Mostly finished results and lots of plots

58 files changed, 3169 insertions, 6 deletions

diff --git a/report/figures/results/1000_max.txt b/report/figures/results/1000_max.txt
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index 0000000..9ba19c3
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diff --git a/report/figures/results/1250_max.txt b/report/figures/results/1250_max.txt
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diff --git a/report/figures/results/1500_max.txt b/report/figures/results/1500_max.txt
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diff --git a/report/figures/results/500_max.txt b/report/figures/results/500_max.txt
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diff --git a/report/figures/results/750_max.txt b/report/figures/results/750_max.txt
new file mode 100644
index 0000000..7d090df
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new file mode 100644
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+ 3.74358974e+00 9.93702908e-04
+ 4.00000000e+00 1.41957558e-03
diff --git a/report/figures/results/N750_L512.txt b/report/figures/results/N750_L512.txt
new file mode 100644
index 0000000..6f3e1c3
--- /dev/null
+++ b/report/figures/results/N750_L512.txt
@@ -0,0 +1,40 @@
+ -6.00000000e+00 2.27471566e-03
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diff --git a/report/figures/results/maxcomp.pdf b/report/figures/results/maxcomp.pdf
new file mode 100644
index 0000000..3f831cf
--- /dev/null
+++ b/report/figures/results/maxcomp.pdf
diff --git a/report/figures/results/maxcomp.tex b/report/figures/results/maxcomp.tex
new file mode 100644
index 0000000..908a045
--- /dev/null
+++ b/report/figures/results/maxcomp.tex
@@ -0,0 +1,340 @@
+% GNUPLOT: LaTeX picture with Postscript
+\begingroup
+  \makeatletter
+  \providecommand\color[2][]{%
+    \GenericError{(gnuplot) \space\space\space\@spaces}{%
+      Package color not loaded in conjunction with
+      terminal option `colourtext'%
+    }{See the gnuplot documentation for explanation.%
+    }{Either use 'blacktext' in gnuplot or load the package
+      color.sty in LaTeX.}%
+    \renewcommand\color[2][]{}%
+  }%
+  \providecommand\includegraphics[2][]{%
+    \GenericError{(gnuplot) \space\space\space\@spaces}{%
+      Package graphicx or graphics not loaded%
+    }{See the gnuplot documentation for explanation.%
+    }{The gnuplot epslatex terminal needs graphicx.sty or graphics.sty.}%
+    \renewcommand\includegraphics[2][]{}%
+  }%
+  \providecommand\rotatebox[2]{#2}%
+  \@ifundefined{ifGPcolor}{%
+    \newif\ifGPcolor
+    \GPcolortrue
+  }{}%
+  \@ifundefined{ifGPblacktext}{%
+    \newif\ifGPblacktext
+    \GPblacktexttrue
+  }{}%
+  % define a \g@addto@macro without @ in the name:
+  \let\gplgaddtomacro\g@addto@macro
+  % define empty templates for all commands taking text:
+  \gdef\gplbacktext{}%
+  \gdef\gplfronttext{}%
+  \makeatother
+  \ifGPblacktext
+    % no textcolor at all
+    \def\colorrgb#1{}%
+    \def\colorgray#1{}%
+  \else
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+    \ifGPcolor
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+      \def\colorgray#1{\color[gray]{#1}}%
+      \expandafter\def\csname LTw\endcsname{\color{white}}%
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+      \expandafter\def\csname LTa\endcsname{\color{black}}%
+      \expandafter\def\csname LT0\endcsname{\color[rgb]{1,0,0}}%
+      \expandafter\def\csname LT1\endcsname{\color[rgb]{0,1,0}}%
+      \expandafter\def\csname LT2\endcsname{\color[rgb]{0,0,1}}%
+      \expandafter\def\csname LT3\endcsname{\color[rgb]{1,0,1}}%
+      \expandafter\def\csname LT4\endcsname{\color[rgb]{0,1,1}}%
+      \expandafter\def\csname LT5\endcsname{\color[rgb]{1,1,0}}%
+      \expandafter\def\csname LT6\endcsname{\color[rgb]{0,0,0}}%
+      \expandafter\def\csname LT7\endcsname{\color[rgb]{1,0.3,0}}%
+      \expandafter\def\csname LT8\endcsname{\color[rgb]{0.5,0.5,0.5}}%
+    \else
+      % gray
+      \def\colorrgb#1{\color{black}}%
+      \def\colorgray#1{\color[gray]{#1}}%
+      \expandafter\def\csname LTw\endcsname{\color{white}}%
+      \expandafter\def\csname LTb\endcsname{\color{black}}%
+      \expandafter\def\csname LTa\endcsname{\color{black}}%
+      \expandafter\def\csname LT0\endcsname{\color{black}}%
+      \expandafter\def\csname LT1\endcsname{\color{black}}%
+      \expandafter\def\csname LT2\endcsname{\color{black}}%
+      \expandafter\def\csname LT3\endcsname{\color{black}}%
+      \expandafter\def\csname LT4\endcsname{\color{black}}%
+      \expandafter\def\csname LT5\endcsname{\color{black}}%
+      \expandafter\def\csname LT6\endcsname{\color{black}}%
+      \expandafter\def\csname LT7\endcsname{\color{black}}%
+      \expandafter\def\csname LT8\endcsname{\color{black}}%
+    \fi
+  \fi
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+    \setlength{\fboxsep}{1pt}%
+    \definecolor{tbcol}{rgb}{1,1,1}%
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+      \put(2453,5889){\makebox(0,0){\strut{}x}}%
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+      \put(6980,6197){\makebox(0,0){\strut{}$0$}}%
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+      \put(7849,6197){\makebox(0,0){\strut{}$4$}}%
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+      \put(8283,6197){\makebox(0,0){\strut{}$6$}}%
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+    \gplgaddtomacro\gplfronttext{%
+      \csname LTb\endcsname%%
+      \put(4496,7203){\rotatebox{-270.00}{\makebox(0,0){\strut{}p(x)}}}%
+      \csname LTb\endcsname%%
+      \put(6763,5889){\makebox(0,0){\strut{}x}}%
+      \csname LTb\endcsname%%
+      \put(6763,8312){\makebox(0,0){\strut{}$N$=750}}%
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+      \put(2453,3016){\makebox(0,0){\strut{}x}}%
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+      \put(2453,143){\makebox(0,0){\strut{}x}}%
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+      \put(8283,450){\makebox(0,0){\strut{}$6$}}%
+    }%
+    \gplgaddtomacro\gplfronttext{%
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+      \put(4496,1457){\rotatebox{-270.00}{\makebox(0,0){\strut{}p(x)}}}%
+      \csname LTb\endcsname%%
+      \put(6763,143){\makebox(0,0){\strut{}x}}%
+      \csname LTb\endcsname%%
+      \put(6763,2566){\makebox(0,0){\strut{}$N$=1750}}%
+    }%
+    \gplbacktext
+    \put(0,0){\includegraphics[width={432.00bp},height={432.00bp}]{maxcomp}}%
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+  \end{picture}%
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diff --git a/report/figures/results/parameters.pdf b/report/figures/results/parameters.pdf
new file mode 100644
index 0000000..5952f53
--- /dev/null
+++ b/report/figures/results/parameters.pdf
diff --git a/report/figures/results/parameters.tex b/report/figures/results/parameters.tex
new file mode 100644
index 0000000..065a156
--- /dev/null
+++ b/report/figures/results/parameters.tex
@@ -0,0 +1,154 @@
+% GNUPLOT: LaTeX picture with Postscript
+\begingroup
+  \makeatletter
+  \providecommand\color[2][]{%
+    \GenericError{(gnuplot) \space\space\space\@spaces}{%
+      Package color not loaded in conjunction with
+      terminal option `colourtext'%
+    }{See the gnuplot documentation for explanation.%
+    }{Either use 'blacktext' in gnuplot or load the package
+      color.sty in LaTeX.}%
+    \renewcommand\color[2][]{}%
+  }%
+  \providecommand\includegraphics[2][]{%
+    \GenericError{(gnuplot) \space\space\space\@spaces}{%
+      Package graphicx or graphics not loaded%
+    }{See the gnuplot documentation for explanation.%
+    }{The gnuplot epslatex terminal needs graphicx.sty or graphics.sty.}%
+    \renewcommand\includegraphics[2][]{}%
+  }%
+  \providecommand\rotatebox[2]{#2}%
+  \@ifundefined{ifGPcolor}{%
+    \newif\ifGPcolor
+    \GPcolortrue
+  }{}%
+  \@ifundefined{ifGPblacktext}{%
+    \newif\ifGPblacktext
+    \GPblacktexttrue
+  }{}%
+  % define a \g@addto@macro without @ in the name:
+  \let\gplgaddtomacro\g@addto@macro
+  % define empty templates for all commands taking text:
+  \gdef\gplbacktext{}%
+  \gdef\gplfronttext{}%
+  \makeatother
+  \ifGPblacktext
+    % no textcolor at all
+    \def\colorrgb#1{}%
+    \def\colorgray#1{}%
+  \else
+    % gray or color?
