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diff --git a/scipy/basic/08_basic_scipy.tex b/scipy/basic/08_basic_scipy.tex new file mode 100644 index 0000000..dca675d --- /dev/null +++ b/scipy/basic/08_basic_scipy.tex @@ -0,0 +1,509 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%Tutorial slides on Python. +% +% Author: FOSSEE +% Copyright (c) 2009-2016, FOSSEE, IIT Bombay +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\documentclass[14pt,compress]{beamer} +%\documentclass[draft]{beamer} +%\documentclass[compress,handout]{beamer} +%\usepackage{pgfpages} +%\pgfpagesuselayout{2 on 1}[a4paper,border shrink=5mm] + +% Modified from: generic-ornate-15min-45min.de.tex +\mode<presentation> +{ + \usetheme{Warsaw} + \useoutertheme{infolines} + \setbeamercovered{transparent} +} + +\usepackage[english]{babel} +\usepackage[latin1]{inputenc} +%\usepackage{times} +\usepackage[T1]{fontenc} +\usepackage{pgf} + +% Taken from Fernando's slides. +\usepackage{ae,aecompl} +\usepackage{mathpazo,courier,euler} +\usepackage[scaled=.95]{helvet} +\usepackage{amsmath} + +\definecolor{darkgreen}{rgb}{0,0.5,0} + +\usepackage{listings} +\lstset{language=Python, + basicstyle=\ttfamily\bfseries, + commentstyle=\color{red}\itshape, + stringstyle=\color{darkgreen}, + showstringspaces=false, + keywordstyle=\color{blue}\bfseries} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Macros +\setbeamercolor{emphbar}{bg=blue!20, fg=black} +\newcommand{\emphbar}[1] +{\begin{beamercolorbox}[rounded=true]{emphbar} + {#1} + \end{beamercolorbox} +} + +\newcommand{\myemph}[1]{\structure{\emph{#1}}} +\newcommand{\PythonCode}[1]{\lstinline{#1}} + +\newcounter{time} +\setcounter{time}{0} +\newcommand{\inctime}[1]{\addtocounter{time}{#1}{\tiny \thetime\ m}} + +\newcommand{\typ}[1]{\lstinline{#1}} + +\newcommand{\kwrd}[1]{ \texttt{\textbf{\color{blue}{#1}}} } + +%%% This is from Fernando's setup. +% \usepackage{color} +% \definecolor{orange}{cmyk}{0,0.4,0.8,0.2} +% % Use and configure listings package for nicely formatted code +% \usepackage{listings} +% \lstset{ +% language=Python, +% basicstyle=\small\ttfamily, +% commentstyle=\ttfamily\color{blue}, +% stringstyle=\ttfamily\color{orange}, +% showstringspaces=false, +% breaklines=true, +% postbreak = \space\dots +% } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Title page +\title[Basic SciPy]{Introductory Scientific Computing with +Python} +\subtitle{Basic SciPy} + +\author[Prabhu] {FOSSEE} + +\institute[FOSSEE -- IITB] {Department of Aerospace Engineering\\IIT Bombay} +\date[] { +Mumbai, India +} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%\pgfdeclareimage[height=0.75cm]{iitmlogo}{iitmlogo} +%\logo{\pgfuseimage{iitmlogo}} + + +%% Delete this, if you do not want the table of contents to pop up at +%% the beginning of each subsection: +\AtBeginSubsection[] +{ + \begin{frame}<beamer> + \frametitle{Outline} + \tableofcontents[currentsection,currentsubsection] + \end{frame} +} + +\AtBeginSection[] +{ + \begin{frame}<beamer> + \frametitle{Outline} + \tableofcontents[currentsection,currentsubsection] + \end{frame} +} + +% If you wish to uncover everything in a step-wise fashion, uncomment +% the following command: +%\beamerdefaultoverlayspecification{<+->} + +%\includeonlyframes{current,current1,current2,current3,current4,current5,current6} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% DOCUMENT STARTS +\begin{document} + +\begin{frame} + \maketitle +\end{frame} + + +\section{Solving linear systems} + +\begin{frame}[fragile] +\frametitle{Solution of equations} +Consider, + \begin{align*} + 3x + 2y - z & = 1 \\ + 2x - 2y + 4z & = -2 \\ + -x + \frac{1}{2}y -z & = 0 + \end{align*} +Solution: + \begin{align*} + x & = 1 \\ + y & = -2 \\ + z & = -2 + \end{align*} +\end{frame} + +\begin{frame}[fragile] +\frametitle{Solving using Matrices} +Let us now look at how to solve this using \kwrd{matrices} + \begin{lstlisting} +In []: A = array([[3,2,-1], + [2,-2,4], + [-1, 0.5, -1]]) +In []: b = array([1, -2, 0]) +In []: x = solve(A, b) + \end{lstlisting} +\end{frame} + +\begin{frame}[fragile] +\frametitle{Solution:} +\begin{lstlisting} +In []: x +Out[]: array([ 1., -2., -2.]) +\end{lstlisting} +\end{frame} + +\begin{frame}[fragile] +\frametitle{Let's check!