diff --git a/Lectures_my/NumMet/Lecture4/Makefile b/Lectures_my/NumMet/Lecture4/Makefile
new file mode 100644
index 0000000..37f4367
--- /dev/null
+++ b/Lectures_my/NumMet/Lecture4/Makefile
@@ -0,0 +1,9 @@
+all: lecture4.pdf
+
+lecture4.pdf: lecture4.tex
+	xelatex lecture4.tex
+#	bibtex lecture3
+	xelatex lecture4.tex
+#	xelatex lecture4.tex
+clean:
+	rm *.aux *.bbl *.blg *.log *.out *.pdf *.sh *.xml *.snm *.toc *.nav 
diff --git a/Lectures_my/NumMet/Lecture4/beamercolorthemeFlip.sty b/Lectures_my/NumMet/Lecture4/beamercolorthemeFlip.sty
new file mode 100755
index 0000000..d6b9c50
--- /dev/null
+++ b/Lectures_my/NumMet/Lecture4/beamercolorthemeFlip.sty
@@ -0,0 +1,179 @@
+% Color style file for the BEAMER FLIP THEME
+% Copyright 2012 by Flip Tanedo
+% This file may be distributed and/or modified
+% 	1. under the LaTeX Project Public License and/or
+% 	2. under the GNU Public License.
+
+
+
+\mode<presentation>
+\newif\ifbeamer@Flip@dark
+\beamer@Flip@darkfalse
+\DeclareOptionBeamer{dark}{\beamer@Flip@darktrue}
+\ProcessOptionsBeamer
+
+
+% These are a few colors that I like.
+
+\definecolor{crimsonred}{RGB}{153,0,0}		% Neurtal red, good for dark or light bg
+\definecolor{darkcharcoal}{RGB}{25,25,25}		% Darker gray
+\definecolor{charcoal}{RGB}{51,51,51}		% Darker gray
+\definecolor{ash}{RGB}{100,100,100}			% medium gray
+\definecolor{paleblue}{RGB}{0,102,102}		% More of an `ocean' color
+\definecolor{turtlegreen}{RGB}{51,153,0}	% A more neutral green
+\definecolor{paleale}{RGB}{204,204,102}		% Only for dark BG
+\definecolor{lager}{RGB}{140,110,10}		% Use instead of pale ale for white BG
+\definecolor{regal}{RGB}{90,0,120}			% A more neutral purple
+\definecolor{jeans}{RGB}{20,30,150}			% A more neutral blue
+
+
+% Define some "invariant colors" so that a color that doen't work with a
+% light/dark background will automatically be substituted with one that does.
+% These will be rewritten in beamercolorthemeFlipDark
+
+\ifbeamer@Flip@dark
+	\definecolor{FlipSand}{RGB}{204,204,102}		% paleale, Only for dark BG
+	\definecolor{FlipGreen}{RGB}{0,255,0}			% Neon-like green, only for dark BG
+\else
+	\definecolor{FlipSand}{RGB}{140,110,10}			% lager, Only for light BG
+	\definecolor{FlipGreen}{RGB}{51,153,0}			% turtlegreen, works for both
+\fi
+
+%% Future revision: It would be nice to define these in terms of the palette.
+
+
+\ifbeamer@Flip@dark
+	\setbeamercolor{normal text}{fg=white, bg=black}		
+	\setbeamercolor{structure}{fg=gray, bg=black}		% fg controls bullets
+	\setbeamercolor*{footerlike}{fg=white,bg=black}		% Controls the footerline
+\else
+	% Default to black on white
+	\setbeamercolor*{footerlike}{fg=white,bg=charcoal}	% for charcoal footer
+	\setbeamercolor{structure}{fg=charcoal, bg=white}	% fg controls bullets
+	% \setbeamercolor{normal text}{fg=white, bg=black}	
+\fi
+
+
+
+
+
+% These defined colors can be used in \textcolor{colorname}{text in that color}.
+
+% Colors for Keynote background: DARK
+\definecolor{keynotebottom}{rgb}{0.32,0.3,0.38}
+\definecolor{keynotemiddle}{rgb}{0.08,0.08,0.16}
+\definecolor{keynotetop}{rgb}{0,0,0}
+%%% Example of usage:
+%% \setbeamertemplate{background canvas}[vertical shading]
+%% [bottom=keynotebottom, middle=keynotemiddle, top=keynotetop]
+
+
+
+
+% ALERT AND COMMENT COLORS (just copies of the above colors)
+% (I don't know how ot do this more elegantly)
+\ifbeamer@Flip@dark
+	\definecolor{ALERT}{RGB}{153,0,0}			% crimsonred
+	\definecolor{Alert}{RGB}{51,153,0}			% turtlegreen
+	\definecolor{alert}{RGB}{204,204,102}		% paleale		
+	\definecolor{charcoal}{RGB}{51,51,51}		% charcoal
+	\definecolor{comment}{RGB}{80,80,80}		% light charcoal
+	\definecolor{Comment}{RGB}{100,100,100}		% ash
+	\definecolor{COMMENT}{RGB}{80,20,120}		% regal
+\else
+%	\definecolor{ALERT}{RGB}{153,0,0}			% crimsonred
+	\definecolor{ALERT}{RGB}{28,134,238}		% DodgerBlue2
+%	\definecolor{Alert}{RGB}{51,153,0}			% turtlegreen
+	\definecolor{Alert}{RGB}{255,127,36}		% chocolate1
+%	\definecolor{alert}{RGB}{140,110,10}		% lager
+	\definecolor{alert}{RGB}{0,205,102} 		% SpringGreen3
+	\definecolor{charcoal}{RGB}{80,80,80}		% light charcoal
+	\definecolor{comment}{RGB}{51,51,51}		% charcoal
+	\definecolor{Comment}{RGB}{100,100,100}		% ash
+	\definecolor{COMMENT}{RGB}{80,20,120}		% regal
+\fi
+
+
+% \definecolor{crimsonred}{RGB}{153,0,0}		% Neurtal red, good for dark or light bg
+% \definecolor{charcoal}{RGB}{51,51,51}		% Darker gray
+% \definecolor{ash}{RGB}{100,100,100}			% medium gray
+% \definecolor{paleblue}{RGB}{0,102,102}		% More of an `ocean' color
+% \definecolor{turtlegreen}{RGB}{51,153,0}	% A more neutral green
