From 44733bf859c288901ad00c55c64497771443400d Mon Sep 17 00:00:00 2001 From: Puneeth Chaganti Date: Thu, 9 Dec 2010 22:37:22 +0530 Subject: Renamed day1/session6 to day1/session4. --HG-- branch : scipyin2010 rename : day1/session6.tex => day1/session4.tex --- day1/session4.tex | 655 ++++++++++++++++++++++++++++++++++++++++++++++++++++++ day1/session6.tex | 655 ------------------------------------------------------ 2 files changed, 655 insertions(+), 655 deletions(-) create mode 100644 day1/session4.tex delete mode 100755 day1/session6.tex diff --git a/day1/session4.tex b/day1/session4.tex new file mode 100644 index 0000000..165957e --- /dev/null +++ b/day1/session4.tex @@ -0,0 +1,655 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%Tutorial slides on Python. +% +% Author: FOSSEE +% Copyright (c) 2009, 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 +{ + \usetheme{Warsaw} + \useoutertheme{infolines} + \setbeamercovered{transparent} +} + +\usepackage[english]{babel} +\usepackage[latin1]{inputenc} +%\usepackage{times} +\usepackage[T1]{fontenc} + +% 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} +} +\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[Solving Equations \& ODEs]{Python for Science and Engg:\\Solving Equations \& ODEs} + +\author[FOSSEE] {FOSSEE} + +\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay} +\date[] {SciPy 2010, Introductory tutorials\\Day 1, Session 6} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%\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} + \frametitle{Outline} + \tableofcontents[currentsection,currentsubsection] + \end{frame} +} + +\AtBeginSection[] +{ + \begin{frame} + \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} + +%% \begin{frame} +%% \frametitle{Outline} +%% \tableofcontents +%% % You might wish to add the option [pausesections] +%% \end{frame} + +\section{Solving linear equations} + +\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*} +\inctime{5} +\end{frame} + +\begin{frame}[fragile] +\frametitle{Solution} +Use \kwrd{solve()} +\begin{align*} + x & = -5\\ + y & = 2\\ + z & = 3\\ + w & = 0\\ +\end{align*} +\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}} +\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 - Function and initial estimate +\item Returns the solution +\end{itemize} +\end{frame} + +\begin{frame}[fragile] +\frametitle{\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}} +Root of $sin(z)+cos^2(z)$ nearest to $0$ +\begin{lstlisting} +In []: fsolve(sin(z)+cos(z)*cos(z), 0) +NameError: name 'z' is not defined +\end{lstlisting} +\end{frame} + +\begin{frame}[fragile] +\frametitle{\typ{fsolve}} +\begin{lstlisting} +In []: z = linspace(-pi, pi) +In []: fsolve(sin(z)+cos(z)*cos(z), 0) +\end{lstlisting} +\begin{small} +\alert{\typ{TypeError:}} +\typ{'numpy.ndarray' object is not callable} +\end{small} +\end{frame} + +\begin{frame}[fragile] +\frametitle{Functions - Definition} +We have been using them all along. Now let's see how to define them. +\begin{lstlisting} +In []: def g(z): + ....: return sin(z)+cos(z)*cos(z) +\end{lstlisting} +\begin{itemize} +\item \typ{def} -- keyword +\item name: \typ{g} +\item arguments: \typ{z} +\item \typ{return} -- keyword +\end{itemize} +\end{frame} + +\begin{frame}[fragile] +\frametitle{Functions - Calling them} +\begin{lstlisting} +In []: g() +--------------------------------------- +\end{lstlisting} +\alert{\typ{TypeError:}}\typ{g() takes exactly 1 argument} +\typ{(0 given)} +\begin{lstlisting} +In []: g(0) +Out[]: 1.0 +In []: g(1) +Out[]: 1.1333975665343254 +\end{lstlisting} +More on Functions later \ldots +\end{frame} + +\begin{frame}[fragile] +\frametitle{\typ{fsolve} \ldots} +Find the root of $sin(z)+cos^2(z)$ nearest to $0$ +\begin{lstlisting} +In []: fsolve(g, 0) +Out[]: -0.66623943249251527 +\end{lstlisting} +\begin{center} +\includegraphics[height=2in, interpolate=true]{data/fsolve} +\end{center} +\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} + \begin{center} +\includegraphics[height=2in, interpolate=true]{data/fsolve_tanx} + \end{center} +\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}[fragile] +\frametitle{Solving