Math 6610 • Analysis of Numerical Methods I

{Interpolation and Integration:}

\partitem{a} Consider \(n+1\) distinct points \(x_0<x_1<\cdots<x_n\) in the interval
\([a,b]\). Let \(f(x)\) be a smooth function defined on \([a,b]\). Show that there is
a unique polynomial \(p(x)\) of degree \(n\) which interpolates \(f\) at all of the
points \(x_j\). 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 \(\eta\in[a,b]\).

\partitem{b} Consider the problem of approximating the integral
\[
  I(f)=\int_a^b f(x)\,dx
\]
by a formula of the type
\[
  I_n(f)=\sum_{j=1}^{n} a_j f(x_j)
\]
where \(x_1,x_2,\ldots,x_n\) are distinct points in the interval \((a,b)\). Derive
formulas for \(a_j\), \(j=1,\ldots,n\) so the \(I_n(f)=I(f)\) when \(f\) is any
polynomial of degree less than or equal to \(n\).

\partitem{c} For the same approximate integration problem as in (b), explain how
to choose the points \(x_1,x_2,\ldots,x_n\) and coefficients \(a_j\),
\(j=1,\ldots,n\), so that \(I_n(f)=I(f)\) for all polynomials of degree less than or
equal to \(2n-1\)? Prove that your proposed choice does give the exact integral
for these polynomials.

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