+    \ifGPcolor
+      \def\colorrgb#1{\color[rgb]{#1}}%
+      \def\colorgray#1{\color[gray]{#1}}%
+      \expandafter\def\csname LTw\endcsname{\color{white}}%
+      \expandafter\def\csname LTb\endcsname{\color{black}}%
+      \expandafter\def\csname LTa\endcsname{\color{black}}%
+      \expandafter\def\csname LT0\endcsname{\color[rgb]{1,0,0}}%
+      \expandafter\def\csname LT1\endcsname{\color[rgb]{0,1,0}}%
+      \expandafter\def\csname LT2\endcsname{\color[rgb]{0,0,1}}%
+      \expandafter\def\csname LT3\endcsname{\color[rgb]{1,0,1}}%
+      \expandafter\def\csname LT4\endcsname{\color[rgb]{0,1,1}}%
+      \expandafter\def\csname LT5\endcsname{\color[rgb]{1,1,0}}%
+      \expandafter\def\csname LT6\endcsname{\color[rgb]{0,0,0}}%
+      \expandafter\def\csname LT7\endcsname{\color[rgb]{1,0.3,0}}%
+      \expandafter\def\csname LT8\endcsname{\color[rgb]{0.5,0.5,0.5}}%
+    \else
+      % gray
+      \def\colorrgb#1{\color{black}}%
+      \def\colorgray#1{\color[gray]{#1}}%
+      \expandafter\def\csname LTw\endcsname{\color{white}}%
+      \expandafter\def\csname LTb\endcsname{\color{black}}%
+      \expandafter\def\csname LTa\endcsname{\color{black}}%
+      \expandafter\def\csname LT0\endcsname{\color{black}}%
+      \expandafter\def\csname LT1\endcsname{\color{black}}%
+      \expandafter\def\csname LT2\endcsname{\color{black}}%
+      \expandafter\def\csname LT3\endcsname{\color{black}}%
+      \expandafter\def\csname LT4\endcsname{\color{black}}%
+      \expandafter\def\csname LT5\endcsname{\color{black}}%
+      \expandafter\def\csname LT6\endcsname{\color{black}}%
+      \expandafter\def\csname LT7\endcsname{\color{black}}%
+      \expandafter\def\csname LT8\endcsname{\color{black}}%
+    \fi
+  \fi
+    \setlength{\unitlength}{0.0500bp}%
+    \ifx\gptboxheight\undefined%
+      \newlength{\gptboxheight}%
+      \newlength{\gptboxwidth}%
+      \newsavebox{\gptboxtext}%
+    \fi%
+    \setlength{\fboxrule}{0.5pt}%
+    \setlength{\fboxsep}{1pt}%
+    \definecolor{tbcol}{rgb}{1,1,1}%
+\begin{picture}(8640.00,4320.00)%
+    \gplgaddtomacro\gplbacktext{%
+      \csname LTb\endcsname%%
+      \put(1181,655){\makebox(0,0)[r]{\strut{}$-0.2$}}%
+      \csname LTb\endcsname%%
+      \put(1181,1160){\makebox(0,0)[r]{\strut{}$0$}}%
+      \csname LTb\endcsname%%
+      \put(1181,1665){\makebox(0,0)[r]{\strut{}$0.2$}}%
+      \csname LTb\endcsname%%
+      \put(1181,2170){\makebox(0,0)[r]{\strut{}$0.4$}}%
+      \csname LTb\endcsname%%
+      \put(1181,2675){\makebox(0,0)[r]{\strut{}$0.6$}}%
+      \csname LTb\endcsname%%
+      \put(1181,3180){\makebox(0,0)[r]{\strut{}$0.8$}}%
+      \csname LTb\endcsname%%
+      \put(1181,3685){\makebox(0,0)[r]{\strut{}$1$}}%
+      \csname LTb\endcsname%%
+      \put(1293,450){\makebox(0,0){\strut{}$400$}}%
+      \csname LTb\endcsname%%
+      \put(2155,450){\makebox(0,0){\strut{}$600$}}%
+      \csname LTb\endcsname%%
+      \put(3017,450){\makebox(0,0){\strut{}$800$}}%
+      \csname LTb\endcsname%%
+      \put(3879,450){\makebox(0,0){\strut{}$1000$}}%
+      \csname LTb\endcsname%%
+      \put(4740,450){\makebox(0,0){\strut{}$1200$}}%
+      \csname LTb\endcsname%%
+      \put(5602,450){\makebox(0,0){\strut{}$1400$}}%
+      \csname LTb\endcsname%%
+      \put(6464,450){\makebox(0,0){\strut{}$1600$}}%
+      \csname LTb\endcsname%%
+      \put(7326,450){\makebox(0,0){\strut{}$1800$}}%
+      \csname LTb\endcsname%%
+      \put(7438,655){\makebox(0,0)[l]{\strut{}$-200$}}%
+      \csname LTb\endcsname%%
+      \put(7438,1034){\makebox(0,0)[l]{\strut{}$-100$}}%
+      \csname LTb\endcsname%%
+      \put(7438,1412){\makebox(0,0)[l]{\strut{}$0$}}%
+      \csname LTb\endcsname%%
+      \put(7438,1791){\makebox(0,0)[l]{\strut{}$100$}}%
+      \csname LTb\endcsname%%
+      \put(7438,2170){\makebox(0,0)[l]{\strut{}$200$}}%
+      \csname LTb\endcsname%%
+      \put(7438,2549){\makebox(0,0)[l]{\strut{}$300$}}%
+      \csname LTb\endcsname%%
+      \put(7438,2928){\makebox(0,0)[l]{\strut{}$400$}}%
+      \csname LTb\endcsname%%
+      \put(7438,3306){\makebox(0,0)[l]{\strut{}$500$}}%
+      \csname LTb\endcsname%%
+      \put(7438,3685){\makebox(0,0)[l]{\strut{}$600$}}%
+    }%
+    \gplgaddtomacro\gplfronttext{%
+      \csname LTb\endcsname%%
+      \put(3780,4115){\makebox(0,0)[r]{\strut{}Skewness}}%
+      \csname LTb\endcsname%%
+      \put(3780,3910){\makebox(0,0)[r]{\strut{}Skewness Error}}%
+      \csname LTb\endcsname%%
+      \put(6661,4115){\makebox(0,0)[r]{\strut{}Excess Kurtosis}}%
+      \csname LTb\endcsname%%
+      \put(6661,3910){\makebox(0,0)[r]{\strut{}Excess Kurtosis Error}}%
+      \csname LTb\endcsname%%
+      \put(546,2170){\rotatebox{-270.00}{\makebox(0,0){\strut{}Skewness/Excess Kurtosis}}}%
+      \csname LTb\endcsname%%
+      \put(8101,2170){\rotatebox{-270.00}{\makebox(0,0){\strut{}Error [\%]}}}%
+      \csname LTb\endcsname%%
+      \put(4309,143){\makebox(0,0){\strut{}$N$}}%
+    }%
+    \gplbacktext
+    \put(0,0){\includegraphics[width={432.00bp},height={216.00bp}]{parameters}}%