ODEs using SciPy} +\begin{itemize} +\item Consider the spread of an epidemic in a population +\item $\frac{dy}{dt} = ky(L-y)$ gives the spread of the disease +\item $L$ is the total population. +\item Use $L = 2.5E5, k = 3E-5, y(0) = 250$ +\item Define a function as below +\end{itemize} +\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=2in, interpolate=true]{data/image} +\end{center} +\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_int(initial, t): + .... theta = initial[0] + .... omega = initial[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_int, + initial,t) +\end{lstlisting} +\end{frame} + +\begin{frame}[fragile] +\frametitle{Result} +\begin{center} +\includegraphics[height=2in, interpolate=true]{data/ode} +\end{center} +\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(y) +In []: freq = 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 = 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(size=500)*0.2 +In []: yn = y + normal(size=500)*0.2 + +In []: plot(t, yr) +In []: figure() +In []: plot(freq[:250], + ...: abs(fft(yn))[: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(yc))[:250]) +\end{lstlisting} +Only scratched the surface here \dots +\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 Random number generation + \item FFTs and basic signal processing + \end{itemize} +\end{frame} + +\end{document} + +%% Questions for Quiz %% +%% ------------------ %% + +\begin{frame} +\frametitle{\incqno } +Given a 4x4 matrix \texttt{A} and a 4-vector \texttt{b}, what command do +you use to solve for the equation \\ +\texttt{Ax = b}? +\end{frame} + +\begin{frame} +\frametitle{\incqno } +What command will you use if you wish to integrate a system of ODEs? +\end{frame} + +\begin{frame} +\frametitle{\incqno } +How do you calculate the roots of the polynomial, $y = 1 + 6x + 8x^2 + +x^3$? +\end{frame} + +\begin{frame} +\frametitle{\incqno } +Two arrays \lstinline+a+ and \lstinline+b+ are numerically almost equal, what command +do you use to check if this is true? +\end{frame} + +%% \begin{frame}[fragile] +%% \frametitle{\incqno } +%% \begin{lstlisting} +%% In []: x = arange(0, 1, 0.25) +%% In []: print x +%% \end{lstlisting} +%% What will be printed? +%% \end{frame} + + +%% \begin{frame}[fragile] +%% \frametitle{\incqno } +%% \begin{lstlisting} +%% from scipy.integrate import quad +%% def f(x): +%% res = x*cos(x) +%% quad(f, 0, 1) +%% \end{lstlisting} +%% What changes will you make to the above code to make it work? +%% \end{frame} + +%% \begin{frame} +%% \frametitle{\incqno } +%% What two commands will you use to create and evaluate a spline given +%% some data? +%% \end{frame} + +%% \begin{frame}[fragile] +%% \frametitle{\incqno } +%% What would be the result? +%% \begin{lstlisting} +%% In []: x +%% array([[0, 1, 2], +%% [3, 4, 5], +%% [6, 7, 8]]) +%% In []: x[::-1,:] +%% \end{lstlisting} +%% Hint: +%% \begin{lstlisting} +%% In []: x = arange(9) +%% In []: x[::-1] +%% array([8, 7, 6, 5, 4, 3, 2, 1, 0]) +%% \end{lstlisting} +%% \end{frame} + +%% \begin{frame}[fragile] +%% \frametitle{\incqno } +%% What would be the result? +%% \begin{lstlisting} +%% In []: y = arange(3) +%% In []: x = linspace(0,3,3) +%% In []: x-y +%% \end{lstlisting} +%% \end{frame} + +%% \begin{frame}[fragile] +%% \frametitle{\incqno } +%% \begin{lstlisting} +%% In []: x +%% array([[ 0, 1, 2, 3], +%% [ 4, 5, 6, 7], +%% [ 8, 9, 10, 11], +%% [12, 13, 14, 15]]) +%% \end{lstlisting} +%% How will you get the following \lstinline+x+? +%% \begin{lstlisting} +%% array([[ 5, 7], +%% [ 9, 11]]) +%% \end{lstlisting} +%% \end{frame} + +%% \begin{frame}[fragile] +%% \frametitle{\incqno } +%% What would be the output? +%% \begin{lstlisting} +%% In []: y = arange(4) +%% In []: x = array(([1,2,3,2],[1,3,6,0])) +%% In []: x + y +%% \end{lstlisting} +%% \end{frame} + +%% \begin{frame}[fragile] +%% \frametitle{\incqno } +%% \begin{lstlisting} +%% In []: line = plot(x, sin(x)) +%% \end{lstlisting} +%% Use the \lstinline+set_linewidth+ method to set