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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[Solving Equations \& ODEs]{Python for Science and Engg:\\Solving Equations \& ODEs}

\author[FOSSEE] {FOSSEE}

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

\date[] {31, October 2009\\Day 1, Session 6}

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\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)

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

  \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{lstlisting}

In []: Ax

Out[]: 

array([[  1.00000000e+00],

       [ -2.00000000e+00],

       [  2.22044605e-16]])

\end{lstlisting}

\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{15}

\end{frame}

\subsection{Exercises}

\begin{frame}[fragile]

\frametitle{Problem 1}

Given the matrix:\\

\begin{center}

$\begin{bmatrix}

-2 & 2 & 3\\

 2 & 1 & 6\\

-1 &-2 & 0\\

\end{bmatrix}$

\end{center}

Find:

\begin{itemize}

  \item[i] Transpose

  \item[ii]Inverse

  \item[iii]Determinant

  \item[iv] Eigenvalues and Eigen vectors

  \item[v] Singular Value decomposition

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Problem 2}

Given 

\begin{center}

A = 

$\begin{bmatrix}

-3 & 1 & 5 \\

1 & 0 & -2 \\

5 & -2 & 4 \\

\end{bmatrix}$

, B = 

$\begin{bmatrix}

0 & 9 & -12 \\

-9 & 0 & 20 \\

12 & -20 & 0 \\

\end{bmatrix}$

\end{center}

Find:

\begin{itemize}

  \item[i] Sum of A and B

  \item[ii]Elementwise Product of A and B

  \item[iii] Matrix product of A and B

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Solution}

Sum: 

$\begin{bmatrix}

-3 & 10 & 7 \\

-8 & 0 & 18 \\

17 & -22 & 4 \\

\end{bmatrix}$

,\\ Elementwise Product:

$\begin{bmatrix}

0 & 9 & -60 \\

-9 & 0 & -40 \\

60 & 40 & 0 \\

\end{bmatrix}$

,\\ Matrix product:

$\begin{bmatrix}

51 & -127 & 56 \\

-24 & 49 & -12 \\

66 & -35 & -100 \\

\end{bmatrix}$

\end{frame}

\begin{frame}[fragile]

\frametitle{Problem 3}

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{10}

\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 Methods - \typ{roots}}

\begin{itemize}

\item Calculates the roots of polynomials

\end{itemize}

\begin{lstlisting}

  In []: coeffs = [1, 6, 13]

  In []: roots(coeffs)

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Scipy Methods - \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}

\begin{lstlisting}

  In []: fsolve(our_f, -pi/2)

\end{lstlisting}

\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{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(10^o)\quad & \&\quad  \omega = 0\ (Initial\ values)\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(initial, t):

  ....     theta, omega = initial

  ....     g, L = -9.81, 0.2

  ....     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{initial} has the initial values

\end{itemize}

\begin{lstlisting}

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

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

\end{lstlisting} 

\end{frame}

\begin{frame}[fragile]

\frametitle{Solving ODEs using SciPy \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}

  \frametitle{Things we have learned}

  \begin{itemize}

  \item Solving ODEs

  \item Finding Roots

  \end{itemize}

\end{frame}

\end{document}