Workshop materials: comparison day2/session2.tex

equal deleted inserted replaced

-:73462010a2fe
+:ab9192cfe870
 \begin{document}
 \begin{frame}
 \maketitle
 \end{frame}
 \begin{frame}[fragile]
 \frametitle{Broadcasting}
-\begin{itemize}
+Try it!
-\item Used so that functions can take inputs that are not of the same shape.
+\begin{lstlisting}
-\item 2 rules -
+>>> a = np.arange(4)
-\begin{enumerate}
+>>> b = np.arange(5)
-\item 1 (repeatedly) pre-pended to shapes of smaller arrays
+>>> a+b
-\item Size 1 in a dimension -> Largest size in that dimension
+>>> a+3
-\end{enumerate}
+>>> c=np.array([3])
-\end{itemize}
+>>> a+c
+>>> b+c
+\end{lstlisting}
+\begin{itemize}
+\item Enter Broadcasting!
+\end{itemize}
+\end{frame}
+\begin{frame}[fragile]
+\frametitle{Broadcasting}
 \begin{columns}
 \column{0.65\textwidth}
 \hspace*{-1.5in}
 \begin{lstlisting}
->>> x = np.arange(4)
+>>> a = np.arange(4)
->>> x+3
+>>> a+3
 array([3, 4, 5, 6])
 \end{lstlisting}
 \column{0.35\textwidth}
 \includegraphics[height=0.7in, interpolate=true]{data/broadcast_scalar}
 \end{columns}
+\begin{itemize}
+\item Allows functions to take inputs not of the same shape
+\item 2 rules -
+\begin{enumerate}
+\item 1 is (repeatedly) prepended to shapes of smaller arrays
+\item Size 1 in a dimension changed to Largest size in that dimension
+\end{enumerate}
+\end{itemize}
 \end{frame}
 \begin{frame}[fragile]
 \frametitle{Broadcasting in 3D}
 \begin{lstlisting}
->>> x = np.zeros((3, 5))
+>>> x = np.ones((3, 5))
->>> y = np.zeros(8)
+>>> y = np.ones(8)
 >>> (x[..., None] + y).shape
 (3, 5, 8)
 \end{lstlisting}
 \begin{figure}
 \begin{center}
 \end{figure}
 \end{frame}
 \begin{frame}[fragile]
 \frametitle{Copies \& Views}
-\begin{lstlisting}
+Try it!
->>> a = array([[1,2,3], [4,5,6],
+\begin{lstlisting}
-[7,8,9]])
+>>> a = np.array([[1,2,3],[4,5,6]])
->>> a[0,1:3]
+>>> b = a
-array([2, 3])
+>>> b is a
->>> a[0::2,0::2]
+>>> b[0,0]=0; print a
-array([[1, 3],
+>>> c = a.view()
-[7, 9]])
+>>> c is a
+>>> c.base is a
+>>> c.flags.owndata
+>>> d = a.copy()
+>>> d.base is a
+>>> d.flags.owndata
+\end{lstlisting}
+\end{frame}
+\begin{frame}[fragile]
+\frametitle{Copies \& Views}
+Try it!
+\begin{lstlisting}
+>>> a = np.arange(1,9)
+>>> a.shape=3,3
+>>> b = a[0,1:3]
+>>> c = a[0::2,0::2]
+>>> a.flags.owndata
+>>> b.flags.owndata
+>>> b.base
+>>> c.base is a
 \end{lstlisting}
 \begin{itemize}
 \item Slicing and Striding just reference the same memory
 \item They produce views of the data, not copies
 \end{itemize}
 \end{frame}
 \begin{frame}[fragile]
 \frametitle{Copies contd \ldots}
 \begin{lstlisting}
->>> a[np.array([0,1,2])]
+>>> b = a[np.array([0,1,2])]
 array([[1, 2, 3],
 [4, 5, 6],
 [7, 8, 9]])
-\end{lstlisting}
+>>> b.flags.owndata
-\begin{itemize}
+>>> abool=np.greater(a,2)
-\item Index arrays or Boolean arrays produce copies
+>>> c = a[abool]
+>>> c.flags.owndata
+\end{lstlisting}
+\begin{itemize}
+\item Indexing arrays or Boolean arrays produce copies
 \end{itemize}
 \inctime{15}
-\end{frame}
-\begin{frame}
-\frametitle{More Numpy Functions \& Methods}
-More functions
-\begin{itemize}
-\item \typ{take}
-\item \typ{choose}
-\item \typ{where}
-\item \typ{compress}
-\item \typ{concatenate}
-\end{itemize}
-Ufunc methods
-\begin{itemize}
-\item \typ{reduce}
-\item \typ{accumulate}
-\item \typ{outer}
-\item \typ{reduceat}
-\end{itemize}
-\inctime{5}
 \end{frame}
 \begin{frame}
 {Intro to SciPy}
 \begin{itemize}
 \item Uses LAPACK, QUADPACK, ODEPACK, FFTPACK etc. from netlib
 \end{itemize}
 \end{frame}
 \begin{frame}[fragile]
+\frametitle{SciPy - Functions \& Submodules}
