Interpolation

interpolation, 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 produces a polynomial function that fits the data:f(x) = a₀ + ^{a₁(x − x₀)}/_h + ^{a₂(x − x₀)(x − x₁)}/_2!h² + … + ^{a_n(x − x₀)⋯(x − x_n − 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).