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1 files changed, 552 insertions, 0 deletions

diff --git a/src/Homework3/Homework3.tex b/src/Homework3/Homework3.tex
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+\documentclass[11pt]{article}
+
+    \usepackage[breakable]{tcolorbox}
+    \usepackage{parskip} % Stop auto-indenting (to mimic markdown behaviour)
+    
+    \usepackage{iftex}
+    \ifPDFTeX
+        \usepackage[T1]{fontenc}
+        \usepackage{mathpazo}
+    \else
+        \usepackage{fontspec}
+    \fi
+
+    % Basic figure setup, for now with no caption control since it's done
+    % automatically by Pandoc (which extracts ![](path) syntax from Markdown).
+    \usepackage{graphicx}
+    % Maintain compatibility with old templates. Remove in nbconvert 6.0
+    \let\Oldincludegraphics\includegraphics
+    % Ensure that by default, figures have no caption (until we provide a
+    % proper Figure object with a Caption API and a way to capture that
+    % in the conversion process - todo).
+    \usepackage{caption}
+    \DeclareCaptionFormat{nocaption}{}
+    \captionsetup{format=nocaption,aboveskip=0pt,belowskip=0pt}
+
+    \usepackage[Export]{adjustbox} % Used to constrain images to a maximum size
+    \adjustboxset{max size={0.9\linewidth}{0.9\paperheight}}
+    \usepackage{float}
+    \floatplacement{figure}{H} % forces figures to be placed at the correct location
+    \usepackage{xcolor} % Allow colors to be defined
+    \usepackage{enumerate} % Needed for markdown enumerations to work
+    \usepackage{geometry} % Used to adjust the document margins
+    \usepackage{amsmath} % Equations
+    \usepackage{amssymb} % Equations
+    \usepackage{textcomp} % defines textquotesingle
+    % Hack from http://tex.stackexchange.com/a/47451/13684:
+    \AtBeginDocument{%
+        \def\PYZsq{\textquotesingle}% Upright quotes in Pygmentized code
+    }
+    \usepackage{upquote} % Upright quotes for verbatim code
+    \usepackage{eurosym} % defines \euro
+    \usepackage[mathletters]{ucs} % Extended unicode (utf-8) support
+    \usepackage{fancyvrb} % verbatim replacement that allows latex
+    \usepackage{grffile} % extends the file name processing of package graphics 
+                         % to support a larger range
+    \makeatletter % fix for grffile with XeLaTeX
+    \def\Gread@@xetex#1{%
+      \IfFileExists{"\Gin@base".bb}%
+      {\Gread@eps{\Gin@base.bb}}%
+      {\Gread@@xetex@aux#1}%
+    }
+    \makeatother
+
+    % The hyperref package gives us a pdf with properly built
+    % internal navigation ('pdf bookmarks' for the table of contents,
+    % internal cross-reference links, web links for URLs, etc.)
+    \usepackage{hyperref}
+    % The default LaTeX title has an obnoxious amount of whitespace. By default,
+    % titling removes some of it. It also provides customization options.
