added the tutorial material - chapter 1.

Lab.tex

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+
+\title{ICTS Graduate Course: Numerical Methods (PHY410.5)}
+\author{P.~Ajith}\email{ajith@icts.res.in}
+\author{Prayush Kumar}\email{prayush@icts.res.in}
+\affiliation{International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India.}
+\bigskip
+\date{\today}
+\maketitle
+
+\section{Numerical differentiation}
+\input{diff.tex}
+
+%\section{Lab 2}
+%\input{rest.tex}
+
+\bibliography{Lab}
+\end{document}

Makefile

+#
+# $Header: /numrelcvs/NumrelGWDA/docs/TemplBank/Makefile,v 1.4 2007/09/23 10:57:22 ajith Exp $
+#
+
+DOCS = Lab.pdf
+
+BYPROD = Lab.ps Lab.blg
+
+TEXS = Lab.tex 
+
+FIGS = 
+
+default: ${DOCS}
+
+#Lab.pdf: Lab.ps
+#	ps2pdf Lab.ps Lab.pdf
+
+#Lab.ps: Lab.dvi ${FIGS}
+#	dvips -t a4 -P pdf -o Lab.ps Lab.dvi
+
+Lab.pdf: Lab.tex ${FIGS} ${TEXS}
+#	latex Lab && latex Lab && latex Lab && bibtex Lab
+	pdflatex Lab  && bibtex Lab && bibtex Lab && pdflatex Lab && pdflatex Lab 
+
+Lab.ps: Lab.pdf ${FIGS}
+	pdftops Lab.pdf
+
+%.pdf: %.eps
+	epstopdf --outfile=$@ ${@:.pdf=.eps}
+
+clean:
+	rm -f *.aux *.log *.out *.dvi *~ ${DOCS} ${BYPROD}
+
+copy:
+	cp Lab.pdf /Users/pajith/Sites/ICTS/P._Ajiths_Homepage/Teaching_files
+publish: 
+	/Users/pajith/Sites/ICTS/P._Ajiths_Homepage/updateweb.sh

diff.tex

+\subsection{Finite differencing, convergence, error estimates}
+
+We derived the following finite-differencing approximants for the derivative of a function $f(x)$: 
+\begin{eqnarray}
+\mathit{Forward~differencing:} & f'(x) \simeq \frac{f(x+h)-f(x)}{h} + \mathcal{O}\,(h) \\
+\mathit{Backward~differencing:} & f'(x) \simeq \frac{f(x)-f(x-h)}{h} + \mathcal{O}\,(h) \\
+\mathit{Central~differencing:} & f'(x) \simeq \frac{f(x+h)-f(x-h)}{2h} + \mathcal{O}\,(h^2)
+\end{eqnarray}
+
+\subsubsection*{Problems:}
+\begin{enumerate}
+\item Write a Python function to compute derivatives using these three finite differencing methods. Compute the derivative of the function $f(x) = e^x \, \sin(x)$ over the range $x = [0, 2\pi]$. Plot the numerically computed derivative $f_{(h)}'(x)$ for three different values of $h$. 
+\item Plot the error $\Delta f_{h}'(x) := |f_{h}'(x) - f'(x)|$ for three different values of $h$, and estimate the order of convergence $n$ of each finite-difference approximation. Plot $n(x)$, where 
+\begin{equation}
+n(x) = \log_2 \frac{f'_{4h}(x) - f'_{2h}(x)}{f'_{2h}(x) - f'_{h}(x)}
+\label{eq:order_convg}
+\end{equation}
+\item Reduce the step size $h$ successively. At what value of $h$ does the round-off error dominate the error budget? 
+%\item Derive a central differencing approximant for the first derivative $f'(x)$ that is accurate to $\mathcal{O}(h^4)$. 
+\end{enumerate}
+
+\subsection{Richardson extrapolation}
+
+We have seen that, if the order of the error in the numerical estimate of a function $f(x)$ is known, Richardson extrapolation provides a powerful way of improving the accuracy of the estimate. If we have two numerical estimates $f_{h}(x)$ and $f_{2h}(x)$ each having an error of $\mathcal{O}\,h^k$, a better estimate is given by 
+\begin{equation}
+f(x) \simeq \frac{2 ~ 2^k\, f_{h}(x) - f_{2h}(x)}{2\,2^k -1 } + \mathcal{O}\,(h^{l}),
+\end{equation}
+where $l$ is the next-to-leading-order error term (for e.g., $l = k+2$ for central differencing, while $l = k+1$ for forward/backward differencing). 
+
+\subsubsection*{Problems:}
+\label{sec:BBH_nr_data_fd}
+\begin{enumerate}
+\item Gravitational-waves (GWs) have two independent polarization states -- called ``plus'' and ``cross'' states. GW signals from the coalescence of black-hole binaries, in the simplest case, are circularly polarized: 
+\begin{eqnarray}
+h_+(t) & = A(t) \, \cos \varphi(t), ~~~ h_\times(t) & = A(t) \, \sin \varphi(t). 
+\end{eqnarray}
+Download the data file~\cite{nrdata} containing $h_+(t)$ and $h_\times(t)$. (This is the reduced form of the data produced by a numerical-relativity simulation of black-hole binaries performed by the SXS collaboration and is publicly available at~\cite{SXScatalog}). Compute the phase evolution $\varphi(t)$, the frequency evolution $\omega(t) := d\varphi(t)/dt$ and the rate of change of frequency $\dot{\omega}(t) := d\omega(t)/dt$ using second-order central difference approximation. 
+\item Estimate the order of convergence of the numerical computation of $\omega(t)$ and $\dot{\omega}(t)$. 
+\item Perform an extrapolation of $\omega(t)$ and $\dot{\omega}(t)$ to the next order using estimates of two different time-resolutions.  
+\item Derive an explicit expression for $f'(x)$ with error $\mathcal{O}(h^4)$ using Richardson extrapolation. This is the fourth-order finite differencing approximant for the derivative, which we will use later. 
+
+\end{enumerate}