+% \definecolor{paleale}{RGB}{204,204,102}		% Only for dark BG
+% \definecolor{lager}{RGB}{140,110,10}		% Use instead of pale ale for white BG
+% \definecolor{regal}{RGB}{90,0,120}			% A more neutral purple
+% \definecolor{jeans}{RGB}{20,30,150}			% A more neutral blue
+
+
+% A few useful colors
+\setbeamercolor{alerted text}{fg=alert}
+
+%% I would like ot use these colors, but there seems to be a bug in Beamer
+%% where the footnote color infects the normal text color
+%% See:  http://tex.stackexchange.com/questions/8264/beamer-changing-the-footnote-color-also-changes-the-normal-text-color/8268#8268
+
+% \setbeamercolor{footnote}{parent=normal text, fg=FlipSand}
+% \setbeamercolor{footnote mark}{parent=normal text, fg=FlipSand}
+
+% Blocks
+\setbeamercolor{block title}{fg=gray!30!white,bg=charcoal}
+\setbeamercolor{block body}{parent=normal text,bg=gray, fg=white}
+
+\setbeamercolor{block title example}{fg=turtlegreen!30!white,bg=turtlegreen!40!black}
+\setbeamercolor{block body example}{parent=normal text,bg=turtlegreen, fg=white}
+% 
+\setbeamercolor{block title alerted}{fg=crimsonred!30!white,bg=crimsonred!40!black}
+\setbeamercolor{block body alerted}{parent=normal text,bg=crimsonred, fg=white}
+
+
+
+% These "setbeamercolor" commands are more standardized. 
+
+
+%% From the Beamer User Guide:
+% Outer themes base the color of navigational elements and, possibly, also of other elements, on the 
+% four palette colors. The “primary” palette should be used for the most important navigational elements, 
+% which are usually the ones that change most often and hence require the most attention by the audience. 
+% The “secondary” and “tertiary” are less important, the “quaternary” one is least important. 
+% By default, the palette colors do not have a background and the foreground ranges from structure.fg to black. 
+% For the sidebar, there is an extra set of palette colors, see palette sidebar primary.
+
+\setbeamercolor*{palette primary}{fg=crimsonred,bg=charcoal}
+\setbeamercolor*{palette secondary}{fg=paleale,bg=lager}
+\setbeamercolor*{palette tertiary}{fg=turtlegreen,bg=jeans}
+\setbeamercolor*{palette quaternary}{fg=paleblue,bg=regal}
+
+% \setbeamercolor*{palette primary}{fg=crimsonred,bg=white}
+% \setbeamercolor*{palette secondary}{fg=charcoal,bg=white}
+% \setbeamercolor*{palette tertiary}{fg=paleblue,bg=white}
+% \setbeamercolor*{palette quaternary}{fg=turtlegreen,bg=white}
+
+
+
+%% These are some structures that I define for use in the outer theme.
+
+\setbeamercolor*{crimsontown}{fg=paleblue,bg=crimsonred}
+\setbeamercolor*{bluetown}{fg=white,bg=DodgerBlue2}
+%\setbeamercolor*{titlelike}{fg=crimsonred}
+\setbeamercolor*{titlelike}{fg=DodgerBlue2}
+\setbeamercolor*{topbarlike}{fg=white, bg=charcoal}
+
+
+
+
+
+%% Some old definitions that I might want to get back to
+%
+% \setbeamercolor*{frametitle}{bg=white,fg=charcoal}
+% \setbeamercolor*{part title}{bg=white,fg=charcoal}
+% \setbeamercolor*{item}{fg=charcoal, bg=white}
+% 
+% \setbeamercolor*{separation line}{}
+% \setbeamercolor*{fine separation line}{}
+
+
+
+
+
+\mode
+<all>
diff --git a/Lectures_my/NumMet/Lecture4/beamerinnerthemeFlip.sty b/Lectures_my/NumMet/Lecture4/beamerinnerthemeFlip.sty
new file mode 100755
index 0000000..8dc12a0
--- /dev/null
+++ b/Lectures_my/NumMet/Lecture4/beamerinnerthemeFlip.sty
@@ -0,0 +1,95 @@
+% Inner style file for the BEAMER FLIP THEME
+% Copyright 2010 by Flip Tanedo
+% This file may be distributed and/or modified
+% 	1. under the LaTeX Project Public License and/or
+% 	2. under the GNU Public License.
+%
+% Based on: beamerinnerthemeFlip by Marco Barisione
+
+\mode<presentation>
+
+% % Use alternative title page style.
+% \DeclareOptionBeamer{alternativetitlepage}[true]{\def\beamer@Flip@alternativetitlepage{#1}}
+
+% Logo to use in the alternative title page.
+% \def\beamer@Flip@titlepagelogo{}
+% \DeclareOptionBeamer{titlepagelogo}{\def\beamer@Flip@titlepagelogo{#1}}
+
+% Bullet shape.
+\DeclareOptionBeamer{bullet}{\def\beamer@Flip@bullet{#1}}
+\DeclareOptionBeamer{shadow}[true]{\def\beamer@themerounded@shadow{#1}}
+\ExecuteOptionsBeamer{shadow=false}
+
+
+% \ExecuteOptionsBeamer{alternativetitlepage=false,bullet=square}
+\ProcessOptionsBeamer
+
+
+
+% Margins.
+\newlength{\beamer@Flip@normalmargin}
+\setlength{\beamer@Flip@normalmargin}{.06\paperwidth}
+\setbeamersize{text margin left=\beamer@Flip@normalmargin}
+\setbeamersize{text margin right=\beamer@Flip@normalmargin}
+\setlength\leftmargini{.6\beamer@Flip@normalmargin}
+\setlength\leftmarginii{.6\beamer@Flip@normalmargin}
+\setlength\leftmarginiii{.6\beamer@Flip@normalmargin}
+
+% Normal title page.
