# 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 It might be outdated or ideologically biased.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.)

## 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).
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References in periodicals archive ?
When the number of the signature shares got by any one participant or signature receiver reaches 2t + 1, he can compute the signature 5 according to the interpolation formula. Thus the signature (r, s) of the message m is got.
Faced with the same task today, high school students could easily implement Lagrange's Interpolation Formula in a spreadsheet.
While Lagrange's interpolation formula can be stated for a general case involving n points and a polynomial of degree n - 1, it is more easily understood by looking at a particular case.
(If we accidentally apply the quadratic form of Lagrange's interpolation formula to three collinear points, the right hand side will gracefully collapse to a linear function.)
The computational implementation and performance of the remaining interpolation formulas of Section 3 are very similar to those shown in this section.
We observe that Eq.(5) and Eq.(6) follow from Shamir's secret reconstruction algorithm using Lagrange interpolation formula and the property of additive homomorphism, respectively.
* Reconstruction phase: Based on Lagrange interpolation formula, the secret verification and reconstruction can be completed by computing the same interpolating coefficients [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (d = 1, ..., t).
In [3] we investigated numerical aspects of the Xu polynomial interpolation formula in the square.
We start by recalling briefly the construction of the Xu interpolation formula of degree n on the square [[-1,1].sup.2].
Rearranging (2.7) in the case that cos([alpha]) = cos([beta]), allows us to give a form of the interpolation formula with pointwise evaluation cost O(N).
The subject of the talks were in the areas of graph theory, mathematical interpolation formulas, estimating data errors in geographic information systems, how to rank computer performance, how to compare of databases of genetic information, how to write programs to make databases from computer surveys, and image feature recognition in remote sensing.
[11] --, On constructing new interpolation formulas using linear operators and an operator type of quadrature rules, J.

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