+    \usepackage{titling}
+    \usepackage{longtable} % longtable support required by pandoc >1.10
+    \usepackage{booktabs}  % table support for pandoc > 1.12.2
+    \usepackage[inline]{enumitem} % IRkernel/repr support (it uses the enumerate* environment)
+    \usepackage[normalem]{ulem} % ulem is needed to support strikethroughs (\sout)
+                                % normalem makes italics be italics, not underlines
+    \usepackage{mathrsfs}
+    
+
+    
+    % Colors for the hyperref package
+    \definecolor{urlcolor}{rgb}{0,.145,.698}
+    \definecolor{linkcolor}{rgb}{.71,0.21,0.01}
+    \definecolor{citecolor}{rgb}{.12,.54,.11}
+
+    % ANSI colors
+    \definecolor{ansi-black}{HTML}{3E424D}
+    \definecolor{ansi-black-intense}{HTML}{282C36}
+    \definecolor{ansi-red}{HTML}{E75C58}
+    \definecolor{ansi-red-intense}{HTML}{B22B31}
+    \definecolor{ansi-green}{HTML}{00A250}
+    \definecolor{ansi-green-intense}{HTML}{007427}
+    \definecolor{ansi-yellow}{HTML}{DDB62B}
+    \definecolor{ansi-yellow-intense}{HTML}{B27D12}
+    \definecolor{ansi-blue}{HTML}{208FFB}
+    \definecolor{ansi-blue-intense}{HTML}{0065CA}
+    \definecolor{ansi-magenta}{HTML}{D160C4}
+    \definecolor{ansi-magenta-intense}{HTML}{A03196}
+    \definecolor{ansi-cyan}{HTML}{60C6C8}
+    \definecolor{ansi-cyan-intense}{HTML}{258F8F}
+    \definecolor{ansi-white}{HTML}{C5C1B4}
+    \definecolor{ansi-white-intense}{HTML}{A1A6B2}
+    \definecolor{ansi-default-inverse-fg}{HTML}{FFFFFF}
+    \definecolor{ansi-default-inverse-bg}{HTML}{000000}
+
+    % commands and environments needed by pandoc snippets
+    % extracted from the output of `pandoc -s`
+    \providecommand{\tightlist}{%
+      \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
+    \DefineVerbatimEnvironment{Highlighting}{Verbatim}{commandchars=\\\{\}}
+    % Add ',fontsize=\small' for more characters per line
+    \newenvironment{Shaded}{}{}
+    \newcommand{\KeywordTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}}
+    \newcommand{\DataTypeTok}[1]{\textcolor[rgb]{0.56,0.13,0.00}{{#1}}}
+    \newcommand{\DecValTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
+    \newcommand{\BaseNTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
+    \newcommand{\FloatTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
+    \newcommand{\CharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
+    \newcommand{\StringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
+    \newcommand{\CommentTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textit{{#1}}}}
+    \newcommand{\OtherTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{{#1}}}
+    \newcommand{\AlertTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}}
+    \newcommand{\FunctionTok}[1]{\textcolor[rgb]{0.02,0.16,0.49}{{#1}}}
+    \newcommand{\RegionMarkerTok}[1]{{#1}}
+    \newcommand{\ErrorTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}}
+    \newcommand{\NormalTok}[1]{{#1}}
+    
+    % Additional commands for more recent versions of Pandoc
+    \newcommand{\ConstantTok}[1]{\textcolor[rgb]{0.53,0.00,0.00}{{#1}}}
+    \newcommand{\SpecialCharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
+    \newcommand{\VerbatimStringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
+    \newcommand{\SpecialStringTok}[1]{\textcolor[rgb]{0.73,0.40,0.53}{{#1}}}
+    \newcommand{\ImportTok}[1]{{#1}}
+    \newcommand{\DocumentationTok}[1]{\textcolor[rgb]{0.73,0.13,0.13}{\textit{{#1}}}}
+    \newcommand{\AnnotationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
+    \newcommand{\CommentVarTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
+    \newcommand{\VariableTok}[1]{\textcolor[rgb]{0.10,0.09,0.49}{{#1}}}
+    \newcommand{\ControlFlowTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}}
+    \newcommand{\OperatorTok}[1]{\textcolor[rgb]{0.40,0.40,0.40}{{#1}}}
+    \newcommand{\BuiltInTok}[1]{{#1}}
+    \newcommand{\ExtensionTok}[1]{{#1}}
+    \newcommand{\PreprocessorTok}[1]{\textcolor[rgb]{0.74,0.48,0.00}{{#1}}}
+    \newcommand{\AttributeTok}[1]{\textcolor[rgb]{0.49,0.56,0.16}{{#1}}}
+    \newcommand{\InformationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
+    \newcommand{\WarningTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
+    
+    
+    % Define a nice break command that doesn't care if a line doesn't already
+    % exist.
