From 82e6cf7d1046d6cee16f7e8b044ec33e7ec6c4b7 Mon Sep 17 00:00:00 2001 From: Nguyễn Gia Phong Date: Sun, 16 Feb 2020 14:26:55 +0700 Subject: [usth] Numerical Method is MATH2.4 --- usth/MATH2.2/labwork/4/cubic.tikz | 521 ----------------------- usth/MATH2.2/labwork/4/f10-linear.tikz | 482 ---------------------- usth/MATH2.2/labwork/4/f10-poly.tikz | 731 --------------------------------- usth/MATH2.2/labwork/4/f10-spline.tikz | 500 ---------------------- usth/MATH2.2/labwork/4/labwork4.pdf | Bin 72397 -> 0 bytes usth/MATH2.2/labwork/4/linear.tikz | 248 ----------- usth/MATH2.2/labwork/4/nearest.tikz | 521 ----------------------- usth/MATH2.2/labwork/4/report.pdf | Bin 159435 -> 0 bytes usth/MATH2.2/labwork/4/report.tex | 123 ------ usth/MATH2.2/labwork/4/spline.tikz | 521 ----------------------- 10 files changed, 3647 deletions(-) delete mode 100644 usth/MATH2.2/labwork/4/cubic.tikz delete mode 100644 usth/MATH2.2/labwork/4/f10-linear.tikz delete mode 100644 usth/MATH2.2/labwork/4/f10-poly.tikz delete mode 100644 usth/MATH2.2/labwork/4/f10-spline.tikz delete mode 100644 usth/MATH2.2/labwork/4/labwork4.pdf delete mode 100644 usth/MATH2.2/labwork/4/linear.tikz delete mode 100644 usth/MATH2.2/labwork/4/nearest.tikz delete mode 100644 usth/MATH2.2/labwork/4/report.pdf delete mode 100644 usth/MATH2.2/labwork/4/report.tex delete mode 100644 usth/MATH2.2/labwork/4/spline.tikz (limited to 'usth/MATH2.2/labwork/4') diff --git a/usth/MATH2.2/labwork/4/cubic.tikz b/usth/MATH2.2/labwork/4/cubic.tikz deleted file mode 100644 index d169c2d..0000000 --- a/usth/MATH2.2/labwork/4/cubic.tikz +++ /dev/null @@ -1,521 +0,0 @@ -% Title: gl2ps_renderer figure -% Creator: GL2PS 1.4.0, (C) 1999-2017 C. 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diff --git a/usth/MATH2.2/labwork/4/report.tex b/usth/MATH2.2/labwork/4/report.tex deleted file mode 100644 index 41d69ba..0000000 --- a/usth/MATH2.2/labwork/4/report.tex +++ /dev/null @@ -1,123 +0,0 @@ -\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} diff --git a/usth/MATH2.2/labwork/4/spline.tikz b/usth/MATH2.2/labwork/4/spline.tikz deleted file mode 100644 index 9c7c9b3..0000000 --- a/usth/MATH2.2/labwork/4/spline.tikz +++ /dev/null @@ -1,521 +0,0 @@ -% Title: gl2ps_renderer figure -% Creator: GL2PS 1.4.0, (C) 1999-2017 C. 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