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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 It might be outdated or ideologically biased.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.
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The present study proposed a meshless improved interpolating boundary node method based on the parameter space.
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We use this simplification in the rest of this paper.) We show that the above formula is a special case of the mentioned general interpolating formula.
Definition 2.5 A sampling operator [S.sup.K.sub.w], defined by (4), and its kernel K are said to be interpolating if K(n) = [[delta].sub.0,n] for n [member of] Z with Kronecker's delta.
In one application, an interpolating DAC was operating as a quadrature upconverter that relied on an internal DDS NCO.
Since [[PI].sup.d.sub.M] is [[PHI].sub.k]([bar.n]) interpolating, hence rank H = m and, thus, rank L [greater than or equal to] rank [L.sub.1] = mN and Q [??] C.
Figure 1 represents an interpolating elastica with natural end conditions and with knots n, four in number.
While interpolating of Minister of State for Cabinet Affairs and Acting Minister of Information Sheikh Mohammad Abdullah Al-Sabah at a parliamentary session, MP Al-Adsani said that reports by the audit bureau showed that the expenses of the CMGS were on the rise, reaching KD 3.6 million (around USD 11.9 million) in the last seven years.
When the interpolant satisfies the conditions, then the resulting interpolating curve or approximating curve preserves the shape of the given data.
He covers fractional monotone approximation theory, approximation by discrete singular operators of Favard style, approximation by interpolating operators induced by neural networks, and approximation theory and functional analysis on time scales.
Mustafa and Rehman [4] unified all existing even-point interpolating and approximating schemes by offering general formula for the mask of (2m + 4)-point even-ary subdivision scheme.

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