+    \def\br{\hspace*{\fill} \\* }
+    % Math Jax compatibility definitions
+    \def\gt{>}
+    \def\lt{<}
+    \let\Oldtex\TeX
+    \let\Oldlatex\LaTeX
+    \renewcommand{\TeX}{\textrm{\Oldtex}}
+    \renewcommand{\LaTeX}{\textrm{\Oldlatex}}
+    % Document parameters
+    % Document title
+    \title{Mathematical Software - Homework 3}
+    \date{Deadline: Sunday, May 9th}
+    
+    
+    
+    
+    
+% Pygments definitions
+\makeatletter
+\def\PY@reset{\let\PY@it=\relax \let\PY@bf=\relax%
+    \let\PY@ul=\relax \let\PY@tc=\relax%
+    \let\PY@bc=\relax \let\PY@ff=\relax}
+\def\PY@tok#1{\csname PY@tok@#1\endcsname}
+\def\PY@toks#1+{\ifx\relax#1\empty\else%
+    \PY@tok{#1}\expandafter\PY@toks\fi}
+\def\PY@do#1{\PY@bc{\PY@tc{\PY@ul{%
+    \PY@it{\PY@bf{\PY@ff{#1}}}}}}}
+\def\PY#1#2{\PY@reset\PY@toks#1+\relax+\PY@do{#2}}
+
+\expandafter\def\csname PY@tok@w\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.73,0.73}{##1}}}
+\expandafter\def\csname PY@tok@c\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
+\expandafter\def\csname PY@tok@cp\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.74,0.48,0.00}{##1}}}
+\expandafter\def\csname PY@tok@k\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
+\expandafter\def\csname PY@tok@kp\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
+\expandafter\def\csname PY@tok@kt\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.69,0.00,0.25}{##1}}}
+\expandafter\def\csname PY@tok@o\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
+\expandafter\def\csname PY@tok@ow\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.67,0.13,1.00}{##1}}}
+\expandafter\def\csname PY@tok@nb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
+\expandafter\def\csname PY@tok@nf\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}}
+\expandafter\def\csname PY@tok@nc\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}}
+\expandafter\def\csname PY@tok@nn\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}}
+\expandafter\def\csname PY@tok@ne\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.82,0.25,0.23}{##1}}}
+\expandafter\def\csname PY@tok@nv\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}}
+\expandafter\def\csname PY@tok@no\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.53,0.00,0.00}{##1}}}
+\expandafter\def\csname PY@tok@nl\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.63,0.63,0.00}{##1}}}
+\expandafter\def\csname PY@tok@ni\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.60,0.60,0.60}{##1}}}
+\expandafter\def\csname PY@tok@na\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.49,0.56,0.16}{##1}}}
+\expandafter\def\csname PY@tok@nt\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
+\expandafter\def\csname PY@tok@nd\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.67,0.13,1.00}{##1}}}
+\expandafter\def\csname PY@tok@s\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
+\expandafter\def\csname PY@tok@sd\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
+\expandafter\def\csname PY@tok@si\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.53}{##1}}}
+\expandafter\def\csname PY@tok@se\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.13}{##1}}}
+\expandafter\def\csname PY@tok@sr\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.53}{##1}}}
+\expandafter\def\csname PY@tok@ss\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}}
+\expandafter\def\csname PY@tok@sx\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
+\expandafter\def\csname PY@tok@m\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
+\expandafter\def\csname PY@tok@gh\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,0.50}{##1}}}
+\expandafter\def\csname PY@tok@gu\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.50,0.00,0.50}{##1}}}
+\expandafter\def\csname PY@tok@gd\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.63,0.00,0.00}{##1}}}
+\expandafter\def\csname PY@tok@gi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.63,0.00}{##1}}}
+\expandafter\def\csname PY@tok@gr\endcsname{\def\PY@tc##1{\textcolor[rgb]{1.00,0.00,0.00}{##1}}}
+\expandafter\def\csname PY@tok@ge\endcsname{\let\PY@it=\textit}
