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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[] {SciPy 2010, Introductory tutorials\\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)

  \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{small}

\begin{lstlisting}

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

In []: Ax

Out[]: array([  1.00000000e+00,  -2.00000000e+00,  -1.11022302e-16])

\end{lstlisting}

\end{small}

\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{10}

\end{frame}

\begin{frame}[fragile]

\frametitle{Problem}

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{5}

\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: \typ{roots}}

\begin{itemize}

\item Calculates the roots of polynomials

\item To calculate the roots of $x^2-5x+6$ 

\end{itemize}

\begin{lstlisting}

  In []: coeffs = [1, -5, 6]

  In []: roots(coeffs)

  Out[]: array([3., 2.])

\end{lstlisting}

\vspace*{-.2in}

\begin{center}

\includegraphics[height=1.6in, interpolate=true]{data/roots}    

\end{center}

\end{frame}

\begin{frame}[fragile]

\frametitle{SciPy: \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}

\end{frame}

\begin{frame}[fragile]

\frametitle{\typ{fsolve}}

Find the root of $sin(z)+cos^2(z)$ nearest to $0$

\vspace{-0.1in}

\begin{center}

\includegraphics[height=2.8in, interpolate=true]{data/fsolve}    

\end{center}

\end{frame}

\begin{frame}[fragile]

\frametitle{\typ{fsolve}}

Root of $sin(z)+cos^2(z)$ nearest to $0$

\begin{lstlisting}

In []: fsolve(sin(z)+cos(z)*cos(z), 0)

NameError: name 'z' is not defined

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{\typ{fsolve}}

\begin{lstlisting}

In []: z = linspace(-pi, pi)

In []: fsolve(sin(z)+cos(z)*cos(z), 0)

\end{lstlisting}

\begin{small}

\alert{\typ{TypeError:}}

\typ{'numpy.ndarray' object is not callable}

\end{small}

\end{frame}

\begin{frame}[fragile]

\frametitle{Functions - Definition}

We have been using them all along. Now let's see how to define them.

\begin{lstlisting}

In []: def g(z):

 ....:     return sin(z)+cos(z)*cos(z)

\end{lstlisting}

\begin{itemize}

\item \typ{def} -- keyword

\item name: \typ{g}

\item arguments: \typ{z}

\item \typ{return} -- keyword

\end{itemize}

\end{frame}

\begin{frame}[fragile]

\frametitle{Functions - Calling them}

\begin{lstlisting}

In []: g()

---------------------------------------

\end{lstlisting}

\alert{\typ{TypeError:}}\typ{g() takes exactly 1 argument}

\typ{(0 given)}

\begin{lstlisting}

In []: g(0)

Out[]: 1.0

In []: g(1)

Out[]: 1.1333975665343254

\end{lstlisting}

More on Functions later \ldots

\end{frame}

\begin{frame}[fragile]

\frametitle{\typ{fsolve} \ldots}

Find the root of $sin(z)+cos^2(z)$ nearest to $0$

\begin{lstlisting}

In []: fsolve(g, 0)

Out[]: -0.66623943249251527

\end{lstlisting}

\begin{center}

\includegraphics[height=2in, interpolate=true]{data/fsolve}    

\end{center}

\end{frame}

\begin{frame}[fragile]

  \frametitle{Exercise Problem}

  Find the root of the equation $x^2 - sin(x) + cos^2(x) = tan(x)$ nearest to $0$

\end{frame}

\begin{frame}[fragile]

  \frametitle{Solution}

  \begin{small}

  \begin{lstlisting}

def g(x):

    return x**2 - sin(x) + cos(x)*cos(x) - tan(x)

fsolve(g, 0)

  \end{lstlisting}

  \end{small}

  \begin{center}

\includegraphics[height=2in, interpolate=true]{data/fsolve_tanx}

  \end{center}

\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{Solving ODEs using SciPy}

\begin{itemize}

\item Consider the spread of an epidemic in a population

\item $\frac{dy}{dt} = ky(L-y)$ gives the spread of the disease

\item $L$ is the total population.

\item Use $L = 2.5E5, k = 3E-5, y(0) = 250$

\item Define a function as below

\end{itemize}

\begin{lstlisting}

In []: from scipy.integrate import odeint

In []: def epid(y, t):

  ....     k = 3.0e-5

  ....     L = 2.5e5

  ....     return k*y*(L-y)

  ....

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{Solving ODEs using SciPy \ldots}

\begin{lstlisting}

In []: t = linspace(0, 12, 61)

In []: y = odeint(epid, 250, t)

In []: plot(t, y)

\end{lstlisting}

%Insert Plot

\end{frame}

\begin{frame}[fragile]

\frametitle{Result}

\begin{center}

\includegraphics[height=2in, interpolate=true]{data/image}  

\end{center}

\end{frame}

\begin{frame}[fragile]

\frametitle{ODEs - Simple Pendulum}

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{ODEs - Simple Pendulum \ldots}

\begin{itemize}

\item Use \typ{odeint} to do the integration

\end{itemize}

\begin{lstlisting}

In []: def pend_int(initial, t):

  ....     theta = initial[0]

  ....     omega = initial[1]

  ....     g = 9.81

  ....     L = 0.2

  ....     F=[omega, -(g/L)*sin(theta)]

  ....     return F

  ....

\end{lstlisting}

\end{frame}

\begin{frame}[fragile]

\frametitle{ODEs - Simple Pendulum \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, 20, 101)

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

\end{lstlisting} 

\end{frame}

\begin{frame}[fragile]

\frametitle{ODEs - Simple Pendulum \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}[fragile]

\frametitle{Result}

\begin{center}

\includegraphics[height=2in, interpolate=true]{data/ode}  

\end{center}

\end{frame}

\section{FFTs}

\begin{frame}[fragile]

\frametitle{The FFT}

\begin{itemize}

    \item We have a simple signal $y(t)$

    \item Find the FFT and plot it

\end{itemize}

\begin{lstlisting}

In []: t = linspace(0, 2*pi, 500)

In []: y = sin(4*pi*t)

In []: f = fft(y)

In []: freq = fftfreq(500, t[1] - t[0])

In []: plot(freq[:250], abs(f)[:250])

In []: grid()

\end{lstlisting} 

\end{frame}

\begin{frame}[fragile]

\frametitle{FFTs cont\dots}

\begin{lstlisting}

In []: y1 = ifft(f) # inverse FFT

In []: allclose(y, y1)

Out[]: True

\end{lstlisting} 

\end{frame}

\begin{frame}[fragile]

\frametitle{FFTs cont\dots}

Let us add some noise to the signal

\begin{lstlisting}

In []: yr = y + random(size=500)*0.2

In []: yn = y + normal(size=500)*0.2

In []: plot(t, yr)

In []: figure()

In []: plot(freq[:250],

  ...:      abs(fft(yn))[:250])

\end{lstlisting}

\begin{itemize}

    \item \typ{random}: produces uniform deviates in $[0, 1)$

    \item \typ{normal}: draws random samples from a Gaussian

        distribution

    \item Useful to create a random matrix of any shape