GuideInterpolationbasic

Interpolation: idea, existence and error

What interpolation is, why polynomials are used, the Weierstrass theorem, uniqueness of the interpolating polynomial and the error bound shared by Newton, Lagrange and Hermite.

What problem interpolation solves

We start from a data table: we know the value of a function at a few points, but not its formula. Interpolation builds a simple function, often a polynomial, through those points to estimate intermediate values we did not measure.

If the point we want to estimate lies inside the data interval, we call it interpolation. If it lies outside, we call it extrapolation; the polynomial is less controlled there, so the estimate carries more risk.

Polynomials are used because their derivatives and integrals are again polynomials and they are easy to evaluate. The generic degree- $n$ form below is the object we are after:

p_n(x)=\sum_{i=0}^{n} a_i x^i,\qquad n\ge 0,\ a_i\in\mathbb{R}

Existence and uniqueness

For interpolation a stronger fact is key: given $n+1$ points with distinct abscissas, there is a unique polynomial of degree at most $n$ through all of them. Newton, Lagrange and Hermite do not give different polynomials: they give the same polynomial written differently, each convenient for a purpose.

The error bound

All families share the same error expression. If $f$ is smooth enough on $[a,b]$ and $p_n$ interpolates at $x_0<\dots<x_n$ , then for each $x$ there is a $\xi$ in the interval such that:

f(x)-p_n(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}\prod_{i=0}^{n}(x-x_i)

The error depends on two things: the $(n+1)$ -th derivative (a property of the function) and the product of distances to the nodes (a property of where you evaluate). Near the nodes the product is small; between widely spaced nodes or outside the interval it can grow a lot.