+\expandafter\def\csname PY@tok@gs\endcsname{\let\PY@bf=\textbf}
+\expandafter\def\csname PY@tok@gp\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,0.50}{##1}}}
+\expandafter\def\csname PY@tok@go\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.53,0.53,0.53}{##1}}}
+\expandafter\def\csname PY@tok@gt\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.27,0.87}{##1}}}
+\expandafter\def\csname PY@tok@err\endcsname{\def\PY@bc##1{\setlength{\fboxsep}{0pt}\fcolorbox[rgb]{1.00,0.00,0.00}{1,1,1}{\strut ##1}}}
+\expandafter\def\csname PY@tok@kc\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
+\expandafter\def\csname PY@tok@kd\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
+\expandafter\def\csname PY@tok@kn\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
+\expandafter\def\csname PY@tok@kr\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
+\expandafter\def\csname PY@tok@bp\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
+\expandafter\def\csname PY@tok@fm\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}}
+\expandafter\def\csname PY@tok@vc\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}}
+\expandafter\def\csname PY@tok@vg\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}}
+\expandafter\def\csname PY@tok@vi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}}
+\expandafter\def\csname PY@tok@vm\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}}
+\expandafter\def\csname PY@tok@sa\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
+\expandafter\def\csname PY@tok@sb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
+\expandafter\def\csname PY@tok@sc\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
+\expandafter\def\csname PY@tok@dl\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
+\expandafter\def\csname PY@tok@s2\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
+\expandafter\def\csname PY@tok@sh\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
+\expandafter\def\csname PY@tok@s1\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
+\expandafter\def\csname PY@tok@mb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
+\expandafter\def\csname PY@tok@mf\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
+\expandafter\def\csname PY@tok@mh\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
+\expandafter\def\csname PY@tok@mi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
+\expandafter\def\csname PY@tok@il\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
+\expandafter\def\csname PY@tok@mo\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
+\expandafter\def\csname PY@tok@ch\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
+\expandafter\def\csname PY@tok@cm\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
+\expandafter\def\csname PY@tok@cpf\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
+\expandafter\def\csname PY@tok@c1\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
+\expandafter\def\csname PY@tok@cs\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
+
+\def\PYZbs{\char`\\}
+\def\PYZus{\char`\_}
+\def\PYZob{\char`\{}
+\def\PYZcb{\char`\}}
+\def\PYZca{\char`\^}
+\def\PYZam{\char`\&}
+\def\PYZlt{\char`\<}
+\def\PYZgt{\char`\>}
+\def\PYZsh{\char`\#}
+\def\PYZpc{\char`\%}
+\def\PYZdl{\char`\$}
+\def\PYZhy{\char`\-}
+\def\PYZsq{\char`\'}
+\def\PYZdq{\char`\"}
+\def\PYZti{\char`\~}
+% for compatibility with earlier versions
+\def\PYZat{@}
+\def\PYZlb{[}
+\def\PYZrb{]}
+\makeatother
+
+
+    % For linebreaks inside Verbatim environment from package fancyvrb. 
+    \makeatletter
+        \newbox\Wrappedcontinuationbox 
+        \newbox\Wrappedvisiblespacebox 
+        \newcommand*\Wrappedvisiblespace {\textcolor{red}{\textvisiblespace}} 
+        \newcommand*\Wrappedcontinuationsymbol {\textcolor{red}{\llap{\tiny$\m@th\hookrightarrow$}}} 
+        \newcommand*\Wrappedcontinuationindent {3ex } 
+        \newcommand*\Wrappedafterbreak {\kern\Wrappedcontinuationindent\copy\Wrappedcontinuationbox} 
+        % Take advantage of the already applied Pygments mark-up to insert 
+        % potential linebreaks for TeX processing. 
