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diff --git a/day1/session5.tex b/day1/session5.tex index 8ed3d1f..3c66ad6 100644 --- a/day1/session5.tex +++ b/day1/session5.tex @@ -79,7 +79,7 @@ \author[FOSSEE] {FOSSEE} \institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay} -\date[] {31, October 2009\\Day 1, Session 4} +\date[] {31, October 2009\\Day 1, Session 5} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\pgfdeclareimage[height=0.75cm]{iitmlogo}{iitmlogo} @@ -96,13 +96,13 @@ \end{frame} } -%%\AtBeginSection[] -%%{ - %%\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: @@ -124,178 +124,277 @@ % \pausesections \end{frame} -\section{Integration} - -\subsection{Quadrature} +\section{Interpolation} \begin{frame}[fragile] -\frametitle{Quadrature} -\begin{itemize} -\item We wish to find area under a curve -\item Area under $(sin(x) + x^2)$ in $(0,1)$ -\item scipy has functions to do that -\end{itemize} -\small{\typ{In []: from scipy.integrate import quad}} +\frametitle{Interpolation} \begin{itemize} -\item Inputs - function to integrate, limits +\item Let us begin with interpolation +\item Let's use the L and T arrays and interpolate this data to obtain data at new points \end{itemize} \begin{lstlisting} -In []: x = 0 -In []: quad(sin(x)+x**2, 0, 1) +In []: L = [] +In []: T = [] +In []: for line in open('pendulum.txt'): + l, t = line.split() + L.append(float(l)) + T.append(float(t)) +In []: L = array(L) +In []: T = array(T) \end{lstlisting} -\alert{\typ{error:}} -\typ{First argument must be a callable function.} \end{frame} +%% \begin{frame}[fragile] +%% \frametitle{Interpolation \ldots} +%% \begin{small} +%% \typ{In []: from scipy.interpolate import interp1d} +%% \end{small} +%% \begin{itemize} +%% \item The \typ{interp1d} function returns a function +%% \begin{lstlisting} +%% In []: f = interp1d(L, T) +%% \end{lstlisting} +%% \item Functions can be assigned to variables +%% \item This function interpolates between known data values to obtain unknown +%% \end{itemize} +%% \end{frame} + +%% \begin{frame}[fragile] +%% \frametitle{Interpolation \ldots} +%% \begin{lstlisting} +%% In []: Ln = arange(0.1,0.99,0.005) +%% # Interpolating! +%% # The new values in range of old data +%% In []: plot(L, T, 'o', Ln, f(Ln), '-') +%% In []: f = interp1d(L, T, kind='cubic') +%% # When kind not specified, it's linear +%% # Others are ... +%% # 'nearest', 'zero', +%% # 'slinear', 'quadratic' +%% \end{lstlisting} +%% \end{frame} + \begin{frame}[fragile] -\frametitle{Functions - Definition} +\frametitle{Spline Interpolation} +\begin{small} \begin{lstlisting} -In []: def f(x): - return sin(x)+x**2 -In []: quad(f, 0, 1) +In []: from scipy.interpolate import splrep +In []: from scipy.interpolate import splev \end{lstlisting} +\end{small} \begin{itemize} -\item \typ{def} -\item arguments -\item \typ{return} +\item Involves two steps + \begin{enumerate} + \item Find out the spline curve, coefficients + \item Evaluate the spline at new points + \end{enumerate} \end{itemize} \end{frame} \begin{frame}[fragile] -\frametitle{Functions - Calling them} +\frametitle{\typ{splrep}} +To find the B-spline representation \begin{lstlisting} -In [15]: f() ---------------------------------------- +In []: tck = splrep(L, T) \end{lstlisting} -\alert{\typ{TypeError:}}\typ{f() takes exactly 1 argument} -\typ{(0 given)} +Returns +\begin{enumerate} +\item the vector of knots, +\item the