+\defbeamertemplate*{title page normal}{Flip theme}[1][]
+{
+  \vbox{}
+  \vfill
+  \begin{centering}
+    \begin{beamercolorbox}[wd=\paperwidth,sep=8pt,center,#1]{title page header}
+      \usebeamerfont{title}\inserttitle\par%
+      \ifx\insertsubtitle\@empty%
+      \else%
+        \vskip0.25em%
+        {\usebeamerfont{subtitle}\usebeamercolor[fg]{subtitle}\insertsubtitle\par}%
+      \fi%     
+    \end{beamercolorbox}%
+    %\vskip0.5em\par
+    \begin{beamercolorbox}[sep=8pt,center,#1]{author}
+      \usebeamerfont{author}\insertauthor
+    \end{beamercolorbox}
+    \begin{beamercolorbox}[sep=8pt,center,#1]{institute}
+      \usebeamerfont{institute}\insertinstitute
+    \end{beamercolorbox}
+    \begin{beamercolorbox}[sep=8pt,center,#1]{date}
+      \usebeamerfont{date}\insertdate
+    \end{beamercolorbox}\vskip0.5em
+    {\usebeamercolor[fg]{titlegraphic}\inserttitlegraphic\par}
+  \end{centering}
+  \vfill
+}
+
+
+% Items.
+\defbeamertemplate{itemize item}{squarealt}%
+{\tiny\raise.5ex\hbox{\donotcoloroutermaths$\blacksquare$}}
+\defbeamertemplate{itemize subitem}{squarealt}%
+{\tiny\raise.4ex\hbox{\donotcoloroutermaths$\square$}}
+\defbeamertemplate{itemize subsubitem}{squarealt}%
+{\tiny\raise.3ex\hbox{\donotcoloroutermaths$\blacksquare$}}
+% 
+\defbeamertemplate{itemize item}{circlealt}%
+{\small\raise.2ex\hbox{\donotcoloroutermaths$\bullet$}}
+\defbeamertemplate{itemize subitem}{circlealt}%
+{\small\raise.1ex\hbox{\donotcoloroutermaths$\circ$}}
+\defbeamertemplate{itemize subsubitem}{circlealt}%
+{\scriptsize\raise.1ex\hbox{\donotcoloroutermaths$\bullet$}}
+\setbeamertemplate{items}[circlealt]
+% 
+\def\circletext{circle}
+\ifx\beamer@Flip@bullet\circletext
+  \setbeamertemplate{items}[circlealt]
+\else
+  \setbeamertemplate{items}[squarealt]
+\fi
+
+\setbeamertemplate{blocks}[rounded][shadow=\beamer@themerounded@shadow]
+
+
+\mode<all>
+
diff --git a/Lectures_my/NumMet/Lecture4/beamerouterthemeFlip.sty b/Lectures_my/NumMet/Lecture4/beamerouterthemeFlip.sty
new file mode 100755
index 0000000..d12f6d6
--- /dev/null
+++ b/Lectures_my/NumMet/Lecture4/beamerouterthemeFlip.sty
@@ -0,0 +1,189 @@
+% Outer style file for the BEAMER FLIP THEME 212
+% Copyright 2012 by Flip Tanedo
+% This file may be distributed and/or modified
+% 	1. under the LaTeX Project Public License and/or
+% 	2. under the GNU Public License.
+%
+% Based on: beamerouterthemesplit by Till Tantau
+
+
+\mode<presentation>
+
+% Watermark
+\DeclareOptionBeamer{watermark}{\def\beamer@Flip@watermark{#1}}
+\ExecuteOptionsBeamer{watermark=} % Default value is empty
+
+
+\newif\ifbeamer@Flip@bigpagenumber
+\beamer@Flip@bigpagenumberfalse
+\DeclareOptionBeamer{bigpagenumber}{\beamer@Flip@bigpagenumbertrue}
+
+
+\def\beamer@Flip@truetext{true} % Kludge for true/false so I can use ifx
+
+\DeclareOptionBeamer{topline}[true]{\def\beamer@Flip@topline{#1}}
+\ExecuteOptionsBeamer{topline=false} % Default
+
+
+
+% \ExecuteOptionsBeamer{topline=} % Default value is empty
+
+
+\ProcessOptionsBeamer
+
+
+\usepackage{tikz}	% For fancy decorations
+
+
+% I should fix these up to be more robust against color changes
+\setbeamercolor{section in head/foot}{parent=footerlike}
+\setbeamercolor{subsection in head/foot}{parent=footerlike}
+\setbeamercolor{author in head/foot}{parent=footerlike}
+\setbeamercolor{title in head/foot}{parent=footerlike}
+
+
+% \setbeamercolor{author in head/foot}{parent=section in head/foot}
+% \setbeamercolor{title in head/foot}{parent=subsection in head/foot}
+
+%\setbeamercolor{topbar}{parent=crimsontown}
+\setbeamercolor{topbar}{parent=bluetown}
+
+
+
+
+\usesectionheadtemplate
+  {\hfill\insertsectionhead}
+  {\hfill\color{fg!50!bg}\insertsectionhead}
+
+
+%%%%%%%%%%%%%%%%%%
+% The footer bar %
+%%%%%%%%%%%%%%%%%%
+
+\defbeamertemplate*{footline}{Flip theme}
+{%
+  \leavevmode% 
+% OBSERVATION: the "%" symbols inside hbox are all very important here.
+% The \hbox enviroment will insert spaces whenever there's whitespace
+% Adding a % at the END of each line ensures that any hard returns aren't
+% interpreted as white space. This allows the color boxes to be flush against
+% one another. If a faint white line appears one could probably do something like
+% hspace{-1px} and add an additional pixel-or-two wide beamercolorbox.	
