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\doclabel{Math 426: Homework 10}
\docauthor{Your name here!}
\docdate{November 4, 2020}

\begin{document}

\begin{exercise}{Supplemental 1}
Consider these three points: $\{(1, 1), (2.5, 8), (4, 5)\}$.
Find the polynomial $P(x)$ of degree 2 which passes through these points. Do this three different ways, by using
\begin{enumerate}
\item[(a)] the Vandermonde matrix method,
\item[(b)] the Newton form and its triangular matrix method, and
\item[(c)] the Lagrange form.
\end{enumerate}
\end{exercise}


\begin{exercise}{Supplemental 2}
Consider the $x$ coordinates $x_0=0$, $x_1=\pi/3$, $x_2=2\pi/3$ and $x_3=\pi$.
\begin{enumerate}
	\item[a)] Plot the four Lagrange basis functions $\phi_k$ $k=0,\ldots,4$ on a single graph with domain $[0,\pi]$.
	\item[b)] Plot the four Newton interpolation basis functions
	$\psi_k$, each on its own individual graph.
	\item[c)] Plot the graph of $\sin(x)$ along with its Lagrange interpolant $p_{\mathrm{Lag}}$.
	\item[d)] Plot the graph of $\sin(x)$ along with its Newton interpolant $p_{\mathrm{Newt}}$.
	\item[e)] What is the relative error of $p_{\mathrm{Lag}}(\pi/4)$?
\end{enumerate}
\end{exercise}

\begin{exercise}{Exercise 8.1}
\end{exercise}

\begin{exercise}{Exercise 8.2}
\end{exercise}

\begin{exercise}{Midterm, problem 6}
\end{exercise}

\end{document}