118 

   122   \begin{itemize}

   123     \item Used so that functions can take inputs that are not of the same shape.

   124     \item 2 rules -

   125       \begin{enumerate}

   126       \item 1 (repeatedly) pre-pended to shapes of smaller arrays

   127       \item Size 1 in a dimension -> Largest size in that dimension

   128       \end{enumerate}

   129   \end{itemize}

   130   \begin{columns}

   131     \column{0.65\textwidth}

   132     \hspace*{-1.5in}

   133     \begin{lstlisting}

   134       >>> x = np.arange(4)

   135       >>> x+3

   136       array([3, 4, 5, 6])

   137     \end{lstlisting}

   138     \column{0.35\textwidth}

   139     \includegraphics[height=0.7in, interpolate=true]{data/broadcast_scalar}

   140   \end{columns}

   141 \end{frame}

   142 

   143 \begin{frame}[fragile]

   144   \frametitle{Broadcasting in 3D}

   145     \begin{lstlisting}

   146       >>> x = np.zeros((3, 5))

   147       >>> y = np.zeros(8)

   148       >>> (x[..., None] + y).shape

   149       (3, 5, 8)

   150     \end{lstlisting}

   151     \begin{figure}

   152       \begin{center}

   153       \includegraphics[height=1.5in, interpolate=true]{data/array_3x5x8}        

   154       \end{center}

   155     \end{figure}

   123 

   160   \begin{lstlisting}

   161     >>> a = array([[1,2,3], [4,5,6],     

   162                    [7,8,9]])

   163     >>> a[0,1:3]

   164     array([2, 3])

   165     >>> a[0::2,0::2]

   166     array([[1, 3],

   167           [7, 9]])

   168   \end{lstlisting}

   169   \begin{itemize}

   170   \item Slicing and Striding just reference the same memory

   171   \item They produce views of the data, not copies

   172   \end{itemize}

   173 \end{frame}

   174 

   175 \begin{frame}[fragile]

   176   \frametitle{Copies contd \ldots}

   177   \begin{lstlisting}

   178     >>> a[np.array([0,1,2])]

   179     array([[1, 2, 3],

   180            [4, 5, 6],

   181            [7, 8, 9]])

   182   \end{lstlisting}

   183   \begin{itemize}

   184   \item Index arrays or Boolean arrays produce copies

   185   \end{itemize}

   177 

   238   \typ{>>> from scipy import linalg}

   178 \end{frame}

   239   \begin{itemize}

   179 \begin{frame}[fragile]

   240     \item \typ{linalg.det, linalg.norm}

   180   \frametitle{ODEs}

   241     \item \typ{linalg.eig, linalg.lu}

   181 

   242     \item \typ{linalg.expm, linalg.logm}

   182 \end{frame}

   243     \item \typ{linalg.sinm, linalg.sinhm}

   244   \end{itemize}

   245 \end{frame}

   246 

   247 \begin{frame}[fragile]

   248   \frametitle{Linear Algebra \ldots}

   249   \begin{align*}

   250     3x + 2y - z  & = 1 \\

   251     2x - 2y + 4z  & = -2 \\

   252     -x + \frac{1}{2}y -z & = 0

   253   \end{align*}

   254   \begin{lstlisting}

   255     >>> linalg.solve(A,B)

   256   \end{lstlisting}

   257 \end{frame}

   258 

   259 \begin{frame}[fragile]

   260   \begin{itemize}

   261     \item Integrating Functions given function object

   262     \item Integrating Functions given fixed samples

   263     \item Numerical integrators of ODE systems

   264   \end{itemize}

   265   \frametitle{Integrate}

   266   Calculate $\int^1_0sin(x) + x^2$

   267   \begin{lstlisting}

   268     >>> def f(x):

   269             return np.sin(x)+x**2

   270     >>> integrate.quad(f, 0, 1)

   271   \end{lstlisting}

   272 \end{frame}

   273 

   274 \begin{frame}[fragile]

   275   \frametitle{Integrate \ldots}

   276   Numerically solve ODEs\\

   277   \begin{align*}

   278   \frac{dx}{dt}&=-e^{(-t)}x^2(t)\\ 

   279            x(0)&=2    

   280   \end{align*}

   281   \begin{lstlisting}

   282     def dx_dt(x,t):

   283         return -np.exp(-t)*x**2

   284 

   285     x=integrate.odeint(dx_dt, 2, t)

   286     plt.plot(x,t)

   287   \end{lstlisting}

   288 \end{frame}

   289 

   185 

   292   \begin{itemize}

   293     \item \typ{interpolate.interp1d, ...}

   294     \item \typ{interpolate.splrep, splev}

   295   \end{itemize}

   296   Cubic Spline of $sin(x)$

   297   \begin{lstlisting}

   298     x = np.arange(0,2*np.pi,np.pi/8)

   299     y = np.sin(x)

   300     t = interpolate.splrep(x,y,s=0)

   301     X = np.arange(0,2*np.pi,np.pi/50)

   302     Y = interpolate.splev(X,t,der=0)

   303 

   304     plt.plot(x,y,'o',x,y,X,Y)

   305     plt.show()

   306   \end{lstlisting}

   195 

   345   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:

   346   \begin{equation*}

   347   \frac{d^2x}{dt^2}+\mu(x^2-1)\frac{dx}{dt}+x= 0

   348   \end{equation*}