# What is the motivation for this theorem on Polynomial Interpolation Error?

My textbook presents this theorem without any sort of introduction. It does cover using the Newton and Lagrange forms of the interpolation polynomial, so I've got that. Anyway, here's the theorem:

Theorem on Polynomial Interpolation Error

Let $f$ be differentiable $n + 1$ times on the interval $[a, b]$, and let $p$ be the polynomial of degree at most $n$ that interpolates the function $f$ at $n + 1$ distinct points $x_0, x_1, \ldots, x_n$ in the interval $[a, b]$. To each $x$ in $[a, b]$ there corresponds a point $\xi_x$ in $(a, b)$ such that $$f(x) - p(x) = \frac{1}{(n + 1)!} f^{n + 1}(\xi_x)\prod_{i = 0}^n(x - x_i).$$

I'm hoping that someone could briefly explain where this equation comes from, or at least where I might find more information.

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## 1 Answer

The formula originates from the natural desire to have an estimate for the error of polynomial interpolation. Since it involves $f^{(n+1)}$ evaluated at an unknown point lying in a certain interval, the practical use of this formula requires one to obtain an upper bound for $|f^{(n+1)}|$ on said interval. If $M$ is such a bound, then $$|f(x) - p(x)| \le \frac{M}{(n + 1)!} \prod_{i = 0}^n |x - x_i|$$ Looking at this estimate, we realize the importance of the placement of the nodes $x_i$. Trying to place them optimally, with $\prod_{i = 0}^n |x - x_i|$ as small as possible on the interval of interpolation, leads us to the Chebyshev nodes.

The proof of the error formula is a clever application of Rolle's theorem: you can read it in ProofWiki.

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