author | Puneeth Chaganti <punchagan@fossee.in> |
Fri, 23 Oct 2009 10:21:27 +0530 (2009-10-23) | |
changeset 141 | fccf675946bf |
parent 137 | 4dea7c5e1bf5 |
child 142 | 57e0f0fd3317 |
permissions | -rw-r--r-- |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%Tutorial slides on Python. |
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% |
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% Author: The FOSSEE Group |
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% Copyright (c) 2009, The FOSSEE Group, IIT Bombay |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\documentclass[14pt,compress]{beamer} |
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%\documentclass[draft]{beamer} |
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%\documentclass[compress,handout]{beamer} |
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%\usepackage{pgfpages} |
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%\pgfpagesuselayout{2 on 1}[a4paper,border shrink=5mm] |
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|
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% Modified from: generic-ornate-15min-45min.de.tex |
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\mode<presentation> |
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{ |
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\usetheme{Warsaw} |
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\useoutertheme{split} |
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\setbeamercovered{transparent} |
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} |
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|
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\usepackage[english]{babel} |
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\usepackage[latin1]{inputenc} |
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%\usepackage{times} |
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\usepackage[T1]{fontenc} |
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|
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% Taken from Fernando's slides. |
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\usepackage{ae,aecompl} |
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\usepackage{mathpazo,courier,euler} |
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\usepackage[scaled=.95]{helvet} |
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\usepackage{amsmath} |
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|
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\definecolor{darkgreen}{rgb}{0,0.5,0} |
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|
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\usepackage{listings} |
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\lstset{language=Python, |
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basicstyle=\ttfamily\bfseries, |
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commentstyle=\color{red}\itshape, |
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stringstyle=\color{darkgreen}, |
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showstringspaces=false, |
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keywordstyle=\color{blue}\bfseries} |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% Macros |
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\setbeamercolor{emphbar}{bg=blue!20, fg=black} |
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\newcommand{\emphbar}[1] |
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{\begin{beamercolorbox}[rounded=true]{emphbar} |
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{#1} |
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\end{beamercolorbox} |
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} |
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\newcounter{time} |
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\setcounter{time}{0} |
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\newcommand{\inctime}[1]{\addtocounter{time}{#1}{\tiny \thetime\ m}} |
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|
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\newcommand{\typ}[1]{\lstinline{#1}} |
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|
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\newcommand{\kwrd}[1]{ \texttt{\textbf{\color{blue}{#1}}} } |
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|
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%%% This is from Fernando's setup. |
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% \usepackage{color} |
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% \definecolor{orange}{cmyk}{0,0.4,0.8,0.2} |
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% % Use and configure listings package for nicely formatted code |
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% \usepackage{listings} |
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% \lstset{ |
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% language=Python, |
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% basicstyle=\small\ttfamily, |
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% commentstyle=\ttfamily\color{blue}, |
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% stringstyle=\ttfamily\color{orange}, |
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% showstringspaces=false, |
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% breaklines=true, |
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% postbreak = \space\dots |
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% } |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% Title page |
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\title[]{Finding Roots} |
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|
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\author[FOSSEE] {FOSSEE} |
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|
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\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay} |
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\date[] {31, October 2009\\Day 1, Session 6} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
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%\pgfdeclareimage[height=0.75cm]{iitmlogo}{iitmlogo} |
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%\logo{\pgfuseimage{iitmlogo}} |
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|
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|
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%% Delete this, if you do not want the table of contents to pop up at |
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%% the beginning of each subsection: |
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\AtBeginSubsection[] |
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{ |
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\begin{frame}<beamer> |
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\frametitle{Outline} |
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\tableofcontents[currentsection,currentsubsection] |
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\end{frame} |
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} |
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|
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\AtBeginSection[] |
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{ |
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\begin{frame}<beamer> |
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\frametitle{Outline} |
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\tableofcontents[currentsection,currentsubsection] |
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\end{frame} |
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} |
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|
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% If you wish to uncover everything in a step-wise fashion, uncomment |
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% the following command: |
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%\beamerdefaultoverlayspecification{<+->} |
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|
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%\includeonlyframes{current,current1,current2,current3,current4,current5,current6} |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% DOCUMENT STARTS |
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\begin{document} |
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|
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\begin{frame} |
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\maketitle |
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|
118 |
\end{frame} |
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Puneeth Chaganti <punchagan@fossee.in>
parents:
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|
119 |
|
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parents:
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120 |
%% \begin{frame} |
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parents:
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|
121 |
%% \frametitle{Outline} |
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parents:
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|
122 |
%% \tableofcontents |
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parents:
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|
123 |
