added part 1 of integration.

Lab.tex

@@ -76,9 +76,9 @@
 \section{Numerical differentiation}
 \section{Numerical differentiation}
 \input{diff.tex}
 \input{diff.tex}
 
 
-% \section{Numerical integration}
-% \label{sec:integr}
-% \input{integration.tex}
+\section{Numerical integration}
+\label{sec:integr}
+\input{integration.tex}
 % 
 % 
 % \section{Ordinary differential equations: Initial value problems}
 % \section{Ordinary differential equations: Initial value problems}
 % \input{ode.tex}
 % \input{ode.tex}

integration.tex

@@ -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:}
 \subsubsection*{Problems:}
 \begin{enumerate}
 \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 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}
 \begin{equation}
 I = \int_{-1}^{1} e^{-x^2} dx.
 I = \int_{-1}^{1} e^{-x^2} dx.
 \end{equation}
 \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 
 \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 
 \begin{equation}
 \begin{equation}