B-spline coefficients +\item the degree of the spline (default=3) +\end{enumerate} +\end{frame} + +\begin{frame}[fragile] +\frametitle{\typ{splev}} +To Evaluate a B-spline and it's derivatives \begin{lstlisting} -In []: f(0) -Out[]: 0.0 -In []: f(1) -Out[]: 1.8414709848078965 +In []: Lnew = arange(0.1,1,0.005) +In []: Tnew = splev(Lnew, tck) + +#To obtain derivatives of the spline +#use der=1, 2,.. for 1st, 2nd,.. order +In []: Tnew = splev(Lnew, tck, der=1) \end{lstlisting} \end{frame} +%% \begin{frame}[fragile] +%% \frametitle{Interpolation \ldots} +%% \begin{itemize} +%% \item +%% \end{itemize} +%% \end{frame} + +\section{Differentiation} + +\begin{frame}[fragile] +\frametitle{Numerical Differentiation} +\begin{itemize} +\item Given function $f(x)$ or data points $y=f(x)$ +\item We wish to calculate $f^{'}(x)$ at points $x$ +\item Taylor series - finite difference approximations +\end{itemize} +\begin{center} +\begin{tabular}{l l} +$f(x+h)=f(x)+h.f^{'}(x)$ &Forward \\ +$f(x-h)=f(x)-h.f^{'}(x)$ &Backward +\end{tabular} +\end{center} +\end{frame} \begin{frame}[fragile] -\frametitle{Functions - Default Arguments} +\frametitle{Forward Difference} \begin{lstlisting} -In []: def f(x=1): - return sin(x)+x**2 -In []: f(10) -Out[]: 99.455978889110625 -In []: f(1) -Out[]: 1.8414709848078965 -In []: f() -Out[]: 1.8414709848078965 +In []: x = linspace(0, 2*pi, 100) +In []: y = sin(x) +In []: deltax = x[1] - x[0] \end{lstlisting} +Obtain the finite forward difference of y \end{frame} \begin{frame}[fragile] -\frametitle{Functions - Keyword Arguments} +\frametitle{Forward Difference \ldots} \begin{lstlisting} -In []: def f(x=1, y=pi): - return sin(y)+x**2 -In []: f() -Out[]: 1.0000000000000002 -In []: f(2) -Out[]: 4.0 -In []: f(y=2) -Out[]: 1.9092974268256817 -In []: f(y=pi/2,x=0) -Out[]: 1.0 +In []: fD = (y[1:] - y[:-1]) / deltax +In []: plot(x, y, x[:-1], fD) \end{lstlisting} +\begin{center} + \includegraphics[height=2in, interpolate=true]{data/fwdDiff} +\end{center} \end{frame} \begin{frame}[fragile] - \frametitle{More on functions} - \begin{itemize} - \item Scope of variables in the function is local - \item Mutable items are \alert{passed by reference} - \item First line after definition may be a documentation string - (\alert{recommended!}) - \item Function definition and execution defines a name bound to the - function - \item You \emph{can} assign a variable to a function! - \end{itemize} +\frametitle{Example} +\begin{itemize} +\item Given x, y positions of a particle in \typ{pos.txt} +\item Find velocity \& acceleration in x, y directions +\end{itemize} +\small{ +\begin{center} +\begin{tabular}{| c | c | c |} +\hline +$X$ & $Y$ \\ \hline +0. & 0.\\ \hline +0.25 & 0.47775\\ \hline +0.5 & 0.931\\ \hline +0.75 & 1.35975\\ \hline +1. & 1.764\\ \hline +1.25 & 2.14375\\ \hline +\vdots & \vdots\\ \hline +\end{tabular} +\end{center}} \end{frame} \begin{frame}[fragile] -\frametitle{Quadrature \ldots} -\begin{lstlisting} -In []: quad(f, 0, 1) -\end{lstlisting} -Returns the integral and an estimate of the absolute error in the result. +\frametitle{Example \ldots} \begin{itemize} -\item Use \typ{dblquad} for Double integrals -\item Use \typ{tplquad} for Triple integrals +\item Read the file +\item Obtain an array of x, y +\item Obtain velocity and acceleration +\item use \typ{deltaT = 0.05} \end{itemize} +\begin{lstlisting} +In []: X = [] +In []: Y = [] +In []: for line in