Math 6610 • Analysis of Numerical Methods I

{Interpolation and Integration}

\textbf{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.

\textbf{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 \(I(f)\).