Math 6610 • Analysis of Numerical Methods I

{Interpolation and Integration:}

\partitem{a} 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
\([a,b]\). 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} Let \(I_n(f)\) denote the result of using the composite Trapezoidal
rule to approximate
\[
  I(f)\equiv \int_a^b f(x)\,dx
\]
using \(n\) equally sized subintervals of length \(h=(b-a)/n\). It can be shown
that the integration error \(E_n(f)\equiv I(f)-I_n(f)\) satisfies
\[
  E_n(f)=d_2h^2+d_4h^4+d_6h^6+\cdots
\]
where \(d_2,d_4,d_6,\ldots\) are known numbers that depend only on the values of
\(f\) and its derivatives at \(a\) and \(b\). Suppose you have a black-box program
that, given \(f\), \(a\), \(b\), and \(n\), calculates \(I_n(f)\). Show how to use this
program to obtain an \(O(h^4)\) approximation and an \(O(h^6)\) approximation to
\(I(f)\).

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