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diff --git a/latex/examples/glimpse-at-scipy.tex b/latex/examples/glimpse-at-scipy.tex deleted file mode 100644 index 8227c34..0000000 --- a/latex/examples/glimpse-at-scipy.tex +++ /dev/null @@ -1,426 +0,0 @@ -\documentclass{article} - -\usepackage{amsmath} -\usepackage{graphicx} -\usepackage{color} -\usepackage{listings} -\usepackage{url} -%\definecolor{darkgreen}{rgb}{0,0.5,0} - -\lstset{language=Python, - basicstyle=\ttfamily\bfseries, - commentstyle=\color{red}\itshape, - stringstyle=\color{green}, - showstringspaces=false, - keywordstyle=\color{blue}\bfseries} - -\title{A Glimpse at Scipy} -\author{FOSSEE} -\date{June 2010} - -\begin{document} -\maketitle - -\begin{abstract} -This document shows a glimpse of the features of Scipy that will be -explored during this course. -\end{abstract} - -\section{Introduction} - -SciPy is open-source software for mathematics, science, and -engineering. - -SciPy (pronounced ``Sigh Pie'') is a collection of mathematical -algorithms and convenience functions built on the Numpy extension for -Python. It adds significant power to the interactive Python session by -exposing the user to high-level commands and classes for the -manipulation and visualization of data. With SciPy, an interactive -Python session becomes a data-processing and system-prototyping -environment rivaling sytems such as \emph{Matlab, IDL, Octave, R-Lab, - and Scilab}. \cite{scipy} - -%% \begin{quote} -%% In 1998, ... I came across Python and its numerical extension -%% (Numeric) while I was looking for ways to analyze large data sets -%% ... using a high-level language. I quickly fell in love with -%% Python programming which is a remarkable statement to make about a -%% programming language. If I had not seen others with the same view, -%% I might have seriously doubted my sanity. -- Travis Oliphant, Creator of Numpy -%% \end{quote} - -\subsection{Sub-packages of Scipy} - -SciPy is organized into subpackages covering different scientific -computing domains. These are summarized in the \underline{table - \ref{subpkg}}. - -\begin{table} - \caption{Sub-packages available in Scipy} - \label{subpkg} -\begin{tabular}{|l|l|} -\hline - -\textbf{Subpackage} & \textbf{Description}\\ - -\hline - -\texttt{cluster} & Clustering algorithms\\ - -\hline - -\texttt{constants} & Physical and mathematical constants\\ - -\hline - -\texttt{fftpack} & Fast Fourier Transform routines\\ - -\hline - -\texttt{integrate} & Integration and ordinary differential equation -solvers\\ - -\hline - -\texttt{interpolate} & Interpolation and smoothing splines\\ - -\hline - -\texttt{io} & Input and Output\\ - -\hline - -\texttt{linalg} & Linear algebra\\ - -\hline - -\texttt{maxentropy} & Maximum entropy methods\\ - -\hline - -\texttt{ndimage} & N-dimensional image processing\\ - -\hline - -\texttt{odr} & Orthogonal distance regression\\ - -\hline - -\texttt{optimize} & Optimization and root-finding routines\\ - -\hline - -\texttt{signal} & Signal processing\\ - -\hline - -\texttt{sparse} & Sparse matrices and associated routines\\ - -\hline - -\texttt{spatial} & Spatial data structures and algorithms\\ - -\hline - -\texttt{special} & Special functions\\ - -\hline - -\texttt{stats} & Statistical distributions and functions\\ - -\hline - -\texttt{weave} & C/C++ integration\\ - -\hline -\end{tabular} -\end{table} - -\subsection{Use of Scipy in this course} -Following is a partial list of tasks we shall perform using Scipy, in -this course. - -\begin{enumerate} - \item Plotting \footnote{using \texttt{pylab} - see Appendix - \ref{mpl}} - \item Matrix Operations - \begin{itemize} - \item Inverse - \item Determinant - \end{itemize} - \item Solving Equations - \begin{itemize} - \item System of Linear equations - \item Polynomials - \item Non-linear equations - \end{itemize} - \item Integration - \begin{itemize} - \item Quadrature - \item ODEs - \end{itemize} -\end{enumerate} -\section{A