width of \lstinline+line+ to 2. +%% \end{frame} + +%% \begin{frame}[fragile] +%% \frametitle{\incqno } +%% What would be the output? +%% \begin{lstlisting} +%% In []: x = arange(9) +%% In []: y = arange(9.) +%% In []: x == y +%% \end{lstlisting} +%% \end{frame} + diff --git a/day1/session6.tex b/day1/session6.tex deleted file mode 100755 index 165957e..0000000 --- a/day1/session6.tex +++ /dev/null @@ -1,655 +0,0 @@ -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%Tutorial slides on Python. -% -% Author: FOSSEE -% Copyright (c) 2009, 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 -{ - \usetheme{Warsaw} - \useoutertheme{infolines} - \setbeamercovered{transparent} -} - -\usepackage[english]{babel} -\usepackage[latin1]{inputenc} -%\usepackage{times} -\usepackage[T1]{fontenc} - -% 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} -} -\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[Solving Equations \& ODEs]{Python for Science and Engg:\\Solving Equations \& ODEs} - -\author[FOSSEE] {FOSSEE} - -\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay} -\date[] {SciPy 2010, Introductory tutorials\\Day 1, Session 6} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%\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} - \frametitle{Outline} - \tableofcontents[currentsection,currentsubsection] - \end{frame} -} - -\AtBeginSection[] -{ - \begin{frame} - \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} - -%% \begin{frame} -%% \frametitle{Outline} -%% \tableofcontents -%% % You might wish to add the option [pausesections] -%% \end{frame} - -\section{Solving linear equations} - -\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*} -\inctime{5} -\end{frame} - -\begin{frame}[fragile] -\frametitle{Solution} -Use \kwrd{solve()} -\begin{align*} - x & = -5\\ - y & = 2\\ - z & = 3\\ - w & = 0\\ -\end{align*} -\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}} -\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 - Function and initial estimate -\item Returns the solution -\end{itemize} -\end{frame} - -\begin{frame}[fragile] -\frametitle{\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}} -Root of $sin(z)+cos^2(z)$ nearest to $0$ -\begin{lstlisting} -In []: fsolve(sin(z)+cos(z)*cos(z), 0) -NameError: name 'z' is not defined -\end{lstlisting} -\end{frame} - -\begin{frame}[fragile] -\frametitle{\typ{fsolve}} -\begin{lstlisting} -In []: z = linspace(-pi, pi) -In []: fsolve(sin(z)+cos(z)*cos(z), 0) -\end{lstlisting} -\begin{small} -\alert{\typ{TypeError:}} -\typ{'numpy.ndarray' object is not callable} -\end{small} -\end{frame} - -\begin{frame}[fragile] -\frametitle{Functions - Definition} -We have been using them all along. Now let's see how to define them. -\begin{lstlisting} -In []: def g(z): - ....: return sin(z)+cos(z)*cos(z) -\end{lstlisting} -\begin{itemize} -\item \typ{def} -- keyword -\item name: \typ{g} -\item arguments: \typ{z} -\item \typ{return} -- keyword -\end{itemize} -\end{frame} - -\begin{frame}[fragile] -\frametitle{Functions - Calling them} -\begin{lstlisting} -In []: g() ---------------------------------------- -\end{lstlisting} -\alert{\typ{TypeError:}}\typ{g() takes exactly 1 argument} -\typ{(0 given)} -\begin{lstlisting} -In []: g(0) -Out[]: 1.0 -In []: g(1) -Out[]: 1.1333975665343254 -\end{lstlisting} -More on Functions later \ldots -\end{frame} - -\begin{frame}[fragile] -\frametitle{\typ{fsolve} \ldots} -Find the root of $sin(z)+cos^2(z)$ nearest to $0$ -\begin{lstlisting} -In []: fsolve(g, 0) -Out[]: -0.66623943249251527 -\end{lstlisting} -\begin{center} -\includegraphics[height=2in, interpolate=true]{data/fsolve} -\end{center} -\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} - \begin{center} -\includegraphics[height=2in, interpolate=true]{data/fsolve_tanx} - \end{center} -\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}[fragile] -\frametitle{Solving ODEs using SciPy} -\begin{itemize} -\item Consider the spread of an epidemic in a population -\item $\frac{dy}{dt} = ky(L-y)$ gives the spread of the disease -\item $L$ is the total