+        %        {, <, #, %, $, ' and ": go to next line. 
+        %        _, }, ^, &, >, - and ~: stay at end of broken line. 
+        % Use of \textquotesingle for straight quote. 
+        \newcommand*\Wrappedbreaksatspecials {% 
+            \def\PYGZus{\discretionary{\char`\_}{\Wrappedafterbreak}{\char`\_}}% 
+            \def\PYGZob{\discretionary{}{\Wrappedafterbreak\char`\{}{\char`\{}}% 
+            \def\PYGZcb{\discretionary{\char`\}}{\Wrappedafterbreak}{\char`\}}}% 
+            \def\PYGZca{\discretionary{\char`\^}{\Wrappedafterbreak}{\char`\^}}% 
+            \def\PYGZam{\discretionary{\char`\&}{\Wrappedafterbreak}{\char`\&}}% 
+            \def\PYGZlt{\discretionary{}{\Wrappedafterbreak\char`\<}{\char`\<}}% 
+            \def\PYGZgt{\discretionary{\char`\>}{\Wrappedafterbreak}{\char`\>}}% 
+            \def\PYGZsh{\discretionary{}{\Wrappedafterbreak\char`\#}{\char`\#}}% 
+            \def\PYGZpc{\discretionary{}{\Wrappedafterbreak\char`\%}{\char`\%}}% 
+            \def\PYGZdl{\discretionary{}{\Wrappedafterbreak\char`\$}{\char`\$}}% 
+            \def\PYGZhy{\discretionary{\char`\-}{\Wrappedafterbreak}{\char`\-}}% 
+            \def\PYGZsq{\discretionary{}{\Wrappedafterbreak\textquotesingle}{\textquotesingle}}% 
+            \def\PYGZdq{\discretionary{}{\Wrappedafterbreak\char`\"}{\char`\"}}% 
+            \def\PYGZti{\discretionary{\char`\~}{\Wrappedafterbreak}{\char`\~}}% 
+        } 
+        % Some characters . , ; ? ! / are not pygmentized. 
+        % This macro makes them "active" and they will insert potential linebreaks 
+        \newcommand*\Wrappedbreaksatpunct {% 
+            \lccode`\~`\.\lowercase{\def~}{\discretionary{\hbox{\char`\.}}{\Wrappedafterbreak}{\hbox{\char`\.}}}% 
+            \lccode`\~`\,\lowercase{\def~}{\discretionary{\hbox{\char`\,}}{\Wrappedafterbreak}{\hbox{\char`\,}}}% 
+            \lccode`\~`\;\lowercase{\def~}{\discretionary{\hbox{\char`\;}}{\Wrappedafterbreak}{\hbox{\char`\;}}}% 
+            \lccode`\~`\:\lowercase{\def~}{\discretionary{\hbox{\char`\:}}{\Wrappedafterbreak}{\hbox{\char`\:}}}% 
+            \lccode`\~`\?\lowercase{\def~}{\discretionary{\hbox{\char`\?}}{\Wrappedafterbreak}{\hbox{\char`\?}}}% 
+            \lccode`\~`\!\lowercase{\def~}{\discretionary{\hbox{\char`\!}}{\Wrappedafterbreak}{\hbox{\char`\!}}}% 
+            \lccode`\~`\/\lowercase{\def~}{\discretionary{\hbox{\char`\/}}{\Wrappedafterbreak}{\hbox{\char`\/}}}% 
+            \catcode`\.\active
+            \catcode`\,\active 
+            \catcode`\;\active
+            \catcode`\:\active
+            \catcode`\?\active
+            \catcode`\!\active
+            \catcode`\/\active 
+            \lccode`\~`\~       
+        }
+    \makeatother
+
+    \let\OriginalVerbatim=\Verbatim
+    \makeatletter
+    \renewcommand{\Verbatim}[1][1]{%
+        %\parskip\z@skip
+        \sbox\Wrappedcontinuationbox {\Wrappedcontinuationsymbol}%
+        \sbox\Wrappedvisiblespacebox {\FV@SetupFont\Wrappedvisiblespace}%
+        \def\FancyVerbFormatLine ##1{\hsize\linewidth
+            \vtop{\raggedright\hyphenpenalty\z@\exhyphenpenalty\z@
+                \doublehyphendemerits\z@\finalhyphendemerits\z@
+                \strut ##1\strut}%
+        }%
+        % If the linebreak is at a space, the latter will be displayed as visible
+        % space at end of first line, and a continuation symbol starts next line.