+    \gplfronttext
+  \end{picture}%
+\endgroup
diff --git a/report/figures/results/pdf_stats.m b/report/figures/results/pdf_stats.m
new file mode 100644
index 0000000..dde2b25
--- /dev/null
+++ b/report/figures/results/pdf_stats.m
@@ -0,0 +1,21 @@
+function [skew, kurtosis] = pdf_stats(x,p)
+
+    % Define some useful bits
+    dx = x(2) - x(1);
+
+    % The expected value (From LibreTexts 4.1)
+    E = @(x) sum(x .* p) * dx;
+
+    % First moment (mean) (From LibreTexts 4.3)
+    mu = E(x);
+    
+    % Second moment (variance) (From LibreTexts 4.3)
+    sigma = sqrt(E((x - mu).^2));
+
+    % Third moment (skewness) (From LibreTexts 4.4)
+    skew = E(((x - mu)/sigma).^3);
+
+    % Fourth moment (excess kurtosis) (From LibreTexts 4.4)
+    kurtosis = E(((x - mu)/sigma).^4) - 3;
+
+end
diff --git a/report/figures/results/plot.gnu b/report/figures/results/plot.gnu
new file mode 100644
index 0000000..53517b3
--- /dev/null
+++ b/report/figures/results/plot.gnu
@@ -0,0 +1,104 @@
+#set term x11 persist
+set term cairolatex pdf size 6in,6in
+set output "size_pts.tex"
+set linetype 1 lc rgb '#E41A1C' # red
+set linetype 2 lc rgb '#377EB8' # blue
+set linetype 3 lc rgb '#4DAF4A' # green
+
+set key samplen 2
+set grid
+set xlabel "x"
+set ylabel "p(x)"
+
+set style fill transparent solid 0.5 border -1
+
+set multiplot layout 3,2
+
+set title "$N$=1750, $L$=128, $S$=$-$0.169, $K$=0.139"
+plot 'N1750_L128.txt' using 1:2 with boxes lt 2 notitle, \
+  '../tracy-widom-approx/tracy_widom_normal.txt' using 1:2 with lines lt 1 lw 5 notitle
+
+set title "$N$=500, $L$=128, $S$=0.227, $K$=0.753"
+plot 'N500_L128.txt' using 1:2 with boxes lt 2 notitle, \
+  '../tracy-widom-approx/tracy_widom_normal.txt' using 1:2 with lines lt 1 lw 5 notitle
+
+set title "$N$=1750, $L$=2048, $S$=$-$0.044, $K$=0.065"
+plot 'N1750_L2048.txt' using 1:2 with boxes lt 2 notitle, \
+  '../tracy-widom-approx/tracy_widom_normal.txt' using 1:2 with lines lt 1 lw 5 notitle
+
+set title "$N$=500, $L$=2048, $S$=0.187, $K$=0.204"
+plot 'N500_L2048.txt' using 1:2 with boxes lt 2 notitle, \
+  '../tracy-widom-approx/tracy_widom_normal.txt' using 1:2 with lines lt 1 lw 5 notitle
+
+set title "$N$=1750, $L$=4096, $S$=$-$0.045, $K$=0.072"
+plot 'N1750_L4096.txt' using 1:2 with boxes lt 2 notitle, \
+  '../tracy-widom-approx/tracy_widom_normal.txt' using 1:2 with lines lt 1 lw 5 notitle
+
+set title "$N$=500, $L$=4096, $S$=0.140, $K$=0.218"
+plot 'N500_L4096.txt' using 1:2 with boxes lt 2 notitle, \
+  '../tracy-widom-approx/tracy_widom_normal.txt' using 1:2 with lines lt 1 lw 5 notitle
+
+unset multiplot
+
+set terminal cairolatex pdf size 6in,3in
+set output "single.tex"
+unset title
+set lmargin at screen 0.15
+set rmargin at screen 0.85
+
+plot 'N1750_L4096.txt' using 1:2 with boxes title '$N=1750,\, L=4096$' lt 2, \
+    '../tracy-widom-approx/tracy_widom_normal.txt' using 1:2 with lines title 'TW\textsubscript{1}' lt 1, \
+    '../tracy-widom-approx/tracy_widom_normal.txt' using 1:3 with lines title 'N(0,1)' dt 2 lt 3
+    
+
+set terminal cairolatex pdf size 6in, 6in
+set output "maxcomp.tex"
+#set term x11 persist
+set multiplot layout 3,2
+
+unset lmargin
+unset rmargin
+
+set title "$N$=500"
+plot '500_max.txt' using 1:2 with boxes notitle lt 3, \
+  '../tracy-widom-approx/tracy_widom_normal.txt' using 1:2 with lines lt 1 lw 5 notitle
+set title "$N$=750"
+plot '750_max.txt' using 1:2 with boxes notitle lt 3, \
+  '../tracy-widom-approx/tracy_widom_normal.txt' using 1:2 with lines lt 1 lw 5 notitle
+set title "$N$=1000"
+plot '1000_max.txt' using 1:2 with boxes notitle lt 3, \
+  '../tracy-widom-approx/tracy_widom_normal.txt' using 1:2 with lines lt 1 lw 5 notitle
+set title "$N$=1250"
+plot '1250_max.txt' using 1:2 with boxes notitle lt 3, \
+  '../tracy-widom-approx/tracy_widom_normal.txt' using 1:2 with lines lt 1 lw 5 notitle
+set title "$N$=1500"
+plot '1500_max.txt' using 1:2 with boxes notitle lt 3, \
+  '../tracy-widom-approx/tracy_widom_normal.txt' using 1:2 with lines lt 1 lw 5 notitle
+set title "$N$=1750"
+plot '1750_max.txt' using 1:2 with boxes notitle lt 3, \
+  '../tracy-widom-approx/tracy_widom_normal.txt' using 1:2 with lines lt 1 lw 5 notitle
+
+unset multiplot
+
+#set term x11 persist
+set terminal cairolatex pdf size 6in,3in
+set output "parameters.tex"
+unset title
+set lmargin at screen 0.15
+set rmargin at screen 0.85
+
+set arrow from 400,0.2935 to 1800,0.2935 nohead lt 1 dt 3 lw 8
+set arrow from 400,0.1292 to 1800,0.1292 nohead lt 2 dt 3 lw 8
+
+set xlabel "$N$"
+set ylabel "Skewness/Excess Kurtosis"
+
+set y2tics
+set ytics nomirror
+set y2label "Error [\\%]"
+set key outside top center horizontal
+
+plot 'props.txt' using 1:2 with linespoints lt 1 pt 7 ps 0.4 title "Skewness" axes x1y1, \
+    'props.txt' using 1:(($2-0.2935)/(0.2935)*100) with lines lt 1 dt 2 axes x1y2 title "Skewness Error", \
+    'props.txt' using 1:3 with linespoints lt 2 pt 7 ps 0.4 title "Excess Kurtosis" axes x1y1, \
+    'props.txt' using 1:(($3-0.1292)/(0.1292)*100) with lines lt 2 dt 2 axes x1y2 title "Excess Kurtosis Error"