population. -\item Use $L = 2.5E5, k = 3E-5, y(0) = 250$ -\item Define a function as below -\end{itemize} -\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=2in, interpolate=true]{data/image} -\end{center} -\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_int(initial, t): - .... theta = initial[0] - .... omega = initial[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_int, - initial,t) -\end{lstlisting} -\end{frame} - -\begin{frame}[fragile] -\frametitle{Result} -\begin{center} -\includegraphics[height=2in, interpolate=true]{data/ode} -\end{center} -\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(y) -In []: freq = 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 = 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(size=500)*0.2 -In []: yn = y + normal(size=500)*0.2 - -In []: plot(t, yr) -In []: figure() -In []: plot(freq[:250], - ...: abs(fft(yn))[: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(yc))[:250]) -\end{lstlisting} -Only scratched the surface here \dots -\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 Random number generation - \item FFTs and basic signal processing - \end{itemize} -\end{frame} - -\end{document} - -%% Questions for Quiz %% -%% ------------------ %% - -\begin{frame} -\frametitle{\incqno } -Given a 4x4 matrix \texttt{A} and a 4-vector \texttt{b}, what command do -you use to solve for the equation \\ -\texttt{Ax = b}? -\end{frame} - -\begin{frame} -\frametitle{\incqno } -What command will you use if you wish to integrate a system of ODEs? -\end{frame} - -\begin{frame} -\frametitle{\incqno } -How do you calculate the roots of the polynomial, $y = 1 + 6x + 8x^2 + -x^3$? -\end{frame} - -\begin{frame} -\frametitle{\incqno } -Two arrays \lstinline+a+ and \lstinline+b+ are numerically almost equal, what command -do you use to check if this is true? -\end{frame} - -%% \begin{frame}[fragile] -%% \frametitle{\incqno } -%% \begin{lstlisting} -%% In []: x = arange(0, 1, 0.25) -%% In []: print x -%% \end{lstlisting} -%% What will be printed? -%% \end{frame} - - -%% \begin{frame}[fragile] -%% \frametitle{\incqno } -%% \begin{lstlisting} -%% from scipy.integrate import quad -%% def f(x): -%% res = x*cos(x) -%% quad(f, 0, 1) -%% \end{lstlisting} -%% What changes will you make to the above code to make it work? -%% \end{frame} - -%% \begin{frame} -%% \frametitle{\incqno } -%% What two commands will you use to create and evaluate a spline given -%% some data? -%% \end{frame} - -%% \begin{frame}[fragile] -%% \frametitle{\incqno } -%% What would be the result? -%% \begin{lstlisting} -%% In []: x -%% array([[0, 1, 2], -%% [3, 4, 5], -%% [6, 7, 8]]) -%% In []: x[::-1,:] -%% \end{lstlisting} -%% Hint: -%% \begin{lstlisting} -%% In []: x = arange(9) -%% In []: x[::-1] -%% array([8, 7, 6, 5, 4, 3, 2, 1, 0]) -%% \end{lstlisting} -%% \end{frame} - -%% \begin{frame}[fragile] -%% \frametitle{\incqno } -%% What would be the result? -%% \begin{lstlisting} -%% In []: y = arange(3) -%% In []: x = linspace(0,3,3) -%% In []: x-y -%% \end{lstlisting} -%% \end{frame} - -%% \begin{frame}[fragile] -%% \frametitle{\incqno } -%% \begin{lstlisting} -%% In []: x -%% array([[ 0, 1, 2, 3], -%% [ 4, 5, 6, 7], -%% [ 8, 9, 10, 11], -%% [12, 13, 14, 15]]) -%% \end{lstlisting} -%% How will you get the following \lstinline+x+? -%% \begin{lstlisting} -%% array([[ 5, 7], -%% [ 9, 11]]) -%% \end{lstlisting} -%% \end{frame} - -%% \begin{frame}[fragile] -%% \frametitle{\incqno } -%% What would be the output? -%% \begin{lstlisting} -%% In []: y = arange(4) -%% In []: x = array(([1,2,3,2],[1,3,6,0])) -%% In []: x + y -%% \end{lstlisting} -%% \end{frame} - -%% \begin{frame}[fragile] -%% \frametitle{\incqno } -%% \begin{lstlisting} -%% In []: line = plot(x, sin(x)) -%% \end{lstlisting} -%% Use the \lstinline+set_linewidth+ method to set width of \lstinline+line+ to 2. -%% \end{frame} - -%% \begin{frame}[fragile] -%% \frametitle{\incqno } -%% What would be the output? -%% \begin{lstlisting} -%% In []: x = arange(9) -%% In []: y = arange(9.) -%% In []: x == y -%% \end{lstlisting} -%% \end{frame} - -- cgit