open('location.txt'): + .... points = line.split() + .... X.append(float(points[0])) + .... Y.append(float(points[1])) +In []: S = array([X, Y]) +\end{lstlisting} \end{frame} -\subsection{ODEs} \begin{frame}[fragile] -\frametitle{ODE Integration} -We shall use the simple ODE of a simple pendulum. -\begin{equation*} -\ddot{\theta} = -\frac{g}{L}sin(\theta) -\end{equation*} +\frametitle{Example \ldots} \begin{itemize} -\item This equation can be written as a system of two first order ODEs +\item use \typ{deltaT = 0.05} \end{itemize} -\begin{align} -\dot{\theta} &= \omega \\ -\dot{\omega} &= -\frac{g}{L}sin(\theta) \\ - \text{At}\ t &= 0 : \nonumber \\ - \theta = \theta_0\quad & \&\quad \omega = 0 \nonumber -\end{align} +\begin{lstlisting} +In []: deltaT = 0.05 + +In []: v = (S[:,1:]-S[:,:-1])/deltaT + +In []: a = (v[:,1:]-v[:,:-1])/deltaT +\end{lstlisting} +Try Plotting the position, velocity \& acceleration. \end{frame} +\section{Quadrature} + \begin{frame}[fragile] -\frametitle{Solving ODEs using SciPy} +\frametitle{Quadrature} +\begin{itemize} +\item We wish to find area under a curve +\item Area under $(sin(x) + x^2)$ in $(0,1)$ +\item scipy has functions to do that +\end{itemize} +\begin{small} + \typ{In []: from scipy.integrate import quad} +\end{small} \begin{itemize} -\item We use the \typ{odeint} function from scipy to do the integration -\item Define a function as below +\item Inputs - function to integrate, limits \end{itemize} \begin{lstlisting} -In []: def pend_int(unknown, t, p): - .... theta, omega = unknown - .... g, L = p - .... f=[omega, -(g/L)*sin(theta)] - .... return f - .... +In []: x = 0 +In []: quad(sin(x)+x**2, 0, 1) \end{lstlisting} +\begin{small} +\alert{\typ{error:}} +\typ{First argument must be a callable function.} +\end{small} \end{frame} \begin{frame}[fragile] -\frametitle{Solving ODEs using SciPy \ldots} +\frametitle{Functions - Definition} +We have been using them all along. Now let's see how to define them. +\begin{lstlisting} +In []: def f(x): + return sin(x)+x**2 +In []: quad(f, 0, 1) +\end{lstlisting} \begin{itemize} -\item \typ{t} is the time variable \\ -\item \typ{p} has the constants \\ -\item \typ{initial} has the initial values +\item \typ{def} +\item name +\item arguments +\item \typ{return} \end{itemize} +\end{frame} + +\begin{frame}[fragile] +\frametitle{Functions - Calling them} +\begin{lstlisting} +In [15]: f() +--------------------------------------- +\end{lstlisting} +\alert{\typ{TypeError:}}\typ{f() takes exactly 1 argument} +\typ{(0 given)} \begin{lstlisting} -In []: t = linspace(0, 10, 101) -In []: p=(-9.81, 0.2) -In []: initial = [10*2*pi/360, 0] +In []: f(0) +Out[]: 0.0 +In []: f(1) +Out[]: 1.8414709848078965 \end{lstlisting} +More on Functions later \ldots \end{frame} \begin{frame}[fragile] -\frametitle{Solving ODEs using SciPy \ldots} - -\small{\typ{In []: from scipy.integrate import odeint}} +\frametitle{Quadrature \ldots} \begin{lstlisting} -In []: pend_sol = odeint(pend_int, - initial,t, - args=(p,)) +In []: quad(f, 0, 1) \end{lstlisting} +Returns the integral and an estimate of the absolute error in the result. +\begin{itemize} +\item Look at \typ{dblquad} for Double integrals +\item Use \typ{tplquad} for Triple integrals +\end{itemize} \end{frame} \begin{frame} \frametitle{Things we have learned} \begin{itemize} + \item Interpolation + \item Differentiation \item Functions \begin{itemize} \item Definition |