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

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\vspace{1em}
\vspace{0.5em}
	\textcolor{normal text.fg!50!Comment}{Numerical Methods, \\ 26. September, 2016}
\end{center}
\end{frame}
}

\begin{frame}{Plan for today}
    \begin{itemize}
        \item \alert{General problem of estimating the integral of a continuous function $f(x)$ on a finite support}\\
        \vfill
        \item \alert{Specific problems in which properties of the integrand can be used to our advantage}\\
            What properties of $f(x)$ can we use make our live easier?
    \end{itemize}
\end{frame}

\begin{frame}{Setup}
\end{frame}

\begin{frame}{Constant approximation}
\end{frame}

\begin{frame}{Linear approximation}
\end{frame}

\begin{frame}{Wait a minute, I know this!}
    Integrate the interpolating polynomial!

    Newton-Coates!

    \begin{equation}
        \sum_k \omega_k = 1\,.
    \end{equation}
\end{frame}

\begin{frame}{Pathological example 1}
    \resizebox{.9\textwidth}{!}{
    \begin{tikzpicture}
        \begin{axis}[%
            axis x line=center,
            axis y line=left,
            ymin=0.0,ymax=1.05,
            xmin=-1,xmax=+1,
            samples=160
        ]
            \addplot[thick,black,domain=-1:0] {
                1 / (1 + 25 * x * x)
            };
            \addplot[thick,black,domain=0:+1] {
                1 / (1 + 25 * x * x)
            };
            \addplot[thick,red,  domain=-1:0] {
                1.0 - 16.8552 * x^2 + 123.36 * x^4 - 381.434 * x^6 + 494.91 * x^8 - 220.942 * x^(10)
            };
            \addplot[thick,red,  domain=0:+1] {
                1.0 - 16.8552 * x^2 + 123.36 * x^4 - 381.434 * x^6 + 494.91 * x^8 - 220.942 * x^(10)
            };
            \addplot[thick,red,only marks,mark=*,mark size=3pt] coordinates {
                (-1.0, 0.0384615)
                (-0.8, 0.0588235)
                (-0.6, 0.1)
                (-0.4, 0.2)
                (-0.2, 0.5)
                ( 0.0, 1.0)
                ( 0.2, 0.5)
                ( 0.4, 0.2)
                ( 0.6, 0.1)
                ( 0.8, 0.0588235)
                ( 1.0, 0.0384615)
            };
        \end{axis}
    \end{tikzpicture}
    }
\end{frame}

\begin{frame}{Conclusion 1: We know how to fix it!}
    \only<1>{
    When interpolating a function, we saw that using splines fixes
    the problem of oscillating interpolating polynomials.\\
    \vfill

    We can now use this to stabilise our interpolation formula as well.\\
    \vfill

    \begin{itemize}
        \item Let $M$ be the number of intervals in which we want to integrate.
        \item Let $N$ be the degree of the interpolating polynomial in each interval.
        \item Enforce that the interpolating function is continous, but not differentiable
            at the interval boundaries.
    \end{itemize}
    \vfill
    The integral can then be approximated as
    \begin{equation}
        I \approx \sum_{k=0}^{N \cdot M + 1} \omega_k f(x_k)
    \end{equation}
    with $x_k = a + (b - a) \frac{k}{N\cdot M}$\,.
    }
    \only<2->{
    \resizebox{.9\textwidth}{!}{
    \begin{tikzpicture}
        \begin{axis}[%
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            axis y line=left,
            ymin=0.0,ymax=1.05,
            xmin=-1,xmax=+1,
            samples=160
        ]
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                1 / (1 + 25 * x * x)
            };
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                1 / (1 + 25 * x * x)
            };
            \only<3>{
                \addplot+[smooth,red,name path=A1,domain=-1:-0.6,mark=none] {
                    0.348416 + 0.570136 * x + 0.260181 * x^2
                };
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                \addplot+[smooth,red,name path=A2,domain=-0.6:-0.2,mark=none] {
                    1. + 3. * x + 2.5 * x^2
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                \addplot+[smooth,red,name path=A3,domain=-0.2:+0.2,mark=none] {
                    1. - 12.5 * x^2
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                \addplot+[smooth,red,name path=A4,domain=+0.2:+0.6,mark=none] {
                    1. - 3. * x + 2.5 * x^2
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                \addplot+[smooth,red,name path=A5,domain=+0.6:+1,mark=none] {
                    0.348416 - 0.570136 * x + 0.260181 * x^2
                };
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                (-1.0, 0.0384615)
                (-0.8, 0.0588235)
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                (-0.2, 0.5)
                ( 0.0, 1.0)
                ( 0.2, 0.5)
                ( 0.4, 0.2)
                ( 0.6, 0.1)
                ( 0.8, 0.0588235)
                ( 1.0, 0.0384615)
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    }
    }
\end{frame}
\begin{frame}{}
    \begin{equation}
        \omega_k = \begin{cases}
            \frac{1}{6}   & k = 0 \lor k = N \cdot M + 1\\
            \frac{4}{6}   & k\,\text{even}\\
            \frac{2}{6}   & k\,\text{odd}
        \end{cases}
    \end{equation}

\end{frame}

\newcommand{\eps}{\varepsilon}
\begin{frame}{Pathological example 2}
    \begin{equation*}
        I \equiv \int_{-3}^{+4} \mathrm{d}x\, \left[1 - \frac{\eps}{x^2 + \eps}\right]
        \, \quad \text{with }\eps = \frac{1}{9}
    \end{equation*}
    \begin{columns}
    \begin{column}{.5\textwidth}
    \resizebox{\textwidth}{!}{
    \begin{tikzpicture}
        \begin{axis}[%
            axis x line=center,
            axis y line=left,
            ymin=0.0,ymax=1.05,
            xmin=-3,xmax=+4,
            samples=160
        ]
            \addplot[thick,black,domain=-3:0] {
                1 - 1/9 / (x^2 + 1/9)
            };
            \addplot[thick,black,domain=0:+4] {
                1 - 1/9 / (x^2 + 1/9)
            };
            \addplot[thick,blue,only marks,mark=+,mark size=5pt] coordinates {
                ( 1, 0.54030 )
            };
        \end{axis}
    \end{tikzpicture}
    }
    \begin{equation*}
    \end{equation*}
    \end{column}
    \begin{column}{.5\textwidth}
        \begin{block}{Exact result}
            ~\\
            \vspace{-2\bigskipamount}
            \begin{align*}
                I      & = 5.57396
            \end{align*}
        \end{block}
        \begin{block}{Entire range, 7 data points}
            ~\\
            \vspace{-2\bigskipamount}
            \begin{align*}
                I_7      & = 6.25746 &
                \delta_7 & = 12.26\%
            \end{align*}
        \end{block}
        \begin{block}{Two integration, 5 data points}
            ~\\
            \vspace{-2\bigskipamount}
            \begin{align*}
                I_5      & = 5.65645 &
                \delta_5 & = 1.47\%
            \end{align*}
        \end{block}
    \end{column}
    \end{columns}
\end{frame}

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