matplotlib Code

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\begin_body

\begin_layout Section
Newton's method
\end_layout

\begin_layout Standard
Consider the problem of solving for 
\begin_inset Formula $t$
\end_inset

 in
\begin_inset Formula \begin{equation}
\int_{o}^{t}f(s)ds=u\end{equation}

\end_inset

 where 
\begin_inset Formula $f(s)$
\end_inset

 is a monotonically increasing function of 
\begin_inset Formula $s$
\end_inset

 and 
\begin_inset Formula $u>0$
\end_inset

.
\end_layout

\begin_layout Standard
This problem can be simply solved if seen as a root finding question.
 Let
\begin_inset Formula \begin{equation}
g(t)=\int_{o}^{t}f(s)ds-u,\end{equation}

\end_inset

then we just need to find the root for 
\begin_inset Formula $g(t),$
\end_inset

 which is guaranteed to be unique given the conditions above.
 
\end_layout

\begin_layout Standard
The SciPy library includes an optimization package that contains a Newton-Raphso
n solver called 
\family typewriter
scipy.optimize.newton.

\family default
 This solver can optionally take a known derivative for the function whose
 roots are being sought, and in this case the derivative is simply 
\begin_inset Formula \begin{equation}
\frac{dg(t)}{dt}=f(t).\end{equation}

\end_inset


\end_layout

\begin_layout Standard
For this exercise, implement the solution for the test function
\begin_inset Formula \[
f(t)=t\sin^{2}(t),\]

\end_inset

 using 
\begin_inset Formula \[
u=\frac{1}{4}.\]

\end_inset


\end_layout

\begin_layout Standard
The listing\InsetSpace ~

\begin_inset LatexCommand \ref{code:quad_newton_skel}

\end_inset

 contains a skeleton that includes for comparison the correct numerical
 value.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Standard


\backslash
lstinputlisting[label=code:quad_newton_skel,caption={IGNORED}]{skel/quad_newton_
skel.py}
\end_layout

\end_inset


\end_layout

\begin_layout Standard

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