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]

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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.

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▪ mathematics

      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 < … < x_n and y₀ = f(x₀),…, y_n = f(x_n) are known, and if x₀ < x < x_n, then the estimated value of f(x) is said to be an interpolation. If x < x₀ or x > x_n, the estimated value of f(x) is said to be an extrapolation.

 If x₀, …, x_n are given, along with corresponding values y₀, …, y_n (see the figure—>), interpolation may be regarded as the determination of a function y = f(x) whose graph passes through the n + 1 points, (x_i, y_i) for i = 0, 1, …, n. There are infinitely many such functions, but the simplest is a polynomial interpolation function y = p(x) = a₀ + a₁x + … + a_nxⁿ with constant  a_i's such that p(x_i) = y_i for i = 0, …, n. There is exactly one such interpolating polynomial of degree n or less. If the x_i'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).

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