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-\documentclass[a4paper,12pt]{article}
-\usepackage[english,vietnamese]{babel}
-\usepackage{amsmath}
-\usepackage{booktabs}
-\usepackage{enumerate}
-\usepackage{lmodern}
-\usepackage{tikz}
-
-\newcommand{\exercise}[1]{\noindent\textbf{#1.}}
-
-\title{Numerical Methods: Labwork 4 Report}
-\author{Nguyễn Gia Phong--BI9-184}
-\date{\dateenglish\today}
-
-\begin{document}
-\maketitle
-\section{Curve Fitting Problems}
-\exercise{3}  From the given table, we define the two vectors
-\begin{verbatim}
-octave> x = [0.00000 0.78540 1.57080 2.35620 ...
->            3.14159 3.92699 4.71239 5.49779 6.28319];
-octave> fx = [0.00000 0.70711 1.00000 0.70711 ...
->             0.00000 -0.70711 -1.00000 -0.70711 0.00000];
-\end{verbatim}
-
-\begin{enumerate}[(a)]
-  \item \verb|f(3.00000)| and \verb|f(4.50000)| can be interpolated by
-\begin{verbatim}
-octave> points = [3.00000 4.50000];
-octave> linear = interp1 (x, fx, points)
-linear =
-   0.12748  -0.92080
-\end{verbatim}
-    To further illustrate this, we can then plot these point along
-    with the linearly interpolated line:
-    \verb|plot (points, linear, "o", x, fx)|
-    \begin{figure}[!h]
-      \centering
-      \scalebox{0.36}{\input{linear.tikz}}
-    \end{figure}
-
-  \item For convenience purposes, we define a thin wrapper around \verb|interp1|
-\begin{verbatim}
-octave> interpolate = @(X, method) interp1 (
-> x, fx, X, method, "extrap");
-\end{verbatim}
-    Anonymous function had to be used because named functions somehow do not
-    support closure.  Now we can use \verb|interpolate (points, method)|
-    to approximate \verb|f(3.00000)| and \verb|f(4.50000)|
-    and obtain the table below
-    \begin{center}
-      \begin{tabular}{c r r r}
-        \toprule
-        method & nearest & cubic & spline \\
-        \midrule
-        f(3.00000) & 0 & 0.13528 & 0.14073 \\
-        f(4.50000) & -1 & -0.96943 & -0.97745\\
-        \bottomrule
-      \end{tabular}
-    \end{center}
-
-    Next, we use some plots to better visualize these interpolation methods.
-\begin{verbatim}
-octave> interplot = @(mark, line, method) plot (
-> mark, interpolate (mark, method), "o",
-> line, interpolate (line, method));
-octave> B = linspace (x(1), x(end));
-octave> interplot (points, B, "nearest")
-\end{verbatim}
-\scalebox{0.61}{\input{nearest.tikz}}
-\begin{verbatim}
-octave> interplot (points, B, "cubic")
-\end{verbatim}
-\scalebox{0.61}{\input{cubic.tikz}}
-\begin{verbatim}
-octave> interplot (points, B, "spline")
-\end{verbatim}
-\scalebox{0.61}{\input{spline.tikz}}
-
-    One can easily notice while \verb|nearest| simply chooses the nearest
-    neighbor, \verb|cubic| and \verb|spline| both try to \textit{smoothen}
-    the curve.  This leads to the fact that \verb|nearest|'s approximations
-    strays from \verb|linear|'s in the opposite dirrection when compared to
-    the other two's.  It also explains why \verb|cubic|'s and \verb|spline|'s
-    results are quite close to each other.
-
-  \item Since we are already extrapolating (by providing the \verb|extrap|
-    argument to \verb|interp1|), interpolating for \verb|f(10)| is rather
-    straightforward:
-\begin{verbatim}
-octave> interpolate (10, "spline")
-ans =  1.4499
-octave> C = linspace (0, 10);
-octave> interplot (10, C, "spline")
-\end{verbatim}
-\scalebox{0.39}{\input{f10-spline.tikz}}
-\begin{verbatim}
-octave> interpolate (10, "linear")
-ans =  3.3463
-octave> interplot (10, C, "linear")
-\end{verbatim}
-\scalebox{0.39}{\input{f10-linear.tikz}}
-
-    From the existing data, we can make a guess that \verb|f|
-    is a cubic function and regression fits quite well:
-\begin{verbatim}
-octave> p = polyfit (x, fx, 3)
-p =
-   0.084488  -0.796282   1.681694  -0.043870
-octave> polyval (p, 10)
-ans =  21.633
-octave> plot (x, fx, "o", C, polyval (p, C))
-\end{verbatim}
-\scalebox{0.62}{\input{f10-poly.tikz}}
-
-    In all these cases, due to the missing data, the value of \verb|f| at 10
-    tends to go \textit{wild}, i.e. far away from the given data in \verb|fx|.
-    If anything, the interpolated/extrapolated ones looks more harmonic,
-    while regression simply fit the curve into the function of the given form.
-    It is not obvious that either technique is better is this case,
-    since the amount of given data is too small.
-\end{enumerate}
-\end{document}