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diff --git a/lecture-notes/advanced-python/slides/scipy.tex b/lecture-notes/advanced-python/slides/scipy.tex deleted file mode 100644 index 1116c3d..0000000 --- a/lecture-notes/advanced-python/slides/scipy.tex +++ /dev/null @@ -1,273 +0,0 @@ -\section{Solving Equations} - -\begin{frame}[fragile] -\frametitle{Solution of linear 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 \texttt{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 \texttt{allclose()} to check. -\end{block} -\begin{lstlisting} -In []: allclose(Ax, b) -Out[]: True -\end{lstlisting} -\end{frame} - -\begin{frame}[fragile] -\frametitle{\texttt{roots} of polynomials} -\begin{itemize} -\item \texttt{roots} function can find 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]{images/roots} -\end{center} -\end{frame} - -\begin{frame}[fragile] -\frametitle{SciPy: \texttt{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{\texttt{fsolve} \ldots} -Find the root of $sin(z)+cos^2(z)$ nearest to $0$ -\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]{images/fsolve} -\end{center} -\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]{images/epid} -\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 \texttt{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 \texttt{t} is the time variable \\ -\item \texttt{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} -\texttt{} -%%\end{small} -\begin{lstlisting} -In []: from scipy.integrate import odeint -In []: pend_sol = odeint(pend_int, - initial,t) -\end{lstlisting} -\end{frame} - -\begin{frame}[fragile] -\frametitle{Result} -\begin{center} -\includegraphics[height=2in, interpolate=true]{images/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 \texttt{random}: produces uniform deviates in $[0, 1)$ -%% \item \texttt{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(yc))[:250]) -%% \end{lstlisting} -%% Only scratched the surface here \dots -%% \end{frame} |