diff --git a/report/figures/results/plot_results.m b/report/figures/results/plot_results.m
new file mode 100644
index 0000000..f6c1713
--- /dev/null
+++ b/report/figures/results/plot_results.m
@@ -0,0 +1,104 @@
+close all;
+clear; clc;
+
+% Get all results files
+files = dir("../../../results/*.out");
+tw = load("../tracy-widom-approx/tracy_widom_normal.txt");
+x = tw(:,1);
+tw1 = tw(:,2);
+nrm = tw(:,3);
+
+N_list = []
+
+%% Make a histogram for every results file
+
+for f = 1:length(files)
+
+    % Get filename
+    filename = files(f).name;
+
+    % Extract N and L
+    tokens = regexp(filename,'eigs_(\d+)_(\d+)\.out', 'tokens');
+    N = str2double(tokens{1}{1});
+    N_list = [N_list, N];
+    L = str2double(tokens{1}{2});
+
+    % Import actual data
+    data = load(strcat("../../../results/",filename));
+    % And clip to just the eigenvalues
+    data = data(:,2);
+
+    % Remove outliers
+    data_clean = data(data > sqrt(N));
+    
+    % Scale and such
+    s_data = (data_clean - 2*sqrt(N))*N^(1/6);
+
+    % Plot them
+    figure(f);
+    title([num2str(N)," matrix size, ",num2str(L)," data points"])
+    hold on
+    [counts, bins] = hist(s_data(s_data > -6), linspace(-6,4,40));
+    bin_width = bins(2)-bins(1);
+    pdf = counts/(sum(counts) * bin_width);
+    bar(bins, pdf);
+    plot(x,tw1);
+    A = [bins', pdf']
+    save("-ascii",["N",num2str(N),"_L",num2str(L),".txt"],"A")
+end
+
+%% Plot largest sample size for each matrix size N
+
+skews = [];
+kurts = [];
+i = 0;
+% Plot totals
+for n = unique(N_list)
+    i = i + 1;
+
+    % Plot all histogram values for this N
+    n_files = dir(strcat("../../../results/eigs_",num2str(n),"_*.out"));
+    n_vals = [];
+    SSE = [];
+    
+    % Get cumulative values
+    for f = 1:length(n_files)
+        filename = n_files(f).name;
+        data = load(strcat("../../../results/",filename));
+        data = data(:,2);
+        data_clean = data(data > sqrt(n));
+        s_data = (data_clean -2*sqrt(n)) * n^(1/6);
+        n_vals = [n_vals, s_data'];
+    end
+
+    % Plot a histogram showing the Gaussian, TW1, and sampled distribution
+    figure(n)
+    title(num2str(length(n_vals)))
+    hold on
+
+    % Generate the histogram
+    [counts, bins] = hist(n_vals(n_vals > -8), linspace(-8,6,70));
+    bin_width = bins(2)-bins(1);
+    pdf = counts/(sum(counts) * bin_width);
+    
+    % And plot it
+    bar(bins,pdf)
+    plot(bins, pdf);
+    plot(x,nrm,'--')
+    plot(x,tw1);
+
+    % Save the histogram to file
+    A = [bins', pdf']
+    save("-ascii",[num2str(n),"_max.txt"], "A")
+
+    % Calculate some statistics 
+    [skews(i),kurts(i)]=pdf_stats(bins,pdf);
+end
+
+figure(n+1)
+plot(unique(N_list),skews)
+hold on
+plot(unique(N_list),kurts)
+set(gca, "xtick", unique(N_list))
+A = [unique(N_list)', skews', kurts']
+save -ascii props.txt A
diff --git a/report/figures/results/props.txt b/report/figures/results/props.txt
new file mode 100644
index 0000000..9c9dc45
--- /dev/null
+++ b/report/figures/results/props.txt
@@ -0,0 +1,6 @@
+ 5.00000000e+02 -8.99346072e-03 8.58814716e-01
+ 7.50000000e+02 -5.74508029e-02 7.66588276e-01
+ 1.00000000e+03 -9.21968249e-02 6.92386742e-01
+ 1.25000000e+03 -1.18510028e-01 6.61642693e-01
+ 1.50000000e+03 -1.18784301e-01 4.56704146e-01
+ 1.75000000e+03 -1.47054483e-01 4.22739414e-01
diff --git a/report/figures/results/single.pdf b/report/figures/results/single.pdf
new file mode 100644
index 0000000..8e3e557
--- /dev/null
+++ b/report/figures/results/single.pdf
diff --git a/report/figures/results/single.tex b/report/figures/results/single.tex
new file mode 100644
index 0000000..e6b85f9
--- /dev/null
+++ b/report/figures/results/single.tex
@@ -0,0 +1,130 @@
+% GNUPLOT: LaTeX picture with Postscript
+\begingroup
+  \makeatletter
+  \providecommand\color[2][]{%
+    \GenericError{(gnuplot) \space\space\space\@spaces}{%
+      Package color not loaded in conjunction with
+      terminal option `colourtext'%
+    }{See the gnuplot documentation for explanation.%
+    }{Either use 'blacktext' in gnuplot or load the package
+      color.sty in LaTeX.}%
+    \renewcommand\color[2][]{}%
+  }%
+  \providecommand\includegraphics[2][]{%
+    \GenericError{(gnuplot) \space\space\space\@spaces}{%
+      Package graphicx or graphics not loaded%
+    }{See the gnuplot documentation for explanation.%
+    }{The gnuplot epslatex terminal needs graphicx.sty or graphics.sty.}%
+    \renewcommand\includegraphics[2][]{}%
+  }%
+  \providecommand\rotatebox[2]{#2}%
+  \@ifundefined{ifGPcolor}{%
+    \newif\ifGPcolor
+    \GPcolortrue
+  }{}%
+  \@ifundefined{ifGPblacktext}{%
+    \newif\ifGPblacktext
+    \GPblacktexttrue
+  }{}%
+  % define a \g@addto@macro without @ in the name:
+  \let\gplgaddtomacro\g@addto@macro
+  % define empty templates for all commands taking text:
+  \gdef\gplbacktext{}%
+  \gdef\gplfronttext{}%
+  \makeatother
+  \ifGPblacktext
+    % no textcolor at all
+    \def\colorrgb#1{}%
+    \def\colorgray#1{}%
+  \else
+    % gray or color?
