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\author{ {\fontspec{Trebuchet MS}Marcin Chrz\k{a}szcz, Danny van Dyk} (UZH)}
\institute{UZH}
\title[Introduction to \\Numerical Methods]{Introduction to \\Numerical Methods}
\date{26. September, 2016}


\begin{document}
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{
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\begin{frame}[c]%{\phantom{title page}} 
\begin{center}
\begin{center}
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		\begin{column}{0.9\textwidth}
			\flushright\fontspec{Trebuchet MS}\bfseries \Huge {Introduction to \\Numerical Methods}
		\end{column}
		\begin{column}{0.2\textwidth}
		  %\includegraphics[width=\textwidth]{SHiP-2}
		\end{column}
	\end{columns}
\end{center}
	\quad
	\vspace{3em}
\begin{columns}
\begin{column}{0.6\textwidth}
\flushright \vspace{-1.8em} {\fontspec{Trebuchet MS} \large Marcin Chrząszcz, Danny van Dyk\\\vspace{-0.1em}\small \href{mailto:mchrzasz@cern.ch}{mchrzasz@cern.ch}, \href{mailto:dany.van.dyk@gmail.com}{danny.van.dyk@gmail.com}}

\end{column}
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\includegraphics[height=1.3cm]{uzh-transp}
\end{column}
\end{columns}

\vspace{1em}
%		\footnotesize\textcolor{gray}{With N. Serra, B. Storaci\\Thanks to the theory support from M. Shaposhnikov, D. Gorbunov}\normalsize\\
\vspace{0.5em}
	\textcolor{normal text.fg!50!Comment}{Numerical Methods, \\ 26. September, 2016}
\end{center}
\end{frame}
}

\begin{frame}{When Interpolation fails}
So far, you have been confronted with examples in which interpolation
works nicely. Let us now discuss an example, which behaves pathologically.\\[\medskipamount]

\begin{columns}
\begin{column}[t]{.5\textwidth}
\begin{overlayarea}{\textwidth}{5cm}
Consider the classical example by Runge:
\begin{equation*}
    f(x) = \left[1 + 25 x^2\right]^{-1}
\end{equation*}

Let us plot
\begin{itemize}
    \item the true function $f(x)$
    \item<2-> {\color{blue}the interpolating polynomial to degree $4$}
    \item<3-> {\color{red}the interpolating polynomial to degree $6$}
\end{itemize}
\end{overlayarea}
\end{column}
\begin{column}[t]{.5\textwidth}
\begin{tikzpicture}
    [
        scale=0.75
    ]
    \begin{axis}[%
        samples=500,
        xmin=-1,xmax=+1,
        ymin=-1,ymax=+1
    ]
        \addplot+[black,mark=none]
            {1.0 / (1.0 + 25.0 * \x^2)};
        \only<2->{
        \addplot+[blue,domain=-1:+1,y domain=-1:+1,mark=none]
            {1 - (3225 * \x^2) / 754 + (1250 * \x^4) / 377};
        }
        \only<3->{
        \addplot+[red,domain=-1:+1,y domain=-1:+1,mark=none]
            {1 - (211600 * \x^2)/24089 + (2019375 * \x^4)/96356 - (1265625 * \x^6)/96356};
        }
    \end{axis}
\end{tikzpicture}
\end{column}
\end{columns}
\end{frame}

\begin{frame}{Interpolation in more than $D=1$ dimensions}
In lecture 2 we discussed various ways to interpolate a
univariate function $f(x)$  A very nice fact for
polynomial interpolations in $D=1$ dimension is that the
interpolating polynomial is \emph{unique}: For two polynomials
$g(x)$ and $h(x)$ of identical degree one has
\begin{equation*}
    f_i = g(x_i) = h(x_i) \Rightarrow g(x) \equiv h(x)\,.
\end{equation*}

In $D=2$ dimensions, this ceases to be true. There can be
two non-identical polynomials $g(x, y)$ and $h(x, y)$
to identical, and fulfilling the same interpolation
conditions
\begin{equation*}
    f_i = g(x_i, y_i) = h(x_i, y_i)\,.
\end{equation*}
\end{frame}

\begin{frame}{Interpolation in $D=2$ dimensions}
    linear splines $\to$ bilinear
    cubic splines $\to$ bicubic splines
\end{frame}

\begin{frame}{Interpolation in $D=3$ dimensions}
\end{frame}

\begin{frame}{Interpolation in $D$ dimensions}
No ``fits it all'' recipe available

Nearest Neighbor
\end{frame}

\begin{frame}{Extrapolation (I): Basics}
    In many numerical applications a common class of problems arises:
    In the valuation of a function $f(x)$, we are interested in the value
    $f(x_0)$. However, at $x_0$ the function $f(x)$ is numerically instable,
    or maybe even ill-posed.\\

    However, in an environment around $x_0$, $x \approx x_0 + h$, we can evaluate $f$.
    Usually, one now discusses $f(h) \equiv f(x_0 + h)$, or rather the limit
    $\lim_{h \to 0} f(h)$. [Note: in all generality we can map problems with
    limits to either $\infty$ or a finite value to limits to $0$.]\\

    Interpolation, as discussed previously, can not directly help, since $h = 0$
    is not part of the domain of data points. Instead, one can take an interpolation
    at finite $h > 0$, $f_\text{int}(h)$, and simply approximate $f(h = 0) \approx f_\text{int}$.
    This step of using the interpolation of $f$ outside the domain of data points is
    called \emph{extrapolation}.\\

    To extrapolate an arbitrary function might work very well, but also might
    fail spectacularly. In this part of the course, we will briefly discuss
    examples of both cases, and what mathematical requirements make extrapolations
    work.
\end{frame}

\begin{frame}{Extrapolation (II): Working example}
$\cos(x)$ as a differential quotient

\begin{tikzpicture}
    [
        scale=0.75
    ]
    \begin{axis}[%
    ]
        \addplot+[black, mark=*, only marks] coordinates {
            (1, 0.877583)
            (2, 1.06024)
            (3, 1.00056)
            (4, 0.99996)
            (5, 1.00000)
            (6, 1.00000)
        };
    \end{axis}
\end{tikzpicture}
\end{frame}

\begin{frame}{Extrapolation (III): Pathological example}
\begin{equation*}
    f(x) = \exp(-x^{-1/2}) / x^4\,,\quad\text{with}\quad \lim_{x\to 0} f(x) = 0
\end{equation*}
show log of $f(x = 2^{-k})$ for $k=1$ to $10$ (with arbitrary numerical precision!).
\begin{tikzpicture}
    [
        scale=0.75
    ]
    \begin{axis}[%
        ymode=log
    ]
        \addplot+[black, mark=*, only marks] coordinates {
            (1,  1.94493)
            (2,  15.3780)
            (3,  64.0244)
            (4,  153.618)
            (5,  169.579)
            (6,  4.61491)
            (7,  94.4333)
            (8,  0.598657)
            (9,  9.77146)
            (10, 1.16233)
        };
    \end{axis}
\end{tikzpicture}
\end{frame}

\begin{frame}{Foundations: Necessary prerequisite for extrapolation}
existance of an asymptotic expansion
\end{frame}

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