248 \section{Integration}

   249 

   250 \subsection{Quadrature}

   251 

   252 \begin{frame}[fragile]

   253 \frametitle{Quadrature}

   254 \begin{itemize}

   255 \item We wish to find area under a curve

   256 \item Area under $(sin(x) + x^2)$ in $(0,1)$

   257 \item scipy has functions to do that

   258 \end{itemize}

   259 \small{\typ{In []: from scipy.integrate import quad}}

   260 \begin{itemize}

   261 \item Inputs - function to integrate, limits

   262 \end{itemize}

   263 \begin{lstlisting}

   264 In []: x = 0

   265 In []: quad(sin(x)+x**2, 0, 1)

   266 \end{lstlisting}

   267 \alert{\typ{error:}}

   268 \typ{First argument must be a callable function.}

   269 \end{frame}

   270 

   271 \begin{frame}[fragile]

   272 \frametitle{Functions - Definition}

   273 \begin{lstlisting}

   274 In []: def f(x):

   275            return sin(x)+x**2

   276 In []: quad(f, 0, 1)

   277 \end{lstlisting}

   278 \begin{itemize}

   279 \item \typ{def}

   280 \item arguments

   281 \item \typ{return}

   282 \end{itemize}

   283 \end{frame}

   284 

   285 \begin{frame}[fragile]

   286 \frametitle{Functions - Calling them}

   287 \begin{lstlisting}

   288 In [15]: f()

   289 ---------------------------------------

   290 \end{lstlisting}

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

   292 \typ{(0 given)}

   293 \begin{lstlisting}

   294 In []: f(0)

   295 Out[]: 0.0

   296 In []: f(1)

   297 Out[]: 1.8414709848078965

   298 \end{lstlisting}

   299 \end{frame}

   300 

   301 

   302 \begin{frame}[fragile]

   303 \frametitle{Functions - Default Arguments}

   304 \begin{lstlisting}

   305 In []: def f(x=1):

   306            return sin(x)+x**2

   307 In []: f(10)

   308 Out[]: 99.455978889110625

   309 In []: f(1)

   310 Out[]: 1.8414709848078965

   311 In []: f()

   312 Out[]: 1.8414709848078965

   313 \end{lstlisting}

   314 \end{frame}

   315 

   316 \begin{frame}[fragile]

   317 \frametitle{Functions - Keyword Arguments}

   318 \begin{lstlisting}

   319 In []: def f(x=1, y=pi):

   320            return sin(y)+x**2

   321 In []: f()

   322 Out[]: 1.0000000000000002

   323 In []: f(2)

   324 Out[]: 4.0

   325 In []: f(y=2)

   326 Out[]: 1.9092974268256817

   327 In []: f(y=pi/2,x=0)

   328 Out[]: 1.0

   329 \end{lstlisting}

   330 \end{frame}

   331 

   332 \begin{frame}[fragile]

   333   \frametitle{More on functions}

   334   \begin{itemize}

   335   \item Scope of variables in the function is local

   336   \item Mutable items are \alert{passed by reference}

   337   \item First line after definition may be a documentation string

   338     (\alert{recommended!})

   339   \item Function definition and execution defines a name bound to the

   340     function

   341   \item You \emph{can} assign a variable to a function!

   342   \end{itemize}

   343 \end{frame}

   344 

   345 \begin{frame}[fragile]

   346 \frametitle{Quadrature \ldots}

   347 \begin{lstlisting}

   348 In []: quad(f, 0, 1)

   349 \end{lstlisting}

   350 Returns the integral and an estimate of the absolute error in the result.

   351 \begin{itemize}

   352 \item Use \typ{dblquad} for Double integrals

   353 \item Use \typ{tplquad} for Triple integrals

   354 \end{itemize}

   355 \end{frame}

   356 

   357 \subsection{ODEs}

   358 

   359 \begin{frame}[fragile]

   360 \frametitle{ODE Integration}

   361 We shall use the simple ODE of a simple pendulum. 

   362 \begin{equation*}

   363 \ddot{\theta} = -\frac{g}{L}sin(\theta)

   364 \end{equation*}

   365 \begin{itemize}

   366 \item This equation can be written as a system of two first order ODEs

   367 \end{itemize}

   368 \begin{align}

   369 \dot{\theta} &= \omega \\

   370 \dot{\omega} &= -\frac{g}{L}sin(\theta) \\

   371  \text{At}\ t &= 0 : \nonumber \\

   372  \theta = \theta_0\quad & \&\quad  \omega = 0 \nonumber

   373 \end{align}

   374 \end{frame}

   375 

   376 \begin{frame}[fragile]

   377 \frametitle{Solving ODEs using SciPy}

   378 \begin{itemize}

   379 \item We use the \typ{odeint} function from scipy to do the integration

   380 \item Define a function as below

   381 \end{itemize}

   382 \begin{lstlisting}

   383 In []: def pend_int(unknown, t, p):

   384   ....     theta, omega = unknown

   385   ....     g, L = p

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

   387   ....     return f

   388   ....

   389 \end{lstlisting}

   390 \end{frame}

   391 

   392 \begin{frame}[fragile]

   393 \frametitle{Solving ODEs using SciPy \ldots}

   394 \begin{itemize}

   395 \item \typ{t} is the time variable \\ 

   396 \item \typ{p} has the constants \\

   397 \item \typ{initial} has the initial values

   398 \end{itemize}

   399 \begin{lstlisting}

   400 In []: t = linspace(0, 10, 101)

   401 In []: p=(-9.81, 0.2)

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

   403 \end{lstlisting}

   404 \end{frame}

   405 

   406 \begin{frame}[fragile]

   407 \frametitle{Solving ODEs using SciPy \ldots}

   408 

   409 \small{\typ{In []: from scipy.integrate import odeint}}

   410 \begin{lstlisting}

   411 In []: pend_sol = odeint(pend_int, 

   412                          initial,t, 

   413                          args=(p,))

   414 \end{lstlisting}

   415 \end{frame}

   416 

   422   \item Functions

   423     \begin{itemize}

   424     \item Definition

   425     \item Calling

   426     \item Default Arguments

   427     \item Keyword Arguments

   428     \end{itemize}

   429     \item Integration

   430     \begin{itemize}

   431       \item Quadrature

   432       \item ODEs

   433     \end{itemize}