+    \ifGPcolor
+      \def\colorrgb#1{\color[rgb]{#1}}%
+      \def\colorgray#1{\color[gray]{#1}}%
+      \expandafter\def\csname LTw\endcsname{\color{white}}%
+      \expandafter\def\csname LTb\endcsname{\color{black}}%
+      \expandafter\def\csname LTa\endcsname{\color{black}}%
+      \expandafter\def\csname LT0\endcsname{\color[rgb]{1,0,0}}%
+      \expandafter\def\csname LT1\endcsname{\color[rgb]{0,1,0}}%
+      \expandafter\def\csname LT2\endcsname{\color[rgb]{0,0,1}}%
+      \expandafter\def\csname LT3\endcsname{\color[rgb]{1,0,1}}%
+      \expandafter\def\csname LT4\endcsname{\color[rgb]{0,1,1}}%
+      \expandafter\def\csname LT5\endcsname{\color[rgb]{1,1,0}}%
+      \expandafter\def\csname LT6\endcsname{\color[rgb]{0,0,0}}%
+      \expandafter\def\csname LT7\endcsname{\color[rgb]{1,0.3,0}}%
+      \expandafter\def\csname LT8\endcsname{\color[rgb]{0.5,0.5,0.5}}%
+    \else
+      % gray
+      \def\colorrgb#1{\color{black}}%
+      \def\colorgray#1{\color[gray]{#1}}%
+      \expandafter\def\csname LTw\endcsname{\color{white}}%
+      \expandafter\def\csname LTb\endcsname{\color{black}}%
+      \expandafter\def\csname LTa\endcsname{\color{black}}%
+      \expandafter\def\csname LT0\endcsname{\color{black}}%
+      \expandafter\def\csname LT1\endcsname{\color{black}}%
+      \expandafter\def\csname LT2\endcsname{\color{black}}%
+      \expandafter\def\csname LT3\endcsname{\color{black}}%
+      \expandafter\def\csname LT4\endcsname{\color{black}}%
+      \expandafter\def\csname LT5\endcsname{\color{black}}%
+      \expandafter\def\csname LT6\endcsname{\color{black}}%
+      \expandafter\def\csname LT7\endcsname{\color{black}}%
+      \expandafter\def\csname LT8\endcsname{\color{black}}%
+    \fi
+  \fi
+    \setlength{\unitlength}{0.0500bp}%
+    \ifx\gptboxheight\undefined%
+      \newlength{\gptboxheight}%
+      \newlength{\gptboxwidth}%
+      \newsavebox{\gptboxtext}%
+    \fi%
+    \setlength{\fboxrule}{0.5pt}%
+    \setlength{\fboxsep}{1pt}%
+    \definecolor{tbcol}{rgb}{1,1,1}%
+\begin{picture}(8640.00,4320.00)%
+    \gplgaddtomacro\gplbacktext{%
+      \csname LTb\endcsname%%
+      \put(1181,655){\makebox(0,0)[r]{\strut{}$0$}}%
+      \csname LTb\endcsname%%
+      \put(1181,1146){\makebox(0,0)[r]{\strut{}$0.05$}}%
+      \csname LTb\endcsname%%
+      \put(1181,1638){\makebox(0,0)[r]{\strut{}$0.1$}}%
+      \csname LTb\endcsname%%
+      \put(1181,2129){\makebox(0,0)[r]{\strut{}$0.15$}}%
+      \csname LTb\endcsname%%
+      \put(1181,2620){\makebox(0,0)[r]{\strut{}$0.2$}}%
+      \csname LTb\endcsname%%
+      \put(1181,3112){\makebox(0,0)[r]{\strut{}$0.25$}}%
+      \csname LTb\endcsname%%
+      \put(1181,3603){\makebox(0,0)[r]{\strut{}$0.3$}}%
+      \csname LTb\endcsname%%
+      \put(1181,4095){\makebox(0,0)[r]{\strut{}$0.35$}}%
+      \csname LTb\endcsname%%
+      \put(1293,450){\makebox(0,0){\strut{}$-6$}}%
+      \csname LTb\endcsname%%
+      \put(2499,450){\makebox(0,0){\strut{}$-4$}}%
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+      \put(3706,450){\makebox(0,0){\strut{}$-2$}}%
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+      \put(7326,450){\makebox(0,0){\strut{}$4$}}%
+    }%
+    \gplgaddtomacro\gplfronttext{%
+      \csname LTb\endcsname%%
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+      \put(6685,3706){\makebox(0,0)[r]{\strut{}TW\textsubscript{1}}}%
+      \csname LTb\endcsname%%
+      \put(6685,3501){\makebox(0,0)[r]{\strut{}N(0,1)}}%
+      \csname LTb\endcsname%%
+      \put(546,2375){\rotatebox{-270.00}{\makebox(0,0){\strut{}p(x)}}}%
+      \csname LTb\endcsname%%
+      \put(4309,143){\makebox(0,0){\strut{}x}}%
+    }%
+    \gplbacktext
+    \put(0,0){\includegraphics[width={432.00bp},height={216.00bp}]{single}}%
+    \gplfronttext
+  \end{picture}%
+\endgroup
diff --git a/report/figures/results/size_pts.pdf b/report/figures/results/size_pts.pdf
new file mode 100644
index 0000000..ae402ab
--- /dev/null
+++ b/report/figures/results/size_pts.pdf
diff --git a/report/figures/results/size_pts.tex b/report/figures/results/size_pts.tex
new file mode 100644
index 0000000..d78b1e6
--- /dev/null
+++ b/report/figures/results/size_pts.tex
@@ -0,0 +1,316 @@
+% GNUPLOT: LaTeX picture with Postscript
+\begingroup
+  \makeatletter
+  \providecommand\color[2][]{%
+    \GenericError{(gnuplot) \space\space\space\@spaces}{%
+      Package color not loaded in conjunction with
+      terminal option `colourtext'%
+    }{See the gnuplot documentation for explanation.%
+    }{Either use 'blacktext' in gnuplot or load the package
+      color.sty in LaTeX.}%
+    \renewcommand\color[2][]{}%
+  }%
+  \providecommand\includegraphics[2][]{%
+    \GenericError{(gnuplot) \space\space\space\@spaces}{%
+      Package graphicx or graphics not loaded%
+    }{See the gnuplot documentation for explanation.%
+    }{The gnuplot epslatex terminal needs graphicx.sty or graphics.sty.}%
+    \renewcommand\includegraphics[2][]{}%
+  }%
+  \providecommand\rotatebox[2]{#2}%
+  \@ifundefined{ifGPcolor}{%
+    \newif\ifGPcolor
+    \GPcolortrue
+  }{}%
+  \@ifundefined{ifGPblacktext}{%
+    \newif\ifGPblacktext
+    \GPblacktexttrue
+  }{}%
+  % define a \g@addto@macro without @ in the name:
+  \let\gplgaddtomacro\g@addto@macro
+  % define empty templates for all commands taking text:
+  \gdef\gplbacktext{}%
+  \gdef\gplfronttext{}%
+  \makeatother
+  \ifGPblacktext
+    % no textcolor at all
+    \def\colorrgb#1{}%
+    \def\colorgray#1{}%
+  \else
+    % gray or color?
