Math 6610 • Analysis of Numerical Methods I

{Interpolation and Integration:}

\partitem{a} Consider equally spaced points \(x_j=a+jh\), \(j=0,\ldots,J+1\) on the
interval \([a,b]\), where \((J+1)h=b-a\). Let \(f(x)\) be a function defined on
\([a,b]\). Consider the problem of finding a cubic spline approximation \(s(x)\) to
\(f(x)\) that interpolates \(f\) at the points \(x_j\); is twice continuously
differentiable, and satisfies \(s''(a)=s''(b)=0\). Does this problem always have a
solution? If your answer is yes, derive formulas by which to determine the
spline. If your answer is no, explain your reasoning.

\partitem{b} Let \(I_n(f)\) denote the result of using the composite Trapezoidal
rule to approximate \(I(f)=\int_a^b f(x)dx\) using \(n\) equally sized subintervals
of length \(h=(b-a)/n\). It can be shown that the integration error
\(E_n(f)=I(f)-I_n(f)\) satisfies
\[
  E_n(f)=d_2h^2+d_4h^4+d_6h^6+\cdots
\]
where \(d_2,d_4,d_6,\ldots\) are 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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