+
+\hbox{%
+\begin{beamercolorbox}[wd=.4\paperwidth,ht=2.5ex,dp=1.125ex,leftskip=.3cm, rightskip=.3cm plus1fil]{author in head/foot}%
+	\usebeamerfont{author in head/foot}\insertshortauthor  \end{beamercolorbox}%
+%
+\begin{beamercolorbox}[wd=.6\paperwidth,ht=2.5ex,dp=1.125ex,leftskip=.3cm,rightskip=.3cm plus1fil]{title in head/foot}%
+    \usebeamerfont{title in head/foot}{\textit \insertshorttitle}%
+    \hskip2ex plus1fill%
+		% \insertpagenumber\,/\,\insertpresentationendpage %% Want FRAME, not SLIDE number
+		\insertframenumber/\inserttotalframenumber
+ \end{beamercolorbox}%
+}%
+
+% This is a complete kludge for the faint white line which appears
+% between the beamer color boxes above. (It doesnt appear in presentation mode
+% but sometimes looks funny when the pdfs are viewed in a window)
+\begin{tikzpicture}[overlay]
+	\usebeamercolor[bg]{author in head/foot}
+	\draw[thick] (0,3.625ex) -- (\paperwidth,3.625ex);
+	\draw (.4\paperwidth,0) -- (.4\paperwidth,3.625ex);
+	\draw[thick] (0,0) -- (\paperwidth,0);
+\end{tikzpicture}
+
+
+
+\ifbeamer@Flip@bigpagenumber
+	\begin{tikzpicture}[overlay]
+	\usebeamercolor[bg]{author in head/foot}
+	\draw[fill] (\paperwidth-5ex,3.625ex) circle (6ex);
+	\usebeamercolor[fg]{author in head/foot}
+	\draw[fill] (\paperwidth-5ex,3.625ex) circle (5ex);
+	\usebeamercolor[bg]{author in head/foot}
+	\node at (\paperwidth-5ex,4ex) {\small$^\text{\insertframenumber}/_\text{\inserttotalframenumber}$};
+	\end{tikzpicture}
+\fi
+}
+
+
+
+\defbeamertemplate*{frametitle}{Flip theme}[1][left]
+{
+% \ifbeamercolorempty[bg]{frametitle}{}{\nointerlineskip}%
+\@tempdima=\textwidth%
+\advance\@tempdima by\beamer@leftmargin%
+\advance\@tempdima by\beamer@rightmargin%
+
+\vbox{}\vskip-3.5ex%
+%
+\begin{beamercolorbox}[sep=0.3cm,#1,wd=\the\@tempdima]{titlelike}
+	\usebeamerfont{frametitle}%
+	\vbox{}\vskip-1ex%
+	\if@tempswa\else\csname beamer@fte#1\endcsname\fi%
+	% \strut{\bf\insertframetitle}\strut\par% 					%% Include the \bf
+	\strut{\insertframetitle}\strut\par%
+	{%
+  		\ifx\insertframesubtitle\@empty%
+  		\else
+			{\usebeamerfont{framesubtitle}
+			 \usebeamercolor[fg]{framesubtitle}
+			\insertframesubtitle\strut\par}%
+  		\fi
+ 	}%
+	%
+  	\vskip-1ex%
+  	\if@tempswa\else\vskip-.3cm\fi% set inside beamercolorbox... evil here...
+\end{beamercolorbox}%
+%%
+
+% \def\beamer@fteright{\vskip0.35cm\advance\leftskip by 1.7cm\advance\rightskip by1.7cm}
+}
+
+
+
+
+% Sidebar right is important: anything placed here will be
+% visible *behind* the main text. This is where to put any
+% interesting watermarks.
+%
+% 
+\defbeamertemplate*{sidebar right}{Flip theme}
+{
+% \llap{\includegraphics[width=50px]{footdecoration}}
+% \llap{\includegraphics[width=\paperwidth,height=\paperheight]{upperleft_watermark_transp}}
+%
+% THIS WOULD BE A NICE WAY TO PUT THE PARTICLE BUBBLE CHAMBER BG
+% ... make a png of the background with TRANSPARENT BG
+% ... then put the image here (use \hfill to get it on the left)
+% Now one can FURTHER apply a slight gradient in the BG
+%
+% 
+% % Top Line 
+\ifx\beamer@Flip@topline\beamer@Flip@truetext
+	\begin{tikzpicture}[overlay]
+	\usebeamercolor[fg]{titlelike}	% alternately, topbarlike would be a good beamercolor
+%	\usebeamercolor[fg]{topbarlike}	% alternately, topbarlike would be a good beamercolor
+	\draw[fill] (-\paperwidth,0) rectangle (0,-3pt);
+	\end{tikzpicture}
+\fi
+%
+% 
+  \vskip.2\beamer@leftmargin%
+   %\llap{\insertlogo\hskip.5\beamer@leftmargin}%
+   \llap{\insertlogo\hskip.08\beamer@leftmargin}%
+    \vfill%
+  \if \beamer@Flip@watermark\@empty
+	% This is kind of a kludge since if I put anything in here, I automatically get errors
+	% Somehow LaTeX wants to read the \else part!
+	\else%
+	\llap{\includegraphics[width=\paperwidth,height=\paperheight]{\beamer@Flip@watermark}}
+  \fi
+}
+
+
+
+\mode<all>
+\endinput
+
+
diff --git a/Lectures_my/NumMet/Lecture4/beamerouterthemedecolines.sty b/Lectures_my/NumMet/Lecture4/beamerouterthemedecolines.sty
new file mode 100755
index 0000000..669e493
--- /dev/null
+++ b/Lectures_my/NumMet/Lecture4/beamerouterthemedecolines.sty
@@ -0,0 +1,168 @@
+% Copyright 2007 by Marco Barisione
+%
+% This file may be distributed and/or modified
+%
+% 1. under the LaTeX Project Public License and/or
+% 2. under the GNU Public License.
+
+\mode<presentation>
+
+% String used between the current page and the total page count.
+\def\beamer@decolines@pageofpages{/}
+\DeclareOptionBeamer{pageofpages}{\def\beamer@decolines@pageofpages{#1}}
+
+% Show a line below the frame title.
+\DeclareOptionBeamer{titleline}[true]{\def\beamer@decolines@titleline{#1}}
+
+% Image used for the watermark.