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+      \def\colorrgb#1{\color[rgb]{#1}}%
+      \def\colorgray#1{\color[gray]{#1}}%
+      \expandafter\def\csname LTw\endcsname{\color{white}}%
+      \expandafter\def\csname LTb\endcsname{\color{black}}%
+      \expandafter\def\csname LTa\endcsname{\color{black}}%
+      \expandafter\def\csname LT0\endcsname{\color[rgb]{1,0,0}}%
+      \expandafter\def\csname LT1\endcsname{\color[rgb]{0,1,0}}%
+      \expandafter\def\csname LT2\endcsname{\color[rgb]{0,0,1}}%
+      \expandafter\def\csname LT3\endcsname{\color[rgb]{1,0,1}}%
+      \expandafter\def\csname LT4\endcsname{\color[rgb]{0,1,1}}%
+      \expandafter\def\csname LT5\endcsname{\color[rgb]{1,1,0}}%
+      \expandafter\def\csname LT6\endcsname{\color[rgb]{0,0,0}}%
+      \expandafter\def\csname LT7\endcsname{\color[rgb]{1,0.3,0}}%
+      \expandafter\def\csname LT8\endcsname{\color[rgb]{0.5,0.5,0.5}}%
+    \else
+      % gray
+      \def\colorrgb#1{\color{black}}%
+      \def\colorgray#1{\color[gray]{#1}}%
+      \expandafter\def\csname LTw\endcsname{\color{white}}%
+      \expandafter\def\csname LTb\endcsname{\color{black}}%
+      \expandafter\def\csname LTa\endcsname{\color{black}}%
+      \expandafter\def\csname LT0\endcsname{\color{black}}%
+      \expandafter\def\csname LT1\endcsname{\color{black}}%
+      \expandafter\def\csname LT2\endcsname{\color{black}}%
+      \expandafter\def\csname LT3\endcsname{\color{black}}%
+      \expandafter\def\csname LT4\endcsname{\color{black}}%
+      \expandafter\def\csname LT5\endcsname{\color{black}}%
+      \expandafter\def\csname LT6\endcsname{\color{black}}%
+      \expandafter\def\csname LT7\endcsname{\color{black}}%
+      \expandafter\def\csname LT8\endcsname{\color{black}}%
+    \fi
+  \fi
+    \setlength{\unitlength}{0.0500bp}%
+    \ifx\gptboxheight\undefined%
+      \newlength{\gptboxheight}%
+      \newlength{\gptboxwidth}%
+      \newsavebox{\gptboxtext}%
+    \fi%
+    \setlength{\fboxrule}{0.5pt}%
+    \setlength{\fboxsep}{1pt}%
+    \definecolor{tbcol}{rgb}{1,1,1}%
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+      \put(2453,2566){\makebox(0,0){\strut{}$N$=1750, $L$=4096, $S$=$-$0.045, $K$=0.072}}%
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+      \put(6459,450){\makebox(0,0){\strut{}$-2$}}%
+      \csname LTb\endcsname%%
+      \put(7067,450){\makebox(0,0){\strut{}$0$}}%
+      \csname LTb\endcsname%%
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+      \csname LTb\endcsname%%
+      \put(8283,450){\makebox(0,0){\strut{}$4$}}%
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+    \gplgaddtomacro\gplfronttext{%
+      \csname LTb\endcsname%%
+      \put(4496,1457){\rotatebox{-270.00}{\makebox(0,0){\strut{}p(x)}}}%
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+      \put(6763,143){\makebox(0,0){\strut{}x}}%
+      \csname LTb\endcsname%%
+      \put(6763,2566){\makebox(0,0){\strut{}$N$=500, $L$=4096, $S$=0.140, $K$=0.218}}%
+    }%
+    \gplbacktext
+    \put(0,0){\includegraphics[width={432.00bp},height={432.00bp}]{size_pts}}%
+    \gplfronttext
+  \end{picture}%
+\endgroup
diff --git a/report/figures/results/sk_tester.m b/report/figures/results/sk_tester.m
new file mode 100644
index 0000000..c396934
--- /dev/null
+++ b/report/figures/results/sk_tester.m
@@ -0,0 +1,14 @@
+close all;
+clear; clc;
+
+N = [1750, 500];
+L = [128, 2048, 4096];
+
+for n = N
+    for l = L
+        % Read the data
+        data = load(["N",num2str(n),"_L",num2str(l),".txt"]);
+        [skew, kurt] = pdf_stats(data(:,1),data(:,2));
+        fprintf("N=%i, L=%i: skew=%f, kurt=%f\n",n,l,skew,kurt)
+    end
+end
diff --git a/report/figures/results/tracywidom_appx.m b/report/figures/results/tracywidom_appx.m
new file mode 100644
index 0000000..bbab815
--- /dev/null
+++ b/report/figures/results/tracywidom_appx.m
@@ -0,0 +1,38 @@
+function [pdftwappx, cdftwappx] = tracywidom_appx(x, i) 
+% 
+%
+% [pdftwappx, cdftwappx]=tracywidom_appx(x, i)
+%
+% SHIFTED GAMMA APPROXIMATION FOR THE TRACY-WIDOM LAWS, by M. Chiani, 2012
+%
+% TW ~ Gamma[k,theta]-alpha   
+%
+% [pdf,cdf]=tracywidom_appx(x,2) gives the approximated pdf(x) and CDF(x) of the TW2   
+%
+% [pdf,cdf]=tracywidom_appx(x,i) for i=1,2,4 gives TW1, TW2, TW4     
+%
+% see paper M.Chiani, "Distribution of the largest eigenvalue for real 
+% Wishart and Gaussian random matrices and a simple approximation for the 
+% Tracy-Widom distribution", submitted 2012, ArXiv
+%
+kappx = [46.44604884387787, 79.6594870666346, 0, 146.0206131050228];   %  K, THETA, ALPHA 
+thetaappx = [0.18605402228279347, 0.10103655775856243, 0, 0.05954454047933292];  
+alphaappx = [9.848007781128567, 9.819607173436484, 0, 11.00161520109004]; 
+cdftwappx = cdfgamma(x+alphaappx(i), thetaappx(i), kappx(i));   % Chiani's appx: Tracy-Widom as a shifted Gamma
