@@ -9,14 +9,14 @@ where $h := x_2 - x_1 = x_1 - x_0$ is the width of one slice. Note that the erro
\subsubsection*{Problems:}
\begin{enumerate}
\item Show that Trapezoidal rule provides an estimate of the integral with local error $\mathcal{O}\,(h^3)$.
\item Show that Trapezoidal rule provides an estimate of the integral with local error $\mathcal{O}\,(h^3)$. For this you can follow the steps similar to the ones that we used in the class to prove that the Midpoint rule has an error of $\mathcal{O}\,(h^3)$.
\item Show that Simpson's 1/3 rule provides an estimate of the integral with local error $\mathcal{O}\,(h^5)$.
\item Compute the following integral numerically using the two explicit methods given above, as well as using Romberg's method:
\item Compute the following integral numerically using the two explicit methods given above (Trapezoidal and Simpson's) %, as well as using Romberg's method:
\begin{equation}
I = \int_{-1}^{1} e^{-x^2} dx.
\end{equation}
You may use the following SciPy functions: \href{http://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.integrate.cumtrapz.html#scipy.integrate.cumtrapz}{\texttt{scipy.integrate.cumtrapz}}, \href{http://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.integrate.simps.html#scipy.integrate.simps}{\texttt{scipy.integrate.simps}} and \href{https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.romb.html}{\texttt{scipy.integrate.romb}}.
\item Repeat the calculation using three different values of $h$. Plot the error $\Delta I$ against $h$. Show that the numerical estimates converge to the exact value according to the expected order of convergence.
You may use the following SciPy functions: \href{https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.trapezoid.html#scipy.integrate.trapezoid}{\texttt{scipy.integrate.trapezoid}} and \href{https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.simpson.html#scipy.integrate.simpson}{\texttt{scipy.integrate.simpson}}. % and \href{https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.romb.html}{\texttt{scipy.integrate.romb}}.
\item Repeat the calculation using three different values of $h$. Plot the error $\Delta I$ against $h$. Show that the numerical estimates converge to the exact value according to the expected order of convergence. To show this, you can plot the error [which will be $\mathcal(O)(h^n)]$ against $h$ as a log-log plot and look at the slope.
\item Using the Trapezoidal and Simpson's methods, compute the energy loss due to GW emission from binary black holes using the numerical-relativity data discussed in Sec.~\ref{sec:BBH_nr_data_fd}. The radiated energy can be computed from the GW polarizations as
