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\noindent \textbf{Physics 567}\\
\noindent \textbf{Homework 1}\\
\noindent Due Monday, January 29, 2024.\\
\begin{enumerate}
   \item In class, I asserted that the truncation error from ``Tres Hermanos''
      quadrature of a single interval is of order $f^{(4)}(x)\, h^5$. Show that this is true, 
      starting from
      \[
         \int_{-\frac{h}{2}}^{\frac{h}{2}} f(x)\,dx  \approx  h \times   \label{eq:th}
      \left( \frac{1}{24} f_1 + \frac{11}{12} f_2 + \frac{1}{24} f_3 \right).
      \]
      Use this result to argue that the extended Tres Hermanos rule should converge as $1/N^4$. 
      For simplicity, you may assume all $N$ steps, including the ends, are evaluated by the 
      above formula.
   \item Evaluate numerically:
      \[
      \int_0^1 \frac{(1-x)^{\frac{1}{3}}}{\ln x}\,dx.
      \]
   \item Plot the rate of convergence ($\log \left| \mathrm{error} \right|$ vs. $\log N$). Use the results to estimate the accuracy you achieved. 
\end{enumerate}
Please turn in electronically to the grader:
\begin{itemize}
   \item Analytic answer to the first question. This is an opportunity to 
      dust off your LaTeX (or MathType, or whatever) skills, which you will 
      put to use for the final project. Note that the latex source for this
      document (\texttt{integration.tex}) is provided as an example. As a last resort, you may turn in
      a scanned pencil and paper solution.
   \item Well-commented source code (one or more files; *.m, *.py, etc.) for questions 2\& 3.
   \item Report a result that is accurate to at least 10 significant figures.\footnote{It is possible to reach 15-figure accuracy by the methods we have been using in class.} 
      You may need to use the command ``\texttt{format long}'' if using Matlab/Octave.
   \item Any commands needed to complete the assignment.  Copy of IO from the terminal would be great for this and the previous bullet.
   \item Well-labeled convergence plot. 
\end{itemize}
\emph{In this and all future homework assignments, I encourage  you to make use of existing code including my examples and the built-in functions in Octave and Matlab.}

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