+\begin{itemize}
+\item All \typ{numpy} functions are in \typ{scipy} namespace
+\item Domain specific functions organized into subpackages
+\item Subpackages need to be imported separately
+\end{itemize}
+\begin{lstlisting}
+>>> from scipy import linalg
+\end{lstlisting}
+\end{frame}
+\begin{frame}[fragile]
 \frametitle{Linear Algebra}
-\typ{>>> from scipy import linalg}
+Try it!
-\begin{itemize}
+\begin{lstlisting}
-\item \typ{linalg.det, linalg.norm}
+>>> import scipy as sp
-\item \typ{linalg.eig, linalg.lu}
+>>> from scipy import linalg
-\item \typ{linalg.expm, linalg.logm}
+>>> A=sp.mat(np.arange(1,10))
-\item \typ{linalg.sinm, linalg.sinhm}
+>>> A.shape=3,3
-\end{itemize}
+>>> linalg.inv(A)
-\end{frame}
+>>> linalg.det(A)
+>>> linalg.norm(A)
-\begin{frame}[fragile]
+>>> linalg.expm(A) #logm
-\frametitle{Linear Algebra \ldots}
+>>> linalg.sinm(A) #cosm, tanm, ...
+\end{lstlisting}
+\end{frame}
+\begin{frame}[fragile]
+\frametitle{Linear Algebra ...}
+Try it!
+\begin{lstlisting}
+>>> A = sp.mat(np.arange(1,10))
+>>> A.shape=3,3
+>>> linalg.lu(A)
+>>> linalg.eig(A)
+>>> linalg.eigvals(A)
+\end{lstlisting}
+\end{frame}
+\begin{frame}[fragile]
+\frametitle{Solving Linear Equations}
 \begin{align*}
 3x + 2y - z  & = 1 \\
 2x - 2y + 4z  & = -2 \\
 -x + \frac{1}{2}y -z & = 0
 \end{align*}
-\begin{lstlisting}
+To Solve this,
+\begin{lstlisting}
+>>> A = sp.mat([[3,2,-1],[2,-2,4]
+,[-1,1/2,-1]])
+>>> B = sp.mat([[1],[-2],[0]])
 >>> linalg.solve(A,B)
 \end{lstlisting}
 \inctime{15}
 \end{frame}
 \begin{frame}[fragile]
+\frametitle{Integrate}
 \begin{itemize}
 \item Integrating Functions given function object
 \item Integrating Functions given fixed samples
 \item Numerical integrators of ODE systems
 \end{itemize}
-\frametitle{Integrate}
+Calculate $\int^1_0(sin(x) + x^2)dx$
-Calculate $\int^1_0sin(x) + x^2$
 \begin{lstlisting}
 >>> def f(x):
 return np.sin(x)+x**2
 >>> integrate.quad(f, 0, 1)
 \end{lstlisting}
 \begin{frame}[fragile]
 \frametitle{Integrate \ldots}
 Numerically solve ODEs\\
 \begin{align*}
-\frac{dx}{dt}&=-e^{(-t)}x^2(t)\\
+\frac{dx}{dt}&=-e^{-t}x^2\\
 x(0)&=2
 \end{align*}
 \begin{lstlisting}
 def dx_dt(x,t):
 return -np.exp(-t)*x**2
 \inctime{10}
 \end{frame}
 \begin{frame}[fragile]
 \frametitle{Interpolation}
-\begin{itemize}
+Try it!
-\item \typ{interpolate.interp1d, ...}
+\begin{lstlisting}
-\item \typ{interpolate.splrep, splev}
+>>> from scipy import interpolate
-\end{itemize}
+>>> interpolate.interp1d?
+>>> x = np.arange(0,2*np.pi,np.pi/4)
+>>> y = np.sin(x)
+>>> fl = interpolate.interp1d(x,y,kind='linear')
+>>> fc = interpolate.interp1d(x,y,kind='cubic')
+>>> fl(np.pi/3)
+>>> fc(np.pi/3)
+\end{lstlisting}
+\end{frame}
+\begin{frame}[fragile]
+\frametitle{Interpolation - Splines}
 Cubic Spline of $sin(x)$
 \begin{lstlisting}
-x = np.arange(0,2*np.pi,np.pi/8)
+x = np.arange(0,2*np.pi,np.pi/4)
 y = np.sin(x)
-t = interpolate.splrep(x,y,s=0)
+tck = interpolate.splrep(x,y)
 X = np.arange(0,2*np.pi,np.pi/50)
-Y = interpolate.splev(X,t,der=0)
+Y = interpolate.splev(X,tck,der=0)
 plt.plot(x,y,'o',x,y,X,Y)
 plt.show()
 \end{lstlisting}
 \inctime{10}
 \end{frame}
 \end{lstlisting}
 Zooming an array - uses spline interpolation
 \begin{lstlisting}
 b=ndimage.zoom(A,0.5)
 imshow(b)
+\end{lstlisting}
 \inctime{5}
-\end{lstlisting}
 \end{frame}
 \begin{frame}[fragile]
 \frametitle{Problems}
 The Van der Pol oscillator is a type of nonconservative oscillator with nonlinear damping. It evolves in time according to the second order differential equation:

changeset 58	ab9192cfe870
parent 46	63704b5650f1
child 66	523f788d2147