%% % You might wish to add the option [pausesections] |
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parents:
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|
124 |
%% \end{frame} |
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parents:
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|
125 |
|
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parents:
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|
126 |
|
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parents:
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|
127 |
\begin{frame}[fragile] |
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parents:
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|
128 |
\frametitle{Roots of $f(x)=0$} |
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parents:
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|
129 |
\begin{itemize} |
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|
130 |
\item Roots --- values of $x$ satisfying $f(x)=0$ |
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parents:
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|
131 |
\item $f(x)=0$ may have real or complex roots |
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parents:
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|
132 |
\item Presently, let's look at real roots |
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parents:
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|
133 |
\end{itemize} |
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parents:
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|
134 |
\end{frame} |
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Puneeth Chaganti <punchagan@fossee.in>
parents:
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changeset
|
135 |
|
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Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
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|
136 |
\begin{frame}[fragile] |
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parents:
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|
137 |
\frametitle{Initial Estimates} |
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Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
138 |
\begin{itemize} |
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parents:
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|
139 |
\item Find the roots of $cosx-x^2$ between $-\pi/2$ and $\pi/2$ |
578db74dfea0
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parents:
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|
140 |
\item We shall use a crude method to get an initial estimate first |
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parents:
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changeset
|
141 |
\end{itemize} |
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Puneeth Chaganti <punchagan@fossee.in>
parents:
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changeset
|
142 |
\begin{enumerate} |
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parents:
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|
143 |
\item Check for change of signs of $f(x)$ in the given interval |
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parents:
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|
144 |
\item A root lies in the interval where the sign change occurs |
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parents:
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changeset
|
145 |
\end{enumerate} |
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Puneeth Chaganti <punchagan@fossee.in>
parents:
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changeset
|
146 |
\end{frame} |
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Puneeth Chaganti <punchagan@fossee.in>
parents:
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|
147 |
|
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Puneeth Chaganti <punchagan@fossee.in>
parents:
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|
148 |
\begin{frame}[fragile] |
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parents:
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|
149 |
\frametitle{Initial Estimates \ldots} |
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parents:
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|
150 |
\begin{lstlisting} |
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Puneeth Chaganti <punchagan@fossee.in>
parents:
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|
151 |
In []: def our_f(x): |
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parents:
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|
152 |
....: return cos(x)-x**2 |
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parents:
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|
153 |
....: |
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Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
154 |
In []: x = linspace(-pi/2, pi/2, 11) |
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parents:
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changeset
|
155 |
\end{lstlisting} |
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Puneeth Chaganti <punchagan@fossee.in>
parents:
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changeset
|
156 |
\begin{itemize} |
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parents:
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|
157 |
\item Get the intervals of x, where sign changes occur |
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Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
158 |
\end{itemize} |
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Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
159 |
\end{frame} |
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Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
160 |
|
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Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
161 |
%% \begin{frame}[fragile] |
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Puneeth Chaganti <punchagan@fossee.in>
parents:
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|
162 |
%% \frametitle{Initial Estimates \ldots} |
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Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
163 |
%% \begin{lstlisting} |
578db74dfea0
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Puneeth Chaganti <punchagan@fossee.in>
parents:
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|
164 |
%% In []: pos = y[:-1]*y[1:] < 0 |
578db74dfea0
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Puneeth Chaganti <punchagan@fossee.in>
parents:
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changeset
|
165 |
%% In []: rpos = zeros(x.shape, dtype=bool) |
578db74dfea0
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Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
166 |
%% In []: rpos[:-1] = pos |
578db74dfea0
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Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
167 |
%% In []: rpos[1:] += pos |
578db74dfea0
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Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
168 |
%% In []: rts = x[rpos] |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
169 |
%% \end{lstlisting} |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
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changeset
|
170 |
%% \end{frame} |
578db74dfea0
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Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
171 |
|
578db74dfea0
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Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
172 |
\begin{frame}[fragile] |
578db74dfea0
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Puneeth Chaganti <punchagan@fossee.in>
parents:
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changeset
|
173 |
\frametitle{Fixed Point Method} |
578db74dfea0
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Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
174 |
\begin{enumerate} |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
175 |
\item Convert $f(x)=0$ to the form $x=g(x)$ |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
176 |
\item Start with an initial value of $x$ and iterate successively |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
177 |
\item $x_{n+1}=g(x_n)$ and $x_0$ is our initial guess |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
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changeset
|
178 |
\item Iterate until $x_{n+1}-x_n \le tolerance$ |
578db74dfea0
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Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
179 |
\end{enumerate} |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
180 |
\end{frame} |
578db74dfea0
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Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
181 |
|
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
182 |
%% \begin{frame}[fragile] |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
183 |
%% \frametitle{Fixed Point \dots} |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
184 |
%% \begin{lstlisting} |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
185 |
%% In []: def our_g(x): |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
186 |
%% ....: return sqrt(cos(x)) |
578db74dfea0