Glimpse of Scipy functions} - -This section gives a brief overview of the tasks that are going to be -performed using Scipy, in future classes of this course. - -\subsection{Matrix Operations} - -Let $\mathbf{A}$ be the matrix -\( -\begin{bmatrix} -1 &3 &5\\ -2 &5 &1\\ -2 &3 &8 -\end{bmatrix} -\) - -To input $\mathbf{A}$ matrix into python, we do the following in -\texttt{ipython}\footnote{\texttt{ipython} must be started with - \texttt{-pylab} flag}\\ - -\begin{lstlisting} - In []: A = array([[1,3,5],[2,5,1],[2,3,8]]) -\end{lstlisting} - -\subsubsection{Inverse} - -The inverse of a matrix $\mathbf{A}$ is the matrix $\mathbf{B}$ such -that $\mathbf{A}\mathbf{B} = \mathbf{I}$ where $\mathbf{I}$ is the -identity matrix consisting of ones down the main diagonal. Usually -$\mathbf{B}$ is denoted $\mathbf{B} = \mathbf{A}^{-1}$ . In SciPy, the -matrix inverse of matrix $\mathbf{A}$ is obtained using - -\lstinline+inv(A)+. -\begin{lstlisting} - In []: inv(A) - Out[]: - array([[-1.48, 0.36, 0.88], - [ 0.56, 0.08, -0.36], - [ 0.16, -0.12, 0.04]]) -\end{lstlisting} - -\subsubsection{Determinant} - -The determinant of a square matrix $\mathbf{A}$ is denoted -$\left|\mathbf{A}\right|$. Suppose $a_{ij}$ are the elements of the -matrix $\mathbf{A}$ and let -$\mathbf{M}_{ij}=\left|\mathbf{A}_{ij}\right|$ be the determinant of -the matrix left by removing the $i^{th}$ row and $j^{th}$ column from -$\mathbf{A}$. Then for any row $i$ - - \[ \left|\mathbf{A}\right|=\sum_{j}\left(-1\right)^{i+j}a_{ij}\mathbf{M}_{ij} \] - -This is a recursive way to define the determinant where the base case -is defined by accepting that the determinant of a $1\times1$ matrix is -the only matrix element. In SciPy the determinant can be calculated -with $det$ . For example, the determinant of - - \[ \mathbf{A}=\begin{bmatrix} 1 & 3 & 5\\ 2 & 5 & 1\\ 2 & 3 & 8\end{bmatrix}\] - -is - - \begin{eqnarray*} - |\mathbf{A}| & = & 1\begin{vmatrix} 5 & 1\\ 3 & 8\end{vmatrix} - -3\begin{vmatrix} 2 & 1\\ 2 & 8\end{vmatrix} - +5\begin{vmatrix}2 & 5\\ 2 & 3\end{vmatrix}\\ - & = & 1(5\cdot8-3\cdot1)-3(2\cdot8-2\cdot1)+5(2\cdot3-2\cdot5)=-25 - \end{eqnarray*} - -In SciPy, this is computed as shown below - -\begin{lstlisting} - In []: A = array([[1, 3, 5], [2, 5, 1], [2, 3, 8]]) - In []: det(A) - Out[]: -25.0 -\end{lstlisting} - -\subsection{Solving Equations} - -\subsubsection{Linear Equations} - -Solving linear systems of equations is straightforward using the scipy -command \lstinline+solve+. This command expects an input matrix and a -right-hand-side vector. The solution vector is then computed. An -option for entering a symmetrix matrix is offered which can speed up -the processing when applicable. As an example, suppose it is desired -to solve the following simultaneous equations: - - \begin{eqnarray} x+3y+5z & = & 10\\ 2x+5y+z & = & 8\\ 2x+3y+8z & = & 3\end{eqnarray} - -We could find the solution vector using a matrix inverse: - - \[ \left[\begin{array}{c} x\\ y\\ z\end{array}\right]=\left[\begin{array}{ccc} 1 & 3 & 5\\ 2 & 5 & 1\\ 2 & 3 & 8\end{array}\right]^{-1}\left[\begin{array}{c} 10\\ 8\\ 3\end{array}\right]=\frac{1}{25}\left[\begin{array}{c} -232\\ 129\\ 19\end{array}\right]=\left[\begin{array}{c} -9.28\\ 5.16\\ 0.76\end{array}\right] \] - -However, it is better to use the solve command which can be faster and -more numerically stable. In this case it however gives the same -answer. - -\begin{lstlisting} - In []: A = array([[1, 3, 5], [2, 5, 1], [2, 3, 8]]) - In []: b = array([[10], [8], [3]]) - In []: dot(inv(A), b) - Out[]: - array([[-9.28], - [ 5.16], - [ 0.76]]) - - In []: solve(A,b) - Out[]: - array([[-9.28], - [ 5.16], - [ 0.76]]) -\end{lstlisting} - -\subsubsection{Polynomials} - -Solving a polynomial is straightforward in scipy using the -\lstinline+roots+ command. It expects the coefficients of the -polynomial in their decreasing order. For example, let's find the -roots of $x^3 - 2x^2 - \frac{1}{2}x + 1$ are $2$, $\sqrt{2}$ and -$-\sqrt{2}$. This is easy to see. - -\begin{align*} -x^3 - 2x^2 - \frac{1}{2}x + 1 = 0\\ -x^2(x-2) - \frac{1}{2}(x-2) = 0\\ -(x-2)(x^2 - \frac{1}{2}) = 0\\ -(x-2)(x - \frac{1}{\sqrt{2}})(x + \frac{1}{\sqrt{2}}) = 0 -\end{align*} - -We do it in scipy as shown below: -\begin{lstlisting} - In []: coeff = array([1, -2, -2, 4]) - In []: roots(coeff) -\end{lstlisting} - -\subsubsection{Non-linear Equations} - -To find a root of a set of non-linear equations, the command -\lstinline+fsolve+ is needed. For example, the following example finds -the roots of the single-variable transcendental equation - - \[ x+2\cos\left(x\right)=0,\] - -and the set of non-linear equations - - \begin{align} - x_{0}\cos\left(x_{1}\right) &= 4,\\ - x_{0}x_{1}-x_{1} &= 5 - \end{align} - -The results are $x=-1.0299$ and $x_{0}=6.5041,\, x_{1}=0.9084$ . - -\begin{lstlisting} -In []: def func(x): - ...: return x + 2*cos(x) - -In []: def func2(x): - ...: out = [x[0]*cos(x[1]) - 4] - ...: out.append(x[1]*x[0] - x[1] - 5) - ...: return out - -In []: from scipy.optimize import fsolve -In []: x0 = fsolve(func, 0.3) -In []: print x0 --1.02986652932 - -In []: x02 = fsolve(func2, [1, 1]) -In []: print x02 -[ 6.50409711 0.90841421] -\end{lstlisting} - -\subsection{Integration} -% To be done in the lab. -\subsubsection{Quadrature} - -The function \texttt{quad} is provided to integrate a function of one -variable between two points. The points can be $\pm\infty$ ($\pm$ -\texttt{inf}) to indicate infinite limits. For example, suppose you -wish to integrate the expression $e^{\sin(x)}$ in the interval -$[0,2\pi]$, i.e. $\int_0^{2\pi}e^{\sin(x)}dx$, it could be computed -using - -\begin{lstlisting} -In []: def func(x): - ...: return exp(sin(x)) - -In []: from scipy.integrate import quad -In []: result = quad(func, 0, 2*pi) -In []: print result -(7.9549265210128457, 4.0521874164521979e-10) -\end{lstlisting} - -\subsubsection{ODE} - -We wish to solve an (a system of) Ordinary Differential Equation. For -this purpose, we shall use \lstinline{odeint}. As an illustration, let -us solve the ODE - -\begin{align} - \frac{dy}{dt} = ky(L-y)\\ - L = 25000,\,k = 0.00003,\,y(0) = 250 \nonumber -\end{align} - -We solve it in scipy as shown below. - -\begin{lstlisting} -In []: from scipy.integrate import odeint -In []: def f(y, t): - ...: k, L = 0.00003, 25000 - ...: return k*y*(L-y) - ...: -In []: t = linspace(0, 12, 60) -In []: y0 = 250 -In []: y = odeint(f, y0, t) -\end{lstlisting} - -Note: To solve a system of ODEs, we need to change the function to -return the right hand side of all the equations and the system and the -pass the required number of initial conditions to the -\lstinline{odeint} function. - -\appendix - -\section{Plotting using Pylab}\label{mpl} - -The following piece of code, produces the plot in Figure \ref{fig:sin} -using \texttt{pylab}\cite{pylab} in \texttt{ipython}\footnote{start - \texttt{ipython} with \texttt{-pylab} flag}\cite{ipy} - -\begin{lstlisting} -In []: x = linspace(0, 2*pi, 50) -In []: plot(x, sin(x)) -In []: title('Sine Curve between 0 and $\pi$') -In []: legend(['sin(x)']) -\end{lstlisting} - -\begin{figure}[h!] - \begin{center} - \includegraphics[scale=0.4]{sine} - \end{center} - \caption{Sine curve} - \label{fig:sin} -\end{figure} - - - -\begin{thebibliography}{9} - \bibitem{scipy} - Eric Jones and Travis Oliphant and Pearu Peterson and others, - \emph{SciPy: Open source scientific tools for Python}, 2001 -- , - \url{http://www.scipy.org/} - - \bibitem{pylab} - John D. Hunter, ``Matplotlib: A 2D Graphics Environment,'' - \emph{Computing in Science and Engineering}, vol. 9, no. 3, - pp. 90-95, May/June 2007, doi:10.1109/MCSE.2007.55 - - \bibitem{ipy} - Fernando Perez, Brian E. Granger, ``IPython: A System for - Interactive Scientific Computing,'' \emph{Computing in Science and - Engineering}, vol.~9, no.~3, pp.~21-29, May/June 2007, - doi:10.1109/MCSE.2007.53. - -\end{thebibliography} -\end{document} |