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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[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}