} +\begin{small} +\begin{lstlisting} +In []: Ax = dot(A, x) +In []: Ax +Out[]: array([ 1.00000000e+00, -2.00000000e+00, -1.11022302e-16]) +\end{lstlisting} +\end{small} +\begin{block}{} +The last term in the matrix is actually \alert{0}!\\ +We can use \kwrd{allclose()} to check. +\end{block} +\begin{lstlisting} +In []: allclose(Ax, b) +Out[]: True +\end{lstlisting} +\inctime{10} +\end{frame} + +\begin{frame}[fragile] +\frametitle{Problem} +Solve the set of equations: +\begin{align*} + x + y + 2z -w & = 3\\ + 2x + 5y - z - 9w & = -3\\ + 2x + y -z + 3w & = -11 \\ + x - 3y + 2z + 7w & = -5\\ +\end{align*} +\end{frame} + +\begin{frame}[fragile] +\frametitle{Solution} +Use \kwrd{solve()} +\begin{align*} + x & = -5\\ + y & = 2\\ + z & = 3\\ + w & = 0\\ +\end{align*} +\inctime{5} +\end{frame} + +\section{Finding Roots} + +\begin{frame}[fragile] +\frametitle{SciPy: \typ{roots}} +\begin{itemize} +\item Calculates the roots of polynomials +\item To calculate the roots of $x^2-5x+6$ +\end{itemize} +\begin{lstlisting} + In []: coeffs = [1, -5, 6] + In []: roots(coeffs) + Out[]: array([3., 2.]) +\end{lstlisting} +\vspace*{-.2in} +\begin{center} +\includegraphics[height=1.6in, interpolate=true]{data/roots} +\end{center} +\end{frame} + +\begin{frame}[fragile] +\frametitle{SciPy: \typ{fsolve}} +Find the root of $sin(z)+cos^2(z)$ nearest to $0$ +\vspace{-0.1in} +\begin{center} +\includegraphics[height=2.8in, interpolate=true]{data/fsolve} +\end{center} +\end{frame} + +\begin{frame}[fragile] +\frametitle{\typ{fsolve}} +\begin{small} +\begin{lstlisting} + In []: from scipy.optimize import fsolve +\end{lstlisting} +\end{small} +\begin{itemize} +\item Finds the roots of a system of non-linear equations +\item Input arguments - \alert{Function} and initial estimate +\item Returns the solution +\end{itemize} +\end{frame} + +\begin{frame}[fragile] +\frametitle{\typ{fsolve} \ldots} +\begin{lstlisting} +In []: def g(z): + ....: return sin(z)+cos(z)*cos(z) +In []: fsolve(g, 0) +Out[]: -0.66623943249251527 +\end{lstlisting} +\begin{center} +\includegraphics[height=2in, interpolate=true]{data/fsolve} +\end{center} +\inctime{10} +\end{frame} + +\begin{frame}[fragile] + \frametitle{Exercise Problem} + Find the root of the equation $x^2 - sin(x) + cos^2(x) = tan(x)$ nearest to $0$ +\end{frame} + +\begin{frame}[fragile] + \frametitle{Solution} + \begin{small} + \begin{lstlisting} +def g(x): + return x**2 - sin(x) + cos(x)*cos(x) - tan(x) +fsolve(g, 0) + \end{lstlisting} + \end{small} + \vspace*{-0.2in} + \begin{center} +\includegraphics[height=2.5in, interpolate=true]{data/fsolve_tanx} +\end{center} +\vspace*{-0.5in} + \inctime{5} +\end{frame} + +%% \begin{frame}[fragile] +%% \frametitle{Scipy Methods \dots} +%% \begin{small} +%% \begin{lstlisting} +%% In []: from scipy.optimize import fixed_point + +%% In []: from scipy.optimize import bisect + +%% In []: from scipy.optimize import newton +%% \end{lstlisting} +%% \end{small} +%% \end{frame} + +\section{ODEs} + +\begin{frame} +\frametitle{Solving ODEs using SciPy} +\begin{itemize} +\item Consider the spread of an epidemic in a population + \vspace*{0.1in} +\item $\frac{dy}{dt} = ky(L-y)$ gives the spread of the disease + \vspace*{0.1in} +\item $L$ is the total population. +\item Use $L = 2.5E5, k = 3E-5, y(0) = 250$ +\end{itemize} +\end{frame} + +\begin{frame}[fragile] +\frametitle{Solving ODEs using SciPy} +Define a function as below +\small +\begin{lstlisting} +In []: from scipy.integrate import odeint +In []: def epid(y, t): + ...: k = 3.0e-5 + ...: L = 2.5e5 + ...: return k*y*(L-y) + ...: +\end{lstlisting} +\end{frame} + +\begin{frame}[fragile] +\frametitle{Solving ODEs using SciPy \ldots} +\begin{lstlisting} +In []: t = linspace(0, 