diff options
Diffstat (limited to 'usth/MATH2.4/labwork/2')
| -rw-r--r-- | usth/MATH2.4/labwork/2/2a.tikz | 515 | ||||
| -rw-r--r-- | usth/MATH2.4/labwork/2/labwork2.pdf | bin | 0 -> 68137 bytes | |||
| -rw-r--r-- | usth/MATH2.4/labwork/2/report.pdf | bin | 0 -> 191189 bytes | |||
| -rw-r--r-- | usth/MATH2.4/labwork/2/report.tex | 88 |
4 files changed, 603 insertions, 0 deletions
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files differdiff --git a/usth/MATH2.4/labwork/2/report.pdf b/usth/MATH2.4/labwork/2/report.pdf new file mode 100644 index 0000000..6711c94 --- /dev/null +++ b/usth/MATH2.4/labwork/2/report.pdf Binary files differdiff --git a/usth/MATH2.4/labwork/2/report.tex b/usth/MATH2.4/labwork/2/report.tex new file mode 100644 index 0000000..c4481fa --- /dev/null +++ b/usth/MATH2.4/labwork/2/report.tex @@ -0,0 +1,88 @@ +\documentclass[a4paper,12pt]{article} +\usepackage[english,vietnamese]{babel} +\usepackage{amsmath} +\usepackage{lmodern} +\usepackage{hyperref} +\usepackage{tikz} + +\newcommand{\exercise}[1]{\noindent\textbf{#1.}} +\renewcommand{\thesection}{\Roman{section}} +\renewcommand*{\thefootnote}{\fnsymbol{footnote}} + +\title{Numerical Method: Labwork 2 Report} +\author{Nguyễn Gia Phong--BI9-184} +\date{Fall 2019} + +\begin{document} +\maketitle +\setcounter{section}{2} +\section{Polynomial} +\exercise{1.c} At the time of writing, function \verb|fzero| +in Octave have not support the \verb|Display| option +just yet\footnote{Bug report: \url{https://savannah.gnu.org/bugs/?56954}}. +However, the implementation of this option is rather trivial, +thus I made a quick patch (which is also attached at the bug report). +Using this, one can easily display all the iterations as followed: + +\begin{verbatim} +octave:1> fzero (@(x) x.^2 - 9, 0, optimset ('display', 'iter')) + +Search for an interval around 0 containing a sign change: +Func-eval 1, how = initial, a = 0, f(a) = -9, b = 0, f(b) = -9 +Func-eval 2, how = search, a = 0, f(a) = -9, b = 0.099, f(b) = -8.9902 +Func-eval 3, how = search, a = 0, f(a) = -9, b = 0.1025, f(b) = -8.98949 +Func-eval 4, how = search, a = 0, f(a) = -9, b = 0.095, f(b) = -8.99098 +Func-eval 5, how = search, a = 0, f(a) = -9, b = 0.11, f(b) = -8.9879 +Func-eval 6, how = search, a = 0, f(a) = -9, b = 0.075, f(b) = -8.99437 +Func-eval 7, how = search, a = 0, f(a) = -9, b = 0.15, f(b) = -8.9775 +Func-eval 8, how = search, a = 0, f(a) = -9, b = 0, f(b) = -9 +Func-eval 9, how = search, a = 0, f(a) = -9, b = 0.35, f(b) = -8.8775 +Func-eval 10, how = search, a = 0, f(a) = -9, b = -0.4, f(b) = -8.84 +Func-eval 11, how = search, a = 0, f(a) = -9, b = 1.1, f(b) = -7.79 +Func-eval 12, how = search, a = 0, f(a) = -9, b = -4.9, f(b) = 15.01 + +Search for a a zero in the interval [-4.9, 0]: +Func-eval 13, how = initial, x = 0, f(x) = -9 +Func-eval 14, how = interpolation, x = -1.83673, f(x) = -5.62641 (NaN%) +Func-eval 15, how = interpolation, x = -3.36837, f(x) = 2.3459 (141.7%) +Func-eval 16, how = interpolation, x = -3.19097, f(x) = 1.1823 (-49.6%) +Func-eval 17, how = interpolation, x = -2.99725, f(x) = -0.0164972 (-101.4%) +Func-eval 18, how = interpolation, x = -3.00258, f(x) = 0.0154927 (193.9%) +Func-eval 19, how = interpolation, x = -3, f(x) = 3.07975e-07 (-100.0%) +Func-eval 20, how = interpolation, x = -3, f(x) = -7.10543e-15 (-100.0%) +Func-eval 21, how = interpolation, x = -3, f(x) = 5.32907e-15 (169.7%) + +Algorithm converged + +ans = -3.0000 +\end{verbatim} + +To answer the question in part b, (since I believe these parts are linked +to each other), the current implementation of \verb|fzero| search for +the second bracket over quantitative chages below if \verb|X0| if it is a +single scalar, thus $[-4.9, 0]$ is gotten and the found solution is negative: + +\begin{verbatim} +[-.01 +.025 -.05 +.10 -.25 +.50 -1 +2.5 -5 +10 -50 +100 -500 +1000] +\end{verbatim} + +\section{Non-linear Systems} +\exercise{1.a} These statements were used to plot the given functions: +\begin{verbatim} +ezplot(@(x1, x2) x1 .^ 2 + x1 .* x2 - 10) +hold on +ezplot(@(x1, x2) x2 + 3 .* x1 .* x2 .^ 2 - 57) +\end{verbatim} + +As shown in the graphs (where $x_1^2 + x_1 x_2 = 10$ are the blue lines +and $x_2 + 3 x_1 x_2 = 57$ are the yellow ones), the solutions of $(x_1, x_2)$ +are quite close to $(2, 3)$ and $(4.5, -2)$. + +\begin{figure}[!h] + \centering + \scalebox{0.37}{\input{2a.tikz}} +\end{figure} + +I would also like to note that I am personally impressed how gnuplot +(which is utilised by Octave) is able to export to TikZ graphics with ease. +\end{document} |