Math 6620 • Analysis of Numerical Methods II

(20 pts)
\begin{enumerate}

\item
(10 pts) Compute coefficients \(A\), \(B\), \(C\) in the 3-point one-sided approximation
\[
u''(x) \approx Au(x) + Bu(x+h) + Cu(x+2h)
\]

that has optimal \(h\)-order of accuracy. What is this order of accuracy?


\item
(10 pts) Compute weights for the following quadrature rule that makes it exact for all
polynomials of degree 2.
\[
\int_{-1}^1 f(x)\,dx = w_0 f(0) + w_1 f'(0) + w_2 f''(0),
\]

\end{enumerate}