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Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
187 |
%% ....: |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
188 |
%% In []: tolerance = 1e-5 |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
189 |
%% In []: while abs(x1-x0)>tolerance: |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
190 |
%% ....: x0 = x1 |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
191 |
%% ....: x1 = our_g(x1) |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
192 |
%% ....: |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
193 |
%% In []: x0 |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
194 |
%% \end{lstlisting} |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
195 |
%% \end{frame} |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
196 |
|
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
197 |
\begin{frame}[fragile] |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
198 |
\frametitle{Bisection Method} |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
199 |
\begin{enumerate} |
137 | 200 |
\item Start with an interval $(a, b)$ within which a root exists |
133
578db74dfea0
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Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
201 |
\item $f(a)\cdot f(b) < 0$ |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
202 |
\item Bisect the interval. $c = \frac{a+b}{2}$ |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
203 |
\item Change the interval to $(a, c)$ if $f(a)\cdot f(c) < 0$ |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
204 |
\item Else, change the interval to $(c, b)$ |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
205 |
\item Go back to 3 until $(b-a) \le$ tolerance |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
206 |
\end{enumerate} |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
207 |
\end{frame} |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
208 |
|
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
209 |
%% \begin{frame}[fragile] |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
210 |
%% \frametitle{Bisection \dots} |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
211 |
%% \begin{lstlisting} |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
212 |
%% In []: tolerance = 1e-5 |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
213 |
%% In []: a = -pi/2 |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
214 |
%% In []: b = 0 |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
215 |
%% In []: while b-a > tolerance: |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
216 |
%% ....: c = (a+b)/2 |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
217 |
%% ....: if our_f(a)*our_f(c) < 0: |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
218 |
%% ....: b = c |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
219 |
%% ....: else: |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
220 |
%% ....: a = c |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
221 |
%% ....: |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
222 |
%% \end{lstlisting} |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
223 |
%% \end{frame} |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
224 |
|
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
225 |
\begin{frame}[fragile] |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
226 |
\frametitle{Newton Raphson Method} |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
227 |
\begin{enumerate} |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
228 |
\item Start with an initial guess of x for the root |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
229 |
\item $\Delta x = -f(x)/f^{'}(x)$ |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
230 |
\item $ x \leftarrow x + \Delta x$ |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
231 |
\item Go back to 2 until $|\Delta x| \le$ tolerance |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
232 |
\end{enumerate} |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
233 |
\end{frame} |
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
234 |
|
578db74dfea0
Added Day1 Session6 - Finding Roots.
Puneeth Chaganti <punchagan@fossee.in>
parents:
diff
changeset
|
235 |
%% \begin{frame}[fragile] |
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%% \frametitle{Newton Raphson \dots} |
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%% \begin{lstlisting} |
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%% In []: def our_df(x): |
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%% ....: return -sin(x)-2*x |
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%% ....: |
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%% In []: delx = 10*tolerance |
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%% In []: while delx > tolerance: |
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%% ....: delx = -our_f(x)/our_df(x) |
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%% ....: x = x + delx |
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%% ....: |
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%% ....: |
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%% \end{lstlisting} |
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%% \end{frame} |
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|
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\begin{frame}[fragile] |
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\frametitle{Newton Raphson \ldots} |
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\begin{itemize} |
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\item What if $f^{'}(x) = 0$? |
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\item Can you write a better version of the Newton Raphson? |
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\item What if the differential is not easy to calculate? |
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\item Look at Secant Method |
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\end{itemize} |
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\end{frame} |
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|
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\begin{frame}[fragile] |
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\frametitle{Scipy Methods - \typ{roots}} |
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\begin{itemize} |
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\item Calculates the roots of polynomials |
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\item Array of coefficients is the only parameter |
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\end{itemize} |
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\begin{lstlisting} |
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In []: coeffs = [1, 6, 13] |
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In []: roots(coeffs) |
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\end{lstlisting} |
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\end{frame} |
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|
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\begin{frame}[fragile] |
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\frametitle{Scipy Methods - \typ{fsolve}} |
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\begin{small} |
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\begin{lstlisting} |
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In []: from scipy.optimize import fsolve |
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\end{lstlisting} |
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\end{small} |
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\begin{itemize} |
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\item Finds the roots of a system of non-linear equations |
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\item Input arguments - Function and initial estimate |
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\item Returns the solution |
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\end{itemize} |
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\begin{lstlisting} |
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In []: fsolve(our_f, -pi/2) |
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\end{lstlisting} |
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\end{frame} |
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|
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\begin{frame}[fragile] |
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\frametitle{Scipy Methods \dots} |
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\small{ |
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\begin{lstlisting} |
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In []: from scipy.optimize import fixed_point |
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|
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In []: from scipy.optimize import bisect |
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|
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In []: from scipy.optimize import newton |
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\end{lstlisting}} |
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\end{frame} |
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|
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|
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\end{document} |