added a new probl.

Lab.tex

@@ -67,8 +67,7 @@
 \author{Parameswaran~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.}
-\author{Koustav Narayan Maity (tutor)}\email{koustav.narayan@icts.res.in}
-\author{Vinay Kumar (tutor)}\email{vinay.kumar@icts.res.in}
+\author{Alorika Kar (tutor)}\email{alorika.kar@icts.res.in}
 \affiliation{International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India.}
 \bigskip
 \date{\today}
@@ -77,35 +76,35 @@
 \section{Numerical differentiation}
 \input{diff.tex}
 
-\section{Numerical integration}
-\label{sec:integr}
-\input{integration.tex}
-
-\section{Ordinary differential equations: Initial value problems}
-\input{ode.tex}
-
-\section{Root finding}
-\input{root.tex}
-
-\section{Ordinary differential equations: Two-point boundary value problems}
-\input{ode_bvp.tex}
-
-\section{Fourier and spectral methods}
-\input{fourier.tex}
-
-\section{Curve fitting}
-\label{sec:curve_fitting}
-\input{curve.tex}
-
-\section{Spectral methods for ODEs and PDEs}
-\input{PDE1.tex}
-
-\section{Statistical inference}
-\label{sec:statinf}
-\input{statinf.tex}
-
-\section{Monte-Carlo methods}
-\input{mc.tex}
+% \section{Numerical integration}
+% \label{sec:integr}
+% \input{integration.tex}
+% 
+% \section{Ordinary differential equations: Initial value problems}
+% \input{ode.tex}
+% 
+% \section{Root finding}
+% \input{root.tex}
+% 
+% \section{Ordinary differential equations: Two-point boundary value problems}
+% \input{ode_bvp.tex}
+% 
+% \section{Fourier and spectral methods}
+% \input{fourier.tex}
+% 
+% \section{Curve fitting}
+% \label{sec:curve_fitting}
+% \input{curve.tex}
+% 
+% \section{Spectral methods for ODEs and PDEs}
+% \input{PDE1.tex}
+% 
+% \section{Statistical inference}
+% \label{sec:statinf}
+% \input{statinf.tex}
+% 
+% \section{Monte-Carlo methods}
+% \input{mc.tex}
 
 %\section{Lab 2}
 %\input{rest.tex}

diff.tex

@@ -16,12 +16,13 @@ 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 You would notice that the truncation error is not constant across $x$. This calls for non-uniform sampling of $f(x)$ in finite differencing. Derive an explicit central differencing formula assuming that $f(x)$ can be sampled at some \emph{non-uniform} samples $x_i$. 
 %\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 
+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}