interpolation

/in terr'peuh lay"sheuhn/, n.
1. the act or process of interpolating or the state of being interpolated.
2. something interpolated, as a passage introduced into a text.
3. Math.
a. the process of determining the value of a function between two points at which it has prescribed values.
b. a similar process using more than two points at which the function has prescribed values.
c. the process of approximating a given function by using its values at a discrete set of points.
[1605-15; < L interpolation- (s. of interpolatio). See INTERPOLATE, -ION]

* * *

In mathematics, estimation of a value between two known data points.

A simple example is calculating the mean (see mean, median, and mode) of two population counts made 10 years apart to estimate the population in the fifth year. Estimating outside the data points (e.g., predicting the population five years after the second population count) is called extrapolation. If more than two data points are available, a curve may fit the data better than a line. The simplest curve that fits is a polynomial curve. Exactly one polynomial of any given degree
an interpolating polynomial
passes through any number of data points.

* * *

      in mathematics, the determination or estimation of the value of f(x), or a function of x, from certain known values of the function. If x0 < … < xn and y0 = f(x0),…, yn = f(xn) are known, and if x0 < x < xn, then the estimated value of f(x) is said to be an interpolation. If x < x0 or x > xn, the estimated value of f(x) is said to be an extrapolation.

 If x0, …, xn are given, along with corresponding values y0, …, yn (see the figure—>), interpolation may be regarded as the determination of a function y = f(x) whose graph passes through the n + 1 points, (xi, yi) for i = 0, 1, …, n. There are infinitely many such functions, but the simplest is a polynomial interpolation function y = p(x) = a0 + a1x + … + anxn with constant  ai's such that p(xi) = yi for i = 0, …, n. There is exactly one such interpolating polynomial of degree n or less. If the xi's are equally spaced, say by some factor h, then the following formula of Isaac Newton (Newton, Sir Isaac) produces a polynomial function that fits the data:
f(x) = a0 + a1(x − x0)/h + a2(x − x0)(x − x1)/2!h2 + … + an(x − x0)⋯(x − xn − 1)/n!hn

      Polynomial approximation is useful even if the actual function f(x) is not a polynomial, for the polynomial p(x) often gives good estimates for other values of f(x).

* * *


Universalium. 2010.

Look at other dictionaries:

  • interpolation — [ ɛ̃tɛrpɔlasjɔ̃ ] n. f. • 1706; « interruption » XIVe; lat. interpolatio 1 ♦ Action d interpoler un texte; résultat de cette action. 2 ♦ (1812) Math. Intercalation de valeurs ou de termes intermédiaires dans une série de valeurs ou de termes… …   Encyclopédie Universelle

  • Interpolation — Interpolation, die Berechnung von Zwischenwerten einer Funktion, wenn für eine endliche Anzahl von Werten des Arguments die zugehörigen Funktionswerte gegeben sind. [206] Bilden die zu den m + 1 gegebenen Funktionswerten u0 u1 ... um gehörigen… …   Lexikon der gesamten Technik

  • Interpolation — (lat. interpolare „auffrischen“, „umgestalten“, „verfälschen“) bezeichnet: in der Mathematik verschiedene Probleme und Verfahren, siehe Interpolation (Mathematik) in der Literatur die Veränderung eines ursprünglichen Textes, siehe Interpolation… …   Deutsch Wikipedia

  • Interpolation — ist ein Verfahren zur Berechnung neuer Bildpunkte (Pixel) aus den Werten der umliegenden Bildpunkte in der elektronischen Bildbearbeitung, und wird eingesetzt, sobald ein Bild vergrößert oder verkleinert wird. So bieten die Programme zur… …   Das Lexikon aus „Bernie's Foto-Programm"

  • Interpolation — (v. lat.), 1) Einschaltung; 2) (Math.), wenn man weiß, y hänge von x nach irgend einem Rechnungsgesetze ab, man kennt aber die Natur dieser Function nicht, sondern kennt nur einige einzelne Werthe y1, y2, y3, ... welche zu eben so vielen Werthen… …   Pierer's Universal-Lexikon

  • interpolation — index expletive, insertion, intervention (interference) Burton s Legal Thesaurus. William C. Burton. 2006 …   Law dictionary

  • Interpolation — In*ter po*la tion, n. [L. interpolatio an alteration made here and there: cf. F. interpolation.] 1. The act of introducing or inserting anything, especially that which is spurious or foreign. [1913 Webster] 2. That which is introduced or inserted …   The Collaborative International Dictionary of English

  • interpolation — (n.) 1610s, from Fr. interpolation (early 17c.), or directly from L. interpolationem (nom. interpolatio), noun of action from pp. stem of interpolare (see INTERPOLATE (Cf. interpolate)) …   Etymology dictionary

  • Interpolation — (lat., »Einschaltung«), in der Mathematik bezeichnet I. ein Verfahren, durch das man zwischen zwei Gliedern einer nach einem bestimmten Gesetz fortschreitenden Reihe von Größen neue Größen derart einschaltet, daß sie sich an dieses Gesetz… …   Meyers Großes Konversations-Lexikon

  • Interpolation — In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points. In engineering and science one often has a number of data points, as obtained… …   Wikipedia

  • Interpolation — Cette page d’homonymie répertorie les différents sujets et articles partageant un même nom. Sur les autres projets Wikimedia : « Interpolation », sur le Wiktionnaire (dictionnaire universel) Le mot interpolation est employé dans… …   Wikipédia en Français

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.