+\def\beamer@decolines@watermarkorig{}
+\DeclareOptionBeamer{watermark}{\def\beamer@decolines@watermarkorig{#1}}
+
+% Height of the watermark.
+\def\beamer@decolines@watermarkheight{100px}
+\DeclareOptionBeamer{watermarkheight}{\def\beamer@decolines@watermarkheight{#1}}
+
+% The original image height is watermarkheightmult * watermarkheight.
+\def\beamer@decolines@watermarkheightmult{1}
+\DeclareOptionBeamer{watermarkheightmult}{\def\beamer@decolines@watermarkheightmult{#1}}
+
+\ExecuteOptionsBeamer{titleline=false}
+\ProcessOptionsBeamer
+
+% Enable/disable the watermark.
+\def\watermarkon{%
+  \def\beamer@decolines@watermark{\beamer@decolines@watermarkorig}%
+}
+\def\watermarkoff{\def\beamer@decolines@watermark{}}
+
+% Initially enable the watermark.
+\watermarkon
+
+% Colors.
+\setbeamercolor*{lineup}{parent=palette primary}
+\setbeamercolor*{linemid}{parent=palette secondary}
+\setbeamercolor*{linebottom}{parent=palette tertiary}
+\setbeamercolor*{page header}{parent=titlelike}
+
+% Lengths
+\newlength{\headerheight}
+\setlength{\headerheight}{.045\paperheight}
+\newlength{\beamer@decolines@lineup}
+\setlength{\beamer@decolines@lineup}{.025\paperheight}
+\newlength{\beamer@decolines@linemid}
+\setlength{\beamer@decolines@linemid}{.015\paperheight}
+\newlength{\beamer@decolines@linebottom}
+\setlength{\beamer@decolines@linebottom}{.01\paperheight}
+
+% The height of the watermark part below the 3 bottom lines.
+\newlength{\beamer@decolines@watermarkheightbottom}
+\addtolength{\beamer@decolines@watermarkheightbottom}{\beamer@decolines@lineup}
+\addtolength{\beamer@decolines@watermarkheightbottom}{\beamer@decolines@linemid}
+\addtolength{\beamer@decolines@watermarkheightbottom}{\beamer@decolines@linebottom}
+
+% The height of the watermark part over the 3 bottom lines before shrinking.
+\newlength{\beamer@decolines@watermarkheightupperorig}
+\setlength{\beamer@decolines@watermarkheightupperorig}{\beamer@decolines@watermarkheight}
+\addtolength{\beamer@decolines@watermarkheightupperorig}{-\beamer@decolines@watermarkheightbottom}
+\multiply\beamer@decolines@watermarkheightupperorig by \beamer@decolines@watermarkheightmult
+
+% % Footer.
+% \defbeamertemplate*{footline}{decolines theme}
+% {
+%   \leavevmode%
+%   % Page number.
+%   \hbox{%
+%   \begin{beamercolorbox}[wd=.2\paperwidth,ht=0ex,dp=0ex,center]{}%
+%     \usebeamerfont{palette primary}\insertframenumber{} \beamer@decolines@pageofpages{} \inserttotalframenumber%
+%   \end{beamercolorbox}%
+%   \begin{beamercolorbox}[wd=.8\paperwidth,ht=0ex,dp=0ex]{}%
+%   \end{beamercolorbox}%
+%   } %
+%   % First line.
+%   \hbox{%
+%   \begin{beamercolorbox}[wd=.2\paperwidth,ht=\beamer@decolines@lineup,dp=0pt]{}%
+%   \end{beamercolorbox}%
+%   \begin{beamercolorbox}[wd=.8\paperwidth,ht=\beamer@decolines@lineup,dp=0pt]{lineup}%
+%   \end{beamercolorbox}%
+%   } %
+%   % Second line.
+%   \hbox{%
+%   \begin{beamercolorbox}[wd=\paperwidth,ht=\beamer@decolines@linemid,dp=0pt]{linemid}%
+%   \end{beamercolorbox}%
+%   } %
+%   % Third line.
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+%   \begin{beamercolorbox}[wd=.1\paperwidth,ht=\beamer@decolines@linebottom,dp=0pt]{}%
+%   \end{beamercolorbox}%
+%   \begin{beamercolorbox}[wd=.9\paperwidth,ht=\beamer@decolines@linebottom,dp=0pt]{linebottom}%
+%   \end{beamercolorbox}%
+%   }%
+%   % This seems to fix some alignment problems with the watermark. It has to be
+%   % always applied if you do not want to see the footer moving up and down when
+%   % moving from a page with watermark to a page without or vice versa.
+%   \vskip-.5px%
+%   % Watermark.
