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%Tutorial slides on Python.

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% Author: FOSSEE 

% Copyright (c) 2009, FOSSEE, IIT Bombay

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% Title page

\title[Basic Python]{Matrices, Solution of equations and Integration\\}

\author[FOSSEE] {FOSSEE}

\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}

\date[] {31, October 2009\\Day 1, Session 4}

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\begin{document}

\begin{frame}

  \titlepage

\end{frame}

\begin{frame}

  \frametitle{Outline}

  \tableofcontents

%  \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 = matrix([[3,2,-1],[2,-2,4],[-1, 0.5, -1]])

    In []: b = matrix([[1], [-2], [0]])

    In []: x = linalg.solve(A, b)

    In []: Ax = dot(A, x)

    In []: allclose(Ax, b)

    Out[]: True

  \end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Solution:}

\begin{lstlisting}

In []: x

Out[]: 

array([[ 1.],

       [-2.],

       [-2.]])

In []: Ax

Out[]: 

matrix([[  1.00000000e+00],

        [ -2.00000000e+00],

        [  2.22044605e-16]])

\end{lstlisting}

\end{frame}

\section{Matrices}

\subsection{Initializing}

\begin{frame}[fragile]

\frametitle{Matrices: Initializing}

\begin{lstlisting}

  In []: a = matrix([[1,2,3],

                     [4,5,6],

                     [7,8,9]])

  In []: a

  Out[]: 

  matrix([[1, 2, 3],

         [4, 5, 6],

         [7, 8, 9]])

\end{lstlisting}

\end{frame}

\subsection{Basic Operations}

\begin{frame}[fragile]

\frametitle{Inverse of a Matrix}

\begin{small}

\begin{lstlisting}

In []: linalg.inv(A)

Out[]: 

matrix([[ 0.07734807,  0.01657459,  0.32044199],

        [ 0.09944751, -0.12154696, -0.01657459],

        [-0.02762431, -0.07734807,  0.17127072]])

\end{lstlisting}

\end{small}

\end{frame}

\begin{frame}[fragile]

\frametitle{Determinant}

\begin{lstlisting}

  In []: linalg.det(a)

  Out[]: -9.5171266700777579e-16

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Computing Norms}

\begin{lstlisting}

  In []: linalg.norm(a)

  Out[]: 16.881943016134134

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Eigen Values and Eigen Matrix}

\begin{small}

\begin{lstlisting}

  In []: linalg.eigvals(a)

  Out[]: array([1.61168440e+01, -1.11684397e+00, -1.22196337e-15])

  In []: linalg.eig(a)

  Out[]: 

  (array([ 1.61168440e+01, -1.11684397e+00, -1.22196337e-15]),

   matrix([[-0.23197069, -0.78583024,  0.40824829],

          [-0.52532209, -0.08675134, -0.81649658],

          [-0.8186735 ,  0.61232756,  0.40824829]]))

\end{lstlisting}

\end{small}

\end{frame}

\section{Integration}

\subsection{Quadrature}

\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}}

\begin{itemize}

\item Inputs - function to integrate, limits

\end{itemize}

\begin{lstlisting}

In []: x = 0

\end{lstlisting}

\alert{\typ{error:}}

\typ{First argument must be a callable function.}

\end{frame}

\begin{frame}[fragile]

\frametitle{Functions - Definition}

\begin{lstlisting}

In []: def f(x):

           return sin(x)+x**2

\end{lstlisting}

\begin{itemize}

\item \typ{def}

\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 []: f(0)

Out[]: 0.0

In []: f(1)

Out[]: 1.8414709848078965

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Functions - Default Arguments}

\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

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Functions - Keyword Arguments}

\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

\end{lstlisting}

\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}

\end{frame}

\begin{frame}[fragile]

\frametitle{Quadrature \ldots}

\begin{lstlisting}

\end{lstlisting}

Returns the integral and an estimate of the absolute error in the result.

\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*}

\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\quad & \&\quad  \omega = 0 \nonumber

\end{align}

\end{frame}

\begin{frame}[fragile]

\frametitle{Solving ODEs using SciPy}

\begin{itemize}

\item We use the \typ{odeint} function from scipy to do the integration

\item Define a function as below

\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

  ....

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Solving ODEs using SciPy \ldots}

\begin{itemize}

\item \typ{t} is the time variable \\ 

\item \typ{p} has the constants \\

\item \typ{initial} has the initial values

\end{itemize}

\begin{lstlisting}

In []: t = linspace(0, 10, 101)

In []: p=(-9.81, 0.2)

In []: initial = [10*2*pi/360, 0]

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Solving ODEs using SciPy \ldots}

\small{\typ{In []: from scipy.integrate import odeint}}

\begin{lstlisting}

In []: pend_sol = odeint(pend_int, 

                         initial,t, 

                         args=(p,))

\end{lstlisting}

\end{frame}

\end{document}