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%Tutorial slides on Python.
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% Author: FOSSEE
% Copyright (c) 2009-2016, FOSSEE, IIT Bombay
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% Title page
\title[Basic SciPy]{Introductory Scientific Computing with
Python}
\subtitle{Basic SciPy}

\author[Prabhu] {FOSSEE}

\institute[FOSSEE -- IITB] {Department of Aerospace Engineering\\IIT Bombay}
\date[] {
Mumbai, India
}
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%% the beginning of each subsection:
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% DOCUMENT STARTS
\begin{document}

\begin{frame}
  \maketitle
\end{frame}


\section{Solving linear systems}

\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*}
\end{frame}

\begin{frame}[fragile]
\frametitle{Solution}
Use \kwrd{solve()}
\begin{align*}
  x & = -5\\
  y & = 2\\
  z & = 3\\
  w & = 0\\
\end{align*}
\inctime{5}
\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}}
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}}
\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 - \alert{Function} and initial estimate
\item Returns the solution
\end{itemize}
\end{frame}

\begin{frame}[fragile]
\frametitle{\typ{fsolve} \ldots}
\begin{lstlisting}
In []: def g(z):
 ....:     return sin(z)+cos(z)*cos(z)
In []: fsolve(g, 0)
Out[]: -0.66623943249251527
\end{lstlisting}
\begin{center}
\includegraphics[height=2in, interpolate=true]{data/fsolve}
\end{center}
\inctime{10}
\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}
  \vspace*{-0.2in}
  \begin{center}
\includegraphics[height=2.5in, interpolate=true]{data/fsolve_tanx}
\end{center}
\vspace*{-0.5in}
  \inctime{5}
\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}
\frametitle{Solving ODEs using SciPy}
\begin{itemize}
\item Consider the spread of an epidemic in a population
  \vspace*{0.1in}
\item $\frac{dy}{dt} = ky(L-y)$ gives the spread of the disease
  \vspace*{0.1in}
\item $L$ is the total population.
\item Use $L = 2.5E5, k = 3E-5, y(0) = 250$
\end{itemize}
\end{frame}

\begin{frame}[fragile]
\frametitle{Solving ODEs using SciPy}
Define a function as below
\small
\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=3in, interpolate=true]{data/image}
\end{center}
\vspace*{-0.5in}
\inctime{5}
\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_rhs(state, t):
  ....     theta = state[0]
  ....     omega = state[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_rhs,
                         initial,t)
\end{lstlisting}
\end{frame}

\begin{frame}[fragile]
\frametitle{Result}
\begin{center}
\includegraphics[height=2in, interpolate=true]{data/ode}
\end{center}
  \inctime{10}
\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.fft(y)
In []: freq = fft.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 = fft.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.random(size=500)*0.2
In []: yn = y +
  ...:   random.normal(size=500)*0.2

In []: plot(t, yr)
In []: figure()
In []: plot(freq[:250],
  ...:      abs(fft.fft(yr))[: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
\end{itemize}
\end{frame}

\begin{frame}[fragile]
\frametitle{FFTs cont\dots}
Filter the noisy signal:
\begin{lstlisting}
In []: from scipy import signal
In []: yc = signal.wiener(yn, 5)
In []: clf()
In []: plot(t, yc)
In []: figure()
In []: plot(freq[:250],
  ...:      abs(fft.fft(yc))[:250])
\end{lstlisting}
Only scratched the surface here \dots

\inctime{10}
\end{frame}


\begin{frame}
  \frametitle{Things we have learned}
  \begin{itemize}
  \item Solving Linear Equations
  \item Defining Functions
  \item Finding Roots
  \item Solving ODEs
  \item FFTs and basic signal processing
  \end{itemize}
\end{frame}


\begin{frame}
    \frametitle{Further reading}
    \begin{itemize}
        \item \url{ipython.readthedocs.io}
        \item \url{matplotlib.org/contents.html}
        \item \url{docs.scipy.org/doc/numpy/user/quickstart.html}
        \item \url{docs.scipy.org/doc/scipy/reference/tutorial}
    \end{itemize}
\end{frame}

\end{document}