+%   \if\beamer@decolines@watermark\@empty\else%
+%     \vskip-\beamer@decolines@watermarkheightbottom%
+%     \llap{\includegraphics[height=\beamer@decolines@watermarkheightbottom,clip=true,%
+%       trim=0pt 0pt 0pt \beamer@decolines@watermarkheightupperorig]{\beamer@decolines@watermark}\hskip-\paperwidth}%
+%   \fi%
+% }
+
+\defbeamertemplate*{headline}{decolines theme}
+{
+  \leavevmode%
+  \hbox{%
+  \begin{beamercolorbox}[wd=\paperwidth,ht=\headerheight,dp=0pt]{page header}%
+  \end{beamercolorbox}%
+  } %
+  \vskip0pt%
+}
+
+\defbeamertemplate*{frametitle}{decolines theme}[1][left]
+{
+  \ifbeamercolorempty[bg]{frametitle}{}{\nointerlineskip}%
+  \@tempdima=\textwidth%
+  \advance\@tempdima by\beamer@leftmargin%
+  \advance\@tempdima by\beamer@rightmargin%
+  \vbox{}\vskip-.5\beamer@leftmargin%
+  \begin{beamercolorbox}[sep=\beamer@leftmargin,#1,wd=\the\@tempdima]{}
+    \usebeamerfont{frametitle}\usebeamercolor[bg]{framesubtitle}%
+    \vbox{}\vskip0ex%
+    \if@tempswa\else\csname beamer@fte#1\endcsname\fi%
+    \strut\insertframetitle\strut\par%
+    {%
+      \ifx\insertframesubtitle\@empty%
+      \else%
+      {\usebeamerfont{framesubtitle}\usebeamercolor[bg]{framesubtitle}\insertframesubtitle\strut\par}%
+      \fi
+    }%
+    \vskip-1ex%
+    \if@tempswa\else\vskip-\beamer@leftmargin\fi
+  \end{beamercolorbox}%
+  \def\beamer@decolines@truetext{true}%
+  \ifx\beamer@decolines@titleline\beamer@decolines@truetext%
+    \vskip-.5\beamer@leftmargin%
+    \begin{beamercolorbox}[wd=\textwidth,ht=.1ex,dp=0ex]{linemid}%
+    \end{beamercolorbox}%
+  \fi
+}
+
+% Frame title continuations, default
+\defbeamertemplate*{frametitle continuation}{decolines theme}{(\insertcontinuationcount)}
+
+
+
+\defbeamertemplate*{sidebar right}{decolines theme}
+{
+  \vskip.1\beamer@leftmargin%
+  \llap{\insertlogo\hskip.5\beamer@leftmargin}%
+  \vfill%
+  \if\beamer@decolines@watermark\@empty\else%
+  \llap{\includegraphics[height=\beamer@decolines@watermarkheight]{\beamer@decolines@watermark}}%
+  \vskip-\beamer@decolines@watermarkheightbottom%
+  \fi
+}
+
+\mode
+<all>
diff --git a/Lectures_my/NumMet/Lecture4/beamerthemeFlip.sty b/Lectures_my/NumMet/Lecture4/beamerthemeFlip.sty
new file mode 100755
index 0000000..a615ff5
--- /dev/null
+++ b/Lectures_my/NumMet/Lecture4/beamerthemeFlip.sty
@@ -0,0 +1,59 @@
+% Main style file for the BEAMER FLIP THEME 2012
+% Copyright 2012 by Flip Tanedo
+% This file may be distributed and/or modified
+% 	1. under the LaTeX Project Public License and/or
+% 	2. under the GNU Public License.
+%
+% Comments: This is still a work in progress.
+% A good way to present: http://www.cs.hmc.edu/~oneill/freesoftware/pdftokeynote.html
+
+% \usepackage{beamerthemesplit}
+\mode<presentation>
+
+
+%% These are the options that are fed in through the driver file
+%% Some of them get passed on to the other theme files
+\DeclareOptionBeamer{bullet}{\PassOptionsToPackage{bullet=#1}{beamerinnerthemeFlip}}
+\DeclareOptionBeamer{bigpagenumber}{\PassOptionsToPackage{bigpagenumber}{beamerouterthemeFlip}}
+\DeclareOptionBeamer{topline}[true]{\PassOptionsToPackage{topline=#1}{beamerouterthemeFlip}}
+\DeclareOptionBeamer{shadow}[false]{\PassOptionsToPackage{shadow=#1}{beamerinnerthemeFlip}}
+
+\DeclareOptionBeamer{watermark}{\PassOptionsToPackage{watermark=#1}{beamerouterthemeFlip}}
+
+\ProcessOptionsBeamer
+
+\useinnertheme{Flip}	% Calls beamerinnerthemeFlip.sty
+\useoutertheme{Flip}	% Calls beamerouterthemeFlip.sty
+\usecolortheme{Flip}	% Calls beamercolorthemeFlip.sty
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+% Background  Gradient %
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%% Usually I would leave this on
+%% However, there seems to be problems with XeLaTeX
+%% http://tex.stackexchange.com/questions/29497/xelatex-preventing-beamer-from-using-different-backgrounds
+\setbeamertemplate{background canvas}[vertical shading][bottom=blue!.1, middle=white, top=white]
+
+%% HOWEVER: it seems to make it impossible to change this later on in the document
+
+
+%% For a dark background: 
+% \setbeamertemplate{background canvas}[vertical shading][bottom=keynotebottom, middle=keynotemiddle, top=keynotetop]
+
+
+%% An alternate way: \beamertemplateshadingbackground{blue!.1}{red!2}
+%% In general: you want the gradient to be subtle!
+
+
+%%%%%%%%%%%%%%%%%%%%%%%
+% Navigation  symbols %
+%%%%%%%%%%%%%%%%%%%%%%%
+
+\setbeamertemplate{navigation symbols}{} 	% Turns off PDF navigation symbols
+
+\mode<all>
+\endinput
\ No newline at end of file
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diff --git a/Lectures_my/NumMet/Lecture4/lecture4.tex b/Lectures_my/NumMet/Lecture4/lecture4.tex
new file mode 100644
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--- /dev/null
+++ b/Lectures_my/NumMet/Lecture4/lecture4.tex
@@ -0,0 +1,321 @@
+% vim: set sts=4 et :
+\documentclass[11 pt,xcolor={dvipsnames,svgnames,x11names,table}]{beamer}
+
+\usetheme[
+	bullet=circle,		% Other option: square
+	bigpagenumber,		% circled page number on lower right
+	topline=true,			% colored bar at the top of the frame 
+	shadow=false,			% Shading for beamer blocks
+	watermark=BG_lower,	% png file for the watermark
+	]{Flip}
+
+\usepackage{bm}
+\usepackage{tikz,pgfplots}
+\usepackage{graphicx}
+
+% SOME COMMANDS THAT I FIND HANDY
+% \renewcommand{\tilde}{\widetilde} % dinky tildes look silly, dosn't work with fontspec
+\newcommand{\comment}[1]{\textcolor{comment}{\footnotesize{#1}\normalsize}} % comment mild
+\newcommand{\Comment}[1]{\textcolor{Comment}{\footnotesize{#1}\normalsize}} % comment bold
+\newcommand{\COMMENT}[1]{\textcolor{COMMENT}{\footnotesize{#1}\normalsize}} % comment crazy bold
+\newcommand{\Alert}[1]{\textcolor{Alert}{#1}} % louder alert
+\newcommand{\ALERT}[1]{\textcolor{ALERT}{#1}} % loudest alert
+%% "\alert" is already a beamer pre-defined
+\newcommand*{\Scale}[2][4]{\scalebox{#1}{$#2$}}%
+
+\graphicspath{{images/}}	% Put all images in this directory. Avoids clutter.