+        % Stretch/shrink are however usually zero for typewriter font.
+        \def\FV@Space {%
+            \nobreak\hskip\z@ plus\fontdimen3\font minus\fontdimen4\font
+            \discretionary{\copy\Wrappedvisiblespacebox}{\Wrappedafterbreak}
+            {\kern\fontdimen2\font}%
+        }%
+        
+        % Allow breaks at special characters using \PYG... macros.
+        \Wrappedbreaksatspecials
+        % Breaks at punctuation characters . , ; ? ! and / need catcode=\active         
+        \OriginalVerbatim[#1,codes*=\Wrappedbreaksatpunct]%
+    }
+    \makeatother
+
+    % Exact colors from NB
+    \definecolor{incolor}{HTML}{303F9F}
+    \definecolor{outcolor}{HTML}{D84315}
+    \definecolor{cellborder}{HTML}{CFCFCF}
+    \definecolor{cellbackground}{HTML}{F7F7F7}
+    
+    % prompt
+    \makeatletter
+    \newcommand{\boxspacing}{\kern\kvtcb@left@rule\kern\kvtcb@boxsep}
+    \makeatother
+    \newcommand{\prompt}[4]{
+        \ttfamily\llap{{\color{#2}[#3]:\hspace{3pt}#4}}\vspace{-\baselineskip}
+    }
+    
+
+    
+    % Prevent overflowing lines due to hard-to-break entities
+    \sloppy 
+    % Setup hyperref package
+    \hypersetup{
+      breaklinks=true,  % so long urls are correctly broken across lines
+      colorlinks=true,
+      urlcolor=urlcolor,
+      linkcolor=linkcolor,
+      citecolor=citecolor,
+      }
+    % Slightly bigger margins than the latex defaults
+    
+    \geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
+    
+    
+
+\begin{document}
+    
+    \maketitle
+    
+    
+
+    \begin{center}
+    \emph{For this exercise you should have received this text in .ipynb
+format. Complete the exercises by modifying this file, and submit the
+modified version}
+        \end{center}
+
+        \vspace{1cm}
+    \textbf{Exercise 1}
+
+Use SageMath to solve the following problems:
+
+\begin{enumerate}
+\def\labelenumi{(\alph{enumi})}
+\item
+  Find the roots of the following polynomial over \(\mathbb Q\):
+  \begin{align*}
+  p = 4 x^{7} + 4 x^{6} + 3 x^{5} - 13 x^{4} - 13 x^{3} - 9 x^{2} + 3 x + 3 \in \mathbb Q[x]
+  \end{align*}
+\item
+  Find the roots of the same polynomial \(p\) over \(\mathbb R\) and
+  over \(\mathbb C\).
+\item
+  Find the determinant, the trace and the characteristic polynomial of
+  the following matrix: \begin{align*}
+  A=\left(\begin{array}{rrrr}
+  -1 & 1 & -1 & 0 \\
+  1 & \frac{1}{2} & 1 & 0 \\
+  \frac{1}{2} & -\frac{1}{2} & -2 & 1 \\
+  0 & 0 & 1 & 1
+  \end{array}\right)
+  \end{align*}
+\item
+  Find a solution to the linear system \(A\mathbf x =\mathbf v\), where
+  \(A\) is the matrix above and \(\mathbf v=(1, 2, 3, 4)\).
+\end{enumerate}
+
+