+pdftwappx = pdfgamma(x+alphaappx(i), thetaappx(i), kappx(i));   % Chiani's appx: Tracy-Widom as a shifted Gamma 
+end
+function [pdfgamma]=pdfgamma(x, ta, ka) % Utility: PDF of a Gamma 
+ if(x > 0)
+     pdfgamma=1/(gamma(ka)*ta^ka) * x.^(ka - 1) .* exp(-x/ta);
+ else
+     pdfgamma=0 ;
+ end
+end
+function[cdfgamma]=cdfgamma(x, ta, ka)  % Utility: CDF of a Gamma
+ if(x > 0) 
+     cdfgamma=gammainc(x/ta,ka); 
+ else
+     cdfgamma=0; 
+ end
+ 
+end
diff --git a/report/refs.bib b/report/refs.bib
index 5fe7b86..8d580dc 100644
--- a/report/refs.bib
+++ b/report/refs.bib
@@ -25,7 +25,7 @@
     title = {{The QR Algorithm: 50 Years Later its Genesis by John Francis and Vera Kublanovskaya and Subsequent Developments}},
     journal = {IMA Journal of Numerical Analysis},
     author = {Golub, G. and Uhlig, F.},
-    note = {},
+    note = {\href{https://atm.org.uk/write/MediaUploads/Resources/Mid_Plenary_FrancisGolub.pdf}{Available online}.},
     url = {http://dx.doi.org/10.1093/imanum/drp012},
     year = {2009},
     volume = {29},
@@ -39,13 +39,15 @@
 @article{wigner1955Charac,
     issn = {0003-486X},
     title = {{Characteristic Vectors of Bordered Matrices With Infinite Dimensions}},
-    pages = {548},
+    pages = {548-564},
     volume = {62},
+    month = {November},
     number = {3},
     author = {Wigner, Eugene P.},
     year = {1955},
     journal = {The Annals of Mathematics},
     url = {http://dx.doi.org/10.2307/1970079},
+    note = {\href{https://web.williams.edu/Mathematics/sjmiller/public_html/RH/Wigner_CharacteristicVectorsOfBorderedMatricesWithInfDim.pdf}{Available online}.},
     doi = {10.2307/1970079},
     publisher = {JSTOR},
 }
@@ -73,10 +75,15 @@
 @article{chiani2012Distri,
     url = {https://arxiv.org/abs/1209.3394},
     copyright = {arXiv.org perpetual, non-exclusive license},
+    journal = {Journal of Multivariate Analysis},
     keywords = {Information Theory (cs.IT), Statistics Theory (math.ST), FOS: Computer and information sciences, FOS: Computer and information sciences, FOS: Mathematics, FOS: Mathematics},
+    volume = {129},
     publisher = {arXiv},
-    year = {2012},
+    month = {April},
+    note = {\href{https://arxiv.org/pdf/1209.3394}{Available online}.},
+    year = {2014},
     author = {Chiani, Marco},
+    pages = {69-81},
     doi = {10.48550/ARXIV.1209.3394},
     title = {{Distribution of the Largest Eigenvalue for Real Wishart and Gaussian Random Matrices and a Simple Approximation for the Tracy-Widom Distribution}},
 }
@@ -87,4 +94,18 @@
     note = {Chapter 4. \href{https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)}{Available online}.},
     author = {Kyle Siegrist},
     title = {{Probability Theory Probability, Mathematical Statistics, and Stochastic Processes (2021 Edition)}},
+}
+
+@article{konig2005,
+    doi = {10.1214/154957805100000177},
+    journal = {Probability Surveys},
+    volume = {2},
+    year = {2005},
+    author = {Wolfgang Konig},
+    number = {},
+    title = {{Orthogonal Polynomial Ensembles in Probability Theory}},
+    pages = {385-447},
+    issn = {1549-5787},
+    month = {December},
+    note = {\href{https://arxiv.org/pdf/math/0403090}{Available online}.},
 }
\ No newline at end of file
diff --git a/report/report.tex b/report/report.tex
index 97dd324..00a8730 100644
--- a/report/report.tex
+++ b/report/report.tex
@@ -6,7 +6,7 @@
 
 \usepackage{xcolor}
 \usepackage{graphicx}
-\graphicspath{{./figures/} {./figures/tracy-widom-approx/}}
+\graphicspath{{./figures/} {./figures/tracy-widom-approx/}, {./figures/results/}}
 \usepackage{amsmath}
 \usepackage{booktabs}
 \usepackage{longtable}
@@ -71,7 +71,7 @@ Compared with a normal distribution, it is also difficult to distinguish between
     \caption{The TW\textsubscript{1} distribution compared with a normal distribution of arbitrary mean and standard deviation. Note that while the peaks are centered, the Tracy-Widom has a `fatter' tail on the $+x$ side and a slimmer one on the $-x$ side. Both distributions are quite similar visually, so this asymmetry will play an important role in quantifying their differences.}
     \label{fig:tracy-widom-vs-normal}
 \end{figure}
-Because of their similarities, a smart approach to validating the generation of a Tracy-Widom distribution should include some type of quantifiable, statistical assurance that a normal distribution was not generated.
+Note that the Tracy-Widoms plotted above have no closed-form expression and must be approximated, typically by using a scaled Gamma function \cite{chiani2012Distri}, or more cleverly by downloading some else's finished approximation off of the \textsc{Matlab} file exchange \cite{marco2013}. Because the distributions are for visual purposes only, a smart approach to validating the generation of a Tracy-Widom should include some type of quantifiable, statistical assurance not based on the simple value of the distribution itself.