+
+% suppress frame numbering for backup slides
+% you always need the appendix for this!
+\newcommand{\backupbegin}{
+	\newcounter{framenumberappendix}
+	\setcounter{framenumberappendix}{\value{framenumber}}
+}
+\newcommand{\backupend}{
+	\addtocounter{framenumberappendix}{-\value{framenumber}}
+	\addtocounter{framenumber}{\value{framenumberappendix}} 
+}
+
+\begin{document}
+\tikzstyle{every picture}+=[remember picture]
+
+{
+\setbeamertemplate{sidebar right}{\llap{\includegraphics[width=\paperwidth,height=\paperheight]{bubble2}}}
+\begin{frame}[c]%{\phantom{title page}} 
+\begin{center}
+\begin{center}
+	\begin{columns}
+		\begin{column}{0.9\textwidth}
+			\flushright%\fontspec{Trebuchet MS}
+			\bfseries \Huge {Root Finding}
+		\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 Chrzaszcz, 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}
+\begin{column}{0.4\textwidth}
+\includegraphics[height=1.3cm]{uzh-transp}
+\end{column}
+\end{columns}
+
+\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 finding the root of a function}\\
+            $\xi$ is a root of $f$ iff $f(\xi) = 0$.
+        \vfill
+        \item \alert{Begin with $D=1$}\\
+            What properties of $f(x)$ can we use to find one, any, or all roots
+            of $f$? What are the requirements on $f(x)$?
+        \vfill
+        \item \alert{How about $D>1$?}\\
+            Can we generalize root finding from $D=1$ do arbitrary $D$?
+    \end{itemize}
+\end{frame}
+
+\begin{frame}{General iterative procedure}
+    Let's assume that
+    \begin{itemize}
+        \item $f(x)$ is a our function of interest,
+        \item $\xi$ is the only root of $f$,
+        \item we have a point $x_0$ close to $\xi$
+    \end{itemize}
+    We now want to find a sequence $\lbrace x_0, x_1, \dots \rbrace$ that
+    \begin{enumerate}
+        \item converges toward $\xi$: $\lim_{k\to \infty} x_k = \xi$,
+        \item is iterative: $x_{k + 1} = \Phi[f](x_k)$
+    \end{enumerate}
+    We can attempt to Taylor expand $f$ around $x_0$ to order $N$, in order
+    to obtain the generator $\Phi[f]$ for the iteration:
+    \begin{block}{Expansion around $x_0$}
+    \begin{equation*}
+        f(\xi) = 0 = \sum_{n=0}^N \frac{(\xi - x_0)^n}{n!} f^{(n)}(x_0)
+    \end{equation*}
+    where $f^{(n)}(x_0)$ is the $n$th derivative of $f$ at the position $x_0$.
+    \end{block}
+\end{frame}
+
+\begin{frame}{Newton-Raphson method \hfill ($N=1$)}
+    \begin{block}{Expansion for $N=1$}
+        \begin{equation}
+            \tag{*}
+            f(\xi) = 0 = f(x_0) + (\xi - x_0) \cdot f'(x_0) + \mathcal{O}((\xi - x_0)^2)
+        \end{equation}
+    \end{block}
+    \vfill
+    Algorithm:
+    \begin{enumerate}
+        \item start with index $k=0$
+        \item solve equation (*), assuming a vanshing approximation error:
+            \begin{gather*}
+                x_{k + 1} \leftarrow x_k - \frac{f(x_k)}{f'(x_k)} \approx \xi + \mathcal{O}((\xi - x_k)^2)\\
+                \Phi[f](x) \equiv x - \frac{f(x)}{f'(x)}
+            \end{gather*}
+        \item if $f(x_{k + 1}) \leq t$, where $t$ is an a-prior threshold, then stop; otherwise
+            jump back to step \#2.
+    \end{enumerate}
+\end{frame}
+
+\begin{frame}{Illustration of Newton-Raphson}
+    \begin{columns}
+    \begin{column}[T]{.5\textwidth}
+    \begin{overlayarea}{\textwidth}{\textheight}
+        $\bm{k=0}$:
+        \only<2->{%
+        \begin{equation*}
+            x_0 := 1
+        \end{equation*}
+        }
+        \only<3->{%
+            $\bm{k=1}$:
+            \begin{align*}
+                x_0     & = 1 &
+                f(x_0)  & = +0.54 \\
+                        &     &
+                f'(x_0) & = -0.84 \\
+            \end{align*}
+        }
+        \only<4->{%
+            \begin{equation*}
+                {\color{red} x_1 \leftarrow 1.64}
+            \end{equation*}
+        }
+        \only<6->{%
+            $\bm{k=2}$:
+            \begin{align*}
+                x_1     & = 1.64 &
+                f(x_0)  & = -0.07 \\
+                        &     &
+                f'(x_0) & = -0.997 \\
+            \end{align*}
+        }
+        \only<7->{%
+            \begin{equation*}
+                {\color{red} x_2 \leftarrow 1.57}
+            \end{equation*}
+        }
+    \end{overlayarea}
+    \end{column}
+    \begin{column}[T]{.5\textwidth}
+    \hspace{-1cm}
+    \resizebox{1.1\textwidth}{!}{
+    \begin{tikzpicture}
+        \begin{axis}[%
+            axis x line=center,
+            axis y line=center,
+            ymin=-0.2,ymax=0.8,
+            xmin=0.0,xmax=1.8,
+            title=$\cos(x)$
+        ]
+            \addplot[thick,black,domain=0:3.14] {
+                cos(deg(x))
+            };
+            \only<2-3>{
+                \addplot[thick,blue,only marks,mark=+,mark size=5pt] coordinates {
+                    ( 1, 0.54030 )
+                };
+            }
