229 \begin{frame}[fragile]

   229 %% \begin{frame}[fragile]

   230 \frametitle{Bisection Method}

   230 %% \frametitle{Bisection Method}

   231 \begin{enumerate}

   231 %% \begin{enumerate}

   232 \item Start with the given interval $(-\pi/2, \pi/2)$ ($(a, b)$)

   232 %% \item Start with the given interval $(-\pi/2, \pi/2)$ ($(a, b)$)

   233 \item $f(a)\cdot f(b) < 0$

   233 %% \item $f(a)\cdot f(b) < 0$

   234 \item Bisect the interval. $c = \frac{a+b}{2}$

   234 %% \item Bisect the interval. $c = \frac{a+b}{2}$

   235 \item Change the interval to $(a, c)$ if $f(a)\cdot f(c) < 0$

   235 %% \item Change the interval to $(a, c)$ if $f(a)\cdot f(c) < 0$

   236 \item Else, change the interval to $(c, b)$

   236 %% \item Else, change the interval to $(c, b)$

   237 \item Go back to 3 until $(b-a) \le$ tolerance

   237 %% \item Go back to 3 until $(b-a) \le$ tolerance

   238 \end{enumerate}

   238 %% \end{enumerate}

   239 \alert{\typ{tolerance = 1e-5}}

   239 %% \alert{\typ{tolerance = 1e-5}}

   240 \end{frame}

   240 %% \end{frame}

   258 \begin{frame}[fragile]

   258 %% \begin{frame}[fragile]

   259 \frametitle{Newton-Raphson Method}

   259 %% \frametitle{Newton-Raphson Method}

   260 \begin{enumerate}

   260 %% \begin{enumerate}

   261 \item Start with an initial guess of x for the root

   261 %% \item Start with an initial guess of x for the root

   262 \item $\Delta x = -f(x)/f^{'}(x)$

   262 %% \item $\Delta x = -f(x)/f^{'}(x)$

   263 \item $ x \leftarrow x + \Delta x$

   263 %% \item $ x \leftarrow x + \Delta x$

   264 \item Go back to 2 until $|\Delta x| \le$ tolerance

   264 %% \item Go back to 2 until $|\Delta x| \le$ tolerance

   265 \end{enumerate}

   265 %% \end{enumerate}

   266 \end{frame}

   266 %% \end{frame}

   293 \begin{frame}[fragile]

   293 %% \begin{frame}[fragile]

   294 \frametitle{Initial Estimates}

   294 %% \frametitle{Initial Estimates}

   295 \begin{itemize}

   295 %% \begin{itemize}

   296 \item Given an interval

   296 %% \item Given an interval

   297 \item How to find \alert{all} the roots?

   297 %% \item How to find \alert{all} the roots?

   298 \end{itemize}

   298 %% \end{itemize}

   299 \begin{enumerate}

   299 %% \begin{enumerate}

   300 \item Check for change of signs of $f(x)$ in the given interval

   300 %% \item Check for change of signs of $f(x)$ in the given interval

   301 \item A root lies in the interval where the sign change occurs

   301 %% \item A root lies in the interval where the sign change occurs

   302 \end{enumerate}

   302 %% \end{enumerate}

   303 \end{frame}

   303 %% \end{frame}

   304 

   304 

   305 \begin{frame}[fragile]

   305 %% \begin{frame}[fragile]

   306 \frametitle{Initial Estimates \ldots}

   306 %% \frametitle{Initial Estimates \ldots}

   307 \begin{lstlisting}

   307 %% \begin{lstlisting}

   308 In []: def our_f(x):

   308 %% In []: def our_f(x):

   309  ....:     return cos(x) - x*x

   309 %%  ....:     return cos(x) - x*x

   310  ....: 

   310 %%  ....: 

   311 In []: x = linspace(-pi/2, pi/2, 11)

   311 %% In []: x = linspace(-pi/2, pi/2, 11)

   312 In []: y = our_f(x)

   312 %% In []: y = our_f(x)

   313 \end{lstlisting}

   313 %% \end{lstlisting}

   314 Get the intervals of x, where sign changes occur

   314 %% Get the intervals of x, where sign changes occur

   315 \end{frame}

   315 %% \end{frame}

   316 

   316 

   317 \begin{frame}[fragile]

   317 %% \begin{frame}[fragile]

   318 \frametitle{Initial Estimates \ldots}

   318 %% \frametitle{Initial Estimates \ldots}

   319 \begin{lstlisting}

   319 %% \begin{lstlisting}

   320 In []: pos = y[:-1]*y[1:] < 0

   320 %% In []: pos = y[:-1]*y[1:] < 0

   321 In []: rpos = zeros(x.shape, dtype=bool)

   321 %% In []: rpos = zeros(x.shape, dtype=bool)

   322 In []: rpos[:-1] = pos

   322 %% In []: rpos[:-1] = pos

   323 In []: rpos[1:] += pos

   323 %% In []: rpos[1:] += pos

   324 In []: rts = x[rpos]

   324 %% In []: rts = x[rpos]

   325 \end{lstlisting}

   325 %% \end{lstlisting}

   326 Now use Newton-Raphson?

   326 %% Now use Newton-Raphson?

   327 \end{frame}

   327 %% \end{frame}

   358 \begin{frame}[fragile]

   358 %% \begin{frame}[fragile]

   359 \frametitle{Scipy Methods \dots}

   359 %% \frametitle{Scipy Methods \dots}

   360 \begin{small}

   360 %% \begin{small}

   361 \begin{lstlisting}

   361 %% \begin{lstlisting}

   362 In []: from scipy.optimize import fixed_point

   362 %% In []: from scipy.optimize import fixed_point

   363 

   363 

   364 In []: from scipy.optimize import bisect

   364 %% In []: from scipy.optimize import bisect

   365 

   365 

   366 In []: from scipy.optimize import newton

   366 %% In []: from scipy.optimize import newton

   367 \end{lstlisting}

   367 %% \end{lstlisting}

   368 \end{small}

   368 %% \end{small}

   369 \end{frame}

   369 %% \end{frame}