 
 \subsection{Statistical Differentiation}
 There are various ways to go about discerning a Tracy-Widom distribution from a normal one. Perhaps the most elegant is to exploit some basic differences in properties between the two. As mentioned earlier, the Tracy-Widom has a fatter side and a slimmer side, which makes it asymmetric. This is not the case with the normal distribution which is identical on either side of the mean value $\mu$. It turns out that the symmetry of a distribution is readily quantifiable using existing statistical tools, and as such that will be the backbone of the statistical analysis. First consider the \emph{expected value} of a probability density function (PDF) $p(x)$, defined as \cite{siegrist2021}:
@@ -95,7 +95,7 @@ Lastly, the 4th moment is the kurtosis of the distribution. Kurtosis comes from
 \begin{equation}
     \mathrm{Excess\ Kurtosis} = E\left[ \frac{(x - \mu)^4}{\sigma} \right] - 3
 \end{equation}
-These tests, while perhaps simple, will do a fine job in the strict context of differentiating a Tracy-Widom distribution from a normal distribution. In the case of the normal distribution, the skewness and excess kurtosis are both zero. For comparison, the approximate Tracy-Widom has a skewness of $\approx 0.2935$ and an excess kurtosis of $\approx 0.1292$.This was calculated using an Octave script\footnote{\href{https://git.hhmoore.ca/mcsc-6030g/p3-tracy-widom/tree/report/figures/tracy-widom-approx/plot_tracy_widom.m}{Available online} as part of the Git repository for the project.} that sampled 50,000 bins in the range $[-9,9]$ for both distributions. This also verified the calculations as it properly reconstructed the exact mean and standard deviation for the normal distribution sampled from its the closed-form equation.
+These tests, while perhaps simple, will do a fine job in the strict context of differentiating a Tracy-Widom distribution from a normal distribution. In the case of the normal distribution, the skewness and excess kurtosis are both zero. For comparison, the approximate Tracy-Widom has a skewness of $\approx 0.2935$ and an excess kurtosis of $\approx 0.1292$.This was calculated using an Octave script\footnote{\href{https://git.hhmoore.ca/mcsc-6030g/p3-tracy-widom/tree/report/figures/tracy-widom-approx/plot_tracy_widom.m}{Available online} as part of the Git repository for the project.} that sampled 50,000 bins in the range $[-9,9]$ for both distributions. Performing calculations with (1,3,4,5) reconstructed the mean and standard deviation for the normal distribution sampled from its the closed-form equations down to an error of $\sim10^{-5}$.
 
 \section{Numerical Methods}
 The natural extension of defining the prerequisite statistical tests to confirm a Tracy-Widom is to actually sample a Tracy-Widom. The issue with this, however, is that it can be quite computationally demanding. The Tracy-Widom is quite similar to the normal except for near the very tails, meaning that most samples (occurring near the peaks) will show agreement between the two. Additionally, it is theorized that the dimension of the matrix $N$ has to be on the order of thousands to get a very Tracy-Widom distribution. The combination of these two points motivates finding a convenient way to calculate the maximum eigenvalue of a large matrix repeated such that thousands of calculations can be performed in a meaningfully short amount of time.
@@ -258,6 +258,61 @@ So far the structure has involved a minimal amount of math. This is only because
 These algorithms compose the entirety of the program for reconstructing the Tracy-Widom distributions discussed earlier.
 
 \section{Results}
+Before the eigenvalues can be directly compared to a Tracy-Widom they must be scaled to match the existing distribution. It is possible to perform the scaling empirically, but a closed-form solution is available that makes things easier \cite{konig2005}:
+\begin{equation}
+    \lambda^\prime = \left(\lambda - 2\sqrt{N} \right)\times N^{1/6}
+\end{equation}
+This specific transform is derived from the Wigner semicircular distribution mentioned earlier in section 2. Although it describes the global distribution, taking the extrema yields a bound for the maximum eigenvalue.
+
+\subsection{Tracy-Widom Convergence}
+Following some software issues related to the OneAPI suite, the author was unable to produce enough samples in time to complete the report. However, a moment is appropriate to thank all of the teammates for being kind enough to share their data for analysis in this report. The following numbers are from Joe's runs but all of the analysis was done individually. Samples here are presented with respect to their matrix dimension $N$, number of samples $L$, skewness $S$ and excess kurtosis $K$. Taking, for example, different sample sizes for $N=1750,500$ yields a half-dozen examples of a clear trend emerging where the histograms tend towards a normal-\emph{ish} shape, shown in Figure \ref{fig:randoms}:
+
+\begin{figure}[H]
+    \centering
+    \input{./figures/results/size_pts.tex}
+    \caption{Progression of the Tracy-Widom convergence as a function of matrix size ($N$) and sample size ($L$). Also included are the skewness ($S$) and excess kurtosis ($K$) for each sample. An increase in both sample size and matrix size yields a distribution comparable with but not necessarily closer to the Tracy-Widom reference. Asymmetry is noted especially for the cases with $N=1750$, but with an incorrect sign that implies a fatter tail in the $-x$ direction.}
+    \label{fig:randoms}
+\end{figure}
+
+Despite the visuals, however, the metrics tell a bleaker story. Recall that the Tracy-Widom is authentically constructed with a fatter tail on the right and thinner on the left, which seems to be the opposite of what is presented above. This is also reflected in the skewness assuming a negative value for the curves associated with $N=1750$, implying that the distribution is more favourable to the left of the mean value. As the sample size increases, the shape of the distribution becomes more refined and tends to that of the expected profile, which seems intuitive. It is possible to consider the maximum number of samples available for each matrix size $N$ to confirm whether or not this can yield better results, shown in Figure \ref{fig:big-ones}:
+
+
+\begin{figure}[H]
+    \centering
+    \input{./figures/results/maxcomp.tex}
+    \caption{Reconstructed distributions from the maximum number of samples $L_{max}$ for each size $N$. Table \ref{tbl:results} provides more information on the behaviour of skewness and excess kurtosis for the high-sample runs. Qualitatively, under filling seems prevalent at larger $N$ and a strong $-x$ tail is visible.}
+    \label{fig:big-ones}
+\end{figure}
+
+\begin{table}[H]
+    \centering
+    \caption{Parameters of the Larger-$L$ Distributions for Various $N$}
+    \begin{tabular}{cccc}
+        \toprule
+        N & L & Skewness & Excess Kurtosis \\
+        \midrule
+        500 & 50,104 & $-8.993\times10^{-3}$ & $8.588\times10^{-1}$ \\
+        750 & 55,397 & $-5.745\times10^{-2}$ & $7.666\times10^{-1}$ \\
+        1,000 & 55,536 & $-9.220\times10^{-2}$ & $6.924\times10^{-1}$ \\ 
+        1,250 & 55,726 & $-1.185\times10^{-1}$ & $6.616\times10^{-1}$ \\
+        1,500 & 55,948 & $-1.188\times10^{-1}$ & $4.567\times10^{-1}$ \\
+        1,750 & 56,097 & $-1.471\times10^{-1}$ & $4.227\times10^{-1}$ \\
+        \bottomrule
+    \end{tabular}
+    \label{tbl:results}
+\end{table}
+
+
+\begin{figure}[H]
+    \centering
+    \input{./figures/results/parameters.tex}
+    \caption{Reconstructed distributions from the maximum number of samples $L_{max}$ for each size $N$. Table \ref{tbl:results} provides more information on the behaviour of skewness and excess kurtosis for the high-sample runs. Qualitatively, under filling seems prevalent at larger $N$ and a strong $-x$ tail is visible.}
+    \label{fig:errors}
+\end{figure}
+
+\subsection{Wall Times}
+
+
 
 \bibliographystyle{ieeetr}
 \bibliography{refs.bib}