199 \frametitle{Initial Estimates}

   199 \frametitle{Roots of $f(x)=0$}

   200 \begin{itemize}

   200 \begin{itemize}

   201 \item Find roots of $cosx-x^2$ in $(-\pi/2, \pi/2)$

   201 \item Given function $cosx-x^2$ 

   202 \item How to get a rough initial estimate?

   202 \item First we find \alert{a} root in $(-\pi/2, \pi/2)$

   203 \end{itemize}

   203 \item Then we find \alert{all} roots in $(-\pi/2, \pi/2)$

   204 \begin{enumerate}

   204 \end{itemize}

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

   205 \end{frame}

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

   206 

   207 \end{enumerate}

   207 %% \begin{frame}[fragile]

   208 \end{frame}

   208 %% \frametitle{Fixed Point Method}

   209 

   209 %% \begin{enumerate}

   210 \begin{frame}[fragile]

   210 %% \item Convert $f(x)=0$ to the form $x=g(x)$

   211 \frametitle{Initial Estimates \ldots}

   211 %% \item Start with an initial value of $x$ and iterate successively

   212 \begin{lstlisting}

   212 %% \item $x_{n+1}=g(x_n)$ and $x_0$ is our initial guess

   213   In []: def our_f(x):

   213 %% \item Iterate until $x_{n+1}-x_n \le tolerance$

   214    ....:     return cos(x)-x**2

   214 %% \end{enumerate}

   215    ....: 

   215 %% \end{frame}

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

   217 \end{lstlisting}

   218 \begin{itemize}

   219 \item Get the intervals of x, where sign changes occur

   220 \end{itemize}

   221 \end{frame}

   222 

   223 %% \begin{frame}[fragile]

   224 %% \frametitle{Initial Estimates \ldots}

   225 %% \begin{lstlisting}

   226 %% In []: pos = y[:-1]*y[1:] < 0

   227 %% In []: rpos = zeros(x.shape, dtype=bool)

   228 %% In []: rpos[:-1] = pos

   229 %% In []: rpos[1:] += pos

   230 %% In []: rts = x[rpos]

   231 %% \end{lstlisting}

   232 %% \end{frame}

   233 

   234 \begin{frame}[fragile]

   235 \frametitle{Fixed Point Method}

   236 \begin{enumerate}

   237 \item Convert $f(x)=0$ to the form $x=g(x)$

   238 \item Start with an initial value of $x$ and iterate successively

   239 \item $x_{n+1}=g(x_n)$ and $x_0$ is our initial guess

   240 \item Iterate until $x_{n+1}-x_n \le tolerance$

   241 \end{enumerate}

   242 \end{frame}

   262 \item Start with an interval $(a, b)$ within which a root exists

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

   312 \begin{frame}[fragile]

   286 %% \begin{frame}[fragile]

   313 \frametitle{Newton-Raphson \ldots}

   287 %% \frametitle{Newton-Raphson \ldots}

   314 \begin{itemize}

   288 %% \begin{itemize}

   315 \item What if $f^{'}(x) = 0$?

   289 %% \item What if $f^{'}(x) = 0$?

   316 \item Can you write a better version of the Newton-Raphson?

   290 %% \item Can you write a better version of the Newton-Raphson?

   317 \item What if the differential is not easy to calculate?

   291 %% \item What if the differential is not easy to calculate?

   318 \item Look at Secant Method

   292 %% \item Look at Secant Method

   319 \end{itemize}

   293 %% \end{itemize}

   320 \end{frame}

   294 %% \end{frame}

   295 

   296 \begin{frame}[fragile]

   297 \frametitle{Initial Estimates}

   298 \begin{itemize}

   299 \item Given an interval

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

   301 \end{itemize}

   302 \begin{enumerate}

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

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

   305 \end{enumerate}

   306 \end{frame}

   307 

   308 \begin{frame}[fragile]

   309 \frametitle{Initial Estimates \ldots}

   310 \begin{lstlisting}

   311   In []: def our_f(x):

   312    ....:     return cos(x)-x**2

   313    ....: 

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

   315 \end{lstlisting}

   316 \begin{itemize}

   317 \item Get the intervals of x, where sign changes occur

   318 \end{itemize}

   319 \end{frame}

   320 

   321 \begin{frame}[fragile]

   322 \frametitle{Initial Estimates \ldots}

   323 \begin{lstlisting}

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

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

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

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

   328 In []: rts = x[rpos]

   329 \end{lstlisting}

   330 Now use Newton-Raphson?

   331 \end{frame}

   332