From e461df7573c2b7b7e26c965d8cf2d8e175d67378 Mon Sep 17 00:00:00 2001 From: Nguyễn Gia Phong Date: Mon, 16 Dec 2019 21:31:18 +0700 Subject: [usth/MATH2.2] Numerical Methods The future starts now. --- 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 0 -> 72397 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 0 -> 159435 bytes usth/MATH2.2/labwork/4/report.tex | 123 ++++++ usth/MATH2.2/labwork/4/spline.tikz | 521 +++++++++++++++++++++++ 10 files changed, 3647 insertions(+) create mode 100644 usth/MATH2.2/labwork/4/cubic.tikz create mode 100644 usth/MATH2.2/labwork/4/f10-linear.tikz create mode 100644 usth/MATH2.2/labwork/4/f10-poly.tikz create mode 100644 usth/MATH2.2/labwork/4/f10-spline.tikz create mode 100644 usth/MATH2.2/labwork/4/labwork4.pdf create mode 100644 usth/MATH2.2/labwork/4/linear.tikz create mode 100644 usth/MATH2.2/labwork/4/nearest.tikz create mode 100644 usth/MATH2.2/labwork/4/report.pdf create mode 100644 usth/MATH2.2/labwork/4/report.tex create 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 new file mode 100644 index 0000000..d169c2d --- /dev/null +++ b/usth/MATH2.2/labwork/4/cubic.tikz @@ -0,0 +1,521 @@ +% Title: gl2ps_renderer figure +% Creator: GL2PS 1.4.0, (C) 1999-2017 C. 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--git a/usth/MATH2.2/labwork/4/report.tex b/usth/MATH2.2/labwork/4/report.tex new file mode 100644 index 0000000..41d69ba --- /dev/null +++ b/usth/MATH2.2/labwork/4/report.tex @@ -0,0 +1,123 @@ +\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 new file mode 100644 index 0000000..9c7c9b3 --- /dev/null +++ b/usth/MATH2.2/labwork/4/spline.tikz @@ -0,0 +1,521 @@ +% Title: gl2ps_renderer figure +% Creator: GL2PS 1.4.0, (C) 1999-2017 C. 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