12, 61) + +In []: y = odeint(epid, 250, t) + +In []: plot(t, y) +\end{lstlisting} +%Insert Plot +\end{frame} + +\begin{frame}[fragile] +\frametitle{Result} +\begin{center} +\includegraphics[height=3in, interpolate=true]{data/image} +\end{center} +\vspace*{-0.5in} +\inctime{5} +\end{frame} + + +\begin{frame}[fragile] +\frametitle{ODEs - Simple Pendulum} +We shall use the simple ODE of a simple pendulum. +\begin{equation*} +\ddot{\theta} = -\frac{g}{L}sin(\theta) +\end{equation*} +\begin{itemize} +\item This equation can be written as a system of two first order ODEs +\end{itemize} +\begin{align} +\dot{\theta} &= \omega \\ +\dot{\omega} &= -\frac{g}{L}sin(\theta) \\ + \text{At}\ t &= 0 : \nonumber \\ + \theta = \theta_0(10^o)\quad & \&\quad \omega = 0\ (Initial\ values)\nonumber +\end{align} +\end{frame} + +\begin{frame}[fragile] +\frametitle{ODEs - Simple Pendulum \ldots} +\begin{itemize} +\item Use \typ{odeint} to do the integration +\end{itemize} +\begin{lstlisting} +In []: def pend_rhs(state, t): + .... theta = state[0] + .... omega = state[1] + .... g = 9.81 + .... L = 0.2 + .... F=[omega, -(g/L)*sin(theta)] + .... return F + .... +\end{lstlisting} +\end{frame} + +\begin{frame}[fragile] +\frametitle{ODEs - Simple Pendulum \ldots} +\begin{itemize} +\item \typ{t} is the time variable \\ +\item \typ{initial} has the initial values +\end{itemize} +\begin{lstlisting} +In []: t = linspace(0, 20, 101) +In []: initial = [10*2*pi/360, 0] +\end{lstlisting} +\end{frame} + +\begin{frame}[fragile] +\frametitle{ODEs - Simple Pendulum \ldots} +%%\begin{small} +\typ{In []: from scipy.integrate import odeint} +%%\end{small} +\begin{lstlisting} +In []: pend_sol = odeint(pend_rhs, + initial,t) +\end{lstlisting} +\end{frame} + +\begin{frame}[fragile] +\frametitle{Result} +\begin{center} +\includegraphics[height=2in, interpolate=true]{data/ode} +\end{center} + \inctime{10} +\end{frame} + +\section{FFTs} + +\begin{frame}[fragile] +\frametitle{The FFT} +\begin{itemize} + \item We have a simple signal $y(t)$ + \item Find the FFT and plot it +\end{itemize} +\begin{lstlisting} +In []: t = linspace(0, 2*pi, 500) +In []: y = sin(4*pi*t) + +In []: f = fft.fft(y) +In []: freq = fft.fftfreq(500, + ...: t[1] - t[0]) + +In []: plot(freq[:250], abs(f)[:250]) +In []: grid() +\end{lstlisting} +\end{frame} + +\begin{frame}[fragile] +\frametitle{FFTs cont\dots} +\begin{lstlisting} +In []: y1 = fft.ifft(f) # inverse FFT +In []: allclose(y, y1) +Out[]: True +\end{lstlisting} +\end{frame} + +\begin{frame}[fragile] +\frametitle{FFTs cont\dots} +Let us add some noise to the signal +\begin{lstlisting} +In []: yr = y + + ...: random.random(size=500)*0.2 +In []: yn = y + + ...: random.normal(size=500)*0.2 + +In []: plot(t, yr) +In []: figure() +In []: plot(freq[:250], + ...: abs(fft.fft(yr))[:250]) +\end{lstlisting} +\begin{itemize} + \item \typ{random}: produces uniform deviates in $[0, 1)$ + \item \typ{normal}: draws random samples from a Gaussian + distribution + \item Useful to create a random matrix of any shape +\end{itemize} +\end{frame} + +\begin{frame}[fragile] +\frametitle{FFTs cont\dots} +Filter the noisy signal: +\begin{lstlisting} +In []: from scipy import signal +In []: yc = signal.wiener(yn, 5) +In []: clf() +In []: plot(t, yc) +In []: figure() +In []: plot(freq[:250], + ...: abs(fft.fft(yc))[:250]) +\end{lstlisting} +Only scratched the surface here \dots + +\inctime{10} +\end{frame} + + +\begin{frame} + \frametitle{Things we have learned} + \begin{itemize} + \item Solving Linear Equations + \item Defining Functions + \item Finding Roots + \item Solving ODEs + \item FFTs and basic signal processing + \end{itemize} +\end{frame} + + +\begin{frame} + \frametitle{Further reading} + \begin{itemize} + \item \url{ipython.readthedocs.io} + \item \url{matplotlib.org/contents.html} + \item \url{docs.scipy.org/doc/numpy/user/quickstart.html} + \item \url{docs.scipy.org/doc/scipy/reference/tutorial} + \end{itemize} +\end{frame} + +\end{document} |