Math 6610 • Analysis of Numerical Methods I

{Interpolation:}

Consider equally spaced points \(x_j=a+jh\), \(j=0,\ldots,n\) on the interval
\([a,b]\), where \(nh=b-a\). Let \(f(x)\) be a smooth function defined on \([a,b]\).

\partitem{a} Show that there is a unique polynomial \(p(x)\) of degree \(n\) which
interpolates \(f\) at all of the points \(x_j\).

\partitem{b} Derive the formula for the interpolation error at an arbitrary
point \(x\) in the interval \([a,b]\):
\[
  f(x)-p(x)\equiv E(x)
  =
  \frac{1}{(n+1)!}(x-x_0)(x-x_1)\cdots(x-x_n)f^{n+1}(\eta).
\]
for some \(\eta\in[a,b]\).

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