogous problem is called an extrapolation problem.

In the simplest case, linear interpolation, the value of f(x) at a point x satisfying the inequality x0 < x1, is taken to be equal to the value

of the linear function coinciding with f(x) at the points x = x0 and x = x1. The interpolation problem is undefined from a strict mathematical viewpoint: if nothing is known about the function f(x) except its values at the points x0, x1, …, xn, then its value at a point x, which is different from all these points, remains completely arbitrary. The interpolation problem acquires a definite meaning if the function f(x) and its derivatives are subject to certain inequalities. If, for example, the values f(x0) and f(x1) are given and it is known that for xo < x < x1 the inequality | f”(x) | ≤ M is fulfilled, then the error of the formula (*) may be estimated with the aid of the inequality

It makes sense to use more complex interpolation formulas only in the case when it is certain that the function is sufficiently “smooth,” that is, when it has a sufficient number of derivatives that do not increase rapidly.

In addition to the computation of values of functions, interpolation has numerous other applications (for example, approximate integration, approximate solution of equations, and, in statistics, the smoothing of distribution series with the aim of eliminating random distortions).

### REFERENCES

Goncharov, V. L. Teoriia interpolirovaniia i priblizheniia funktsii, 2nd ed. Moscow, 1954.
Krylov, A. N. Lektsii o pribiizhennykh vychisleniiakh, 6th ed. Moscow, 1954.
Yule, G. U., and M. G. Kendall. Teoriia statistiki, 14th ed. Moscow, 1960. (Translated from English.)
The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.

## interpolation

This article is provided by FOLDOC - Free Online Dictionary of Computing (foldoc.org)

## interpolation

In computer graphics, interpolation is the creation of new values that lie between known values. For example, when objects are rasterized into two-dimensional images from their corner points (vertices), all the pixels between those points are filled in by an interpolation algorithm, which determines their color and other attributes (see graphics pipeline).

Another example is when a video image in a low resolution is upscaled to display on a monitor with a higher resolution, the missing lines are created by interpolation. In a digital camera, the optical zoom is based on the physical lenses, but the digital zoom is accomplished by algorithms (see interpolated resolution).
Copyright © 1981-2019 by The Computer Language Company Inc. All Rights reserved. THIS DEFINITION IS FOR PERSONAL USE ONLY. All other reproduction is strictly prohibited without permission from the publisher.
References in periodicals archive ?
In Tables 3.1-3.2, finally, we report the interpolation errors on such domains for two functions with different degree of regularity, correspondingly to a sequence of Xu-like nodal sets.
The interpolation error in the discrete [H.sup.1]--norm is not convergent on anisotropic rectangular meshes.
If the initial estimate has missed detections, the obtained small interpolation sectors fail to cover the missed sources inside, which leads to big interpolation errors in the directions of the missed sources, thus causing big DOA estimate bias for the missed sources, while these detected sources that are covered in the obtained small interpolation sectors can still be detected with increased DOA estimation bias.
The kernel function approximation simulation showed that the improved method acquires higher accuracy whose interpolation error is lesser by one or two magnitudes than the Pravin's.
involving six samples; by this kernel, the interpolation error has a fourth-order convergence rate.
Interpolation error m was calculated using following formula:
(3) Because [[??].sub.1] (t) and [[??].sub.2] (t) are uncorrelated by construction, we can make use of the Pythagorean theorem to express the interpolation error at [[sigma].sub.t]
Sources of uncertainty in PSI angle measurements Source of uncertainty Conditions Standard uncertainty (arcsecond) Scale error Range <[+ or -]2.5" 0.0025 Small-scale nonlinearity Range <[+ or -]2.5" 0.007 and measurement noise Pyramid error Pyramid angle < 15" 0.003 Eccentricity Runout < 0.2 mm 0.004 Diffraction and edge effects x tilt <2.5", good focus 0.004 Periodic interpolation error 20 phase averages 0.005 Bull's-eye patterns Moderately clean optics 0.002 Aberrations when viewing <150 nm P-V form error 0.004 non-flat faces of faces Combined standard uncertainty 0.012 Expanded uncertainty 0.024 (k = 2)
This set of procedures, termed "topographically informed interpolation" (TII), reduces interpolation error relative to more traditional techniques (Willmott and Matsuura, 1995).
From standard results in approximation theory, we know that if this Hermite-Birkhoff interpolant is used to approximate a sufficiently differentiable function, then the associated interpolation error is O([h.sup.d+1]).
The consistency of the schemes follows directly from the estimate for the interpolation error of the exponential sum.

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