Math 6610 • Analysis of Numerical Methods I

\textbf{Interpolation.} Suppose you are given an arbitrarily smooth function \(f\)
of a single variable \(x\), and points
\[
  x_0<x_1<\cdots<x_n.
\]
Let \(p\) be the interpolating polynomial of degree \(n\) satisfying
\[
  p(x_i)=f(x_i),\qquad i=0,1,\ldots,n.
\]
Derive an expression for the error
\[
  f(x)-p(x)
\]
for a general value of \(x\), in terms of the derivatives of \(f\).