+            \only<3-4>{
+                \addplot+[thick,blue,domain=0:3.14,mark=none] {
+                    0.54030 - 0.84147 * (x - 1)
+                };
+            }
+            \only<4-5>{
+                \addplot[thick,red,only marks,mark=o,mark size=5pt] coordinates {
+                    ( 1.6421, 0.0 )
+                };
+                \draw[thick,red] (axis cs:1.6421, 0) -- (axis cs:1.6421, -0.0712);
+            }
+            \only<5-6>{
+                \addplot[thick,blue,only marks,mark=+,mark size=5pt] coordinates {
+                    ( 1.6421, -0.0712 )
+                };
+            }
+            \only<6-7>{
+                \addplot+[thick,blue,domain=0:3.14,mark=none] {
+                    -0.0712 - 0.99746 * (x - 1.6421)
+                };
+            }
+            \only<7->{
+                \addplot[thick,red,only marks,mark=o,mark size=5pt] coordinates {
+                    ( 1.5707, 7.8e-5 )
+                };
+            }
+        \end{axis}
+    \end{tikzpicture}
+    }
+    \vfill
+    \begin{overlayarea}{\textwidth}{.3\textheight}
+        \only<8->{
+        \begin{tabular}{c|cc}
+            $k$ & $x_k$ & $f(x_k)$\\
+            \hline
+            $0$ & $1.00$ & $+0.54$\\
+            $1$ & $1.64$ & $-0.07$\\
+            $2$ & $1.57$ & $+0.00$\\
+            \hline
+            $\xi$&$1.5708$ & $0$
+        \end{tabular}
+        }
+    \end{overlayarea}
+    \end{column}
+    \end{columns}
+\end{frame}
+
+\begin{frame}{Modified Newton-Raphson method \hfill ($N=2$)}
+    \begin{block}{Expansion for $N=2$}
+        \vspace{-2\medskipamount}
+        \begin{equation}
+            \tag{*}
+            f(\xi) = 0 = f(x_0) + (\xi - x_0) \cdot f'(x_0) + \frac{1}{2} (\xi - x_0)^2 f''(x_0) + \mathcal{O}((\xi - x_0)^3)
+        \end{equation}
+    \end{block}
+    \vfill
+    Algorithm:
+    \begin{enumerate}
+        \item start with index $k=0$
+        \item solve equation (*), assuming a vanshing approximation error:
+            \begin{align*}
+                a_\pm
+                    & \leftarrow x_k - \frac{f'(x_k) \pm \sqrt{[f'(x_k)]^2 - 2 f(x_k) f''(x_k)}}{f''(x_k)}\\
+                    & \approx \xi + \mathcal{O}((\xi - x_k)^3)
+            \end{align*}
+        \item if $|f(a_+)| < |f(a_-)|$, then $x_{k + 1} \leftarrow a_+$; otherwise $x_{k + 1} \leftarrow a_-$
+        \item if $f(x_{k + 1}) \leq t$, where $t$ is an a-prior threshold, then stop; otherwise
+            jump back to step \#2.
+    \end{enumerate}
+\end{frame}
+
+\begin{frame}{Rate of convergence}
+    Does this iterative procedure converge? If yes, how fast? How can we quantify the rate of convergence?
+
+    \begin{itemize}
+        \item we have \alert{local} convergence of order $p \leq 1$, if for all $x \in U(\xi)$
+        \begin{equation*}
+            || \Phi[f](x) - \xi|| \leq C \cdot || x - \xi||^p\,,\quad\text{with}\quad C \geq 0\,.
+        \end{equation*}
+        Note: if $p = 1$ then we must have $C < 1$.
+        \item we have \alert{global} convergence if $U(\xi) = \mathbb{R}$
+        \item for $D=1$ we can calculate $p$ is $\Phi[f](x)$ differentiable to sufficient degree:
+        \begin{align*}
+            \Phi(x) - \xi = \Phi(x) - \Phi(\xi) = \frac{(x - \xi)^p}{p!} + o(||x - \xi||^p)
+        \end{align*}
+    \end{itemize}
+\end{frame}
+
+\begin{frame}{Rate of convergence for Newton-Raphson}
+    We have $\Phi(x) \equiv \Phi[f](x)$:
+    \begin{itemize}
+        \item $\Phi(\xi) = \xi$ by construction
+        \item $\Phi'(\xi) = \frac{f(\xi) \cdot f''(\xi)}{[f'(\xi)]^2} = 0$ \alert{(*)} by construction
+        \item $\Phi''(\xi) = \frac{f''(\xi)}{f'(\xi)}$
+    \end{itemize}
+    We can therefore write
+    \begin{equation}
+        \Phi(x) - \Phi(\xi) = \frac{(x - \xi)^2}{2!} \Phi''(\xi) + o(||x - \xi||^2)\,.
+    \end{equation}
+    The Newton-Raphson method converges therefore \emph{at least quadratically}, i.e.: it is a
+    second-order method.
+    \vfill
+    \alert{(*): only if $f'(\xi) \neq 0$}, which is equivalent to $\xi$ is a simple root of $f$.
+\end{frame}
+
+\begin{frame}{Honorable mention: Horner Scheme}
+    If $f(x) \equiv p_n(x)$ is a polynomial of degree $n$ in $x$, there might be multiple roots of $f$, i.e.
+    roots $\xi_{n}$ with $f'(\xi_n) = 0$.\\
+    \vfill
+    The Horner Scheme allows to efficiently calculate all the (multiple) roots
+    of the polynomial $p_n(x)$.\\
+    \vfill
+    However, this is of limited use: These polynomials usually only arise in computing
+    the characteric polynomial $\chi_M$ of a matrix $M$. In this case, the roots of $\chi_M$
+    correspond to eigenvalue of $M$. However, the are better/more stables ways to \alert{numerically} compute
+    all eigenvalues of $M$.
+\end{frame}
+
+\begin{frame}{Newton-Raphson for $D > 1$}
+\end{frame}
+
+\backupbegin
+
+\begin{frame}\frametitle{Backup}
+
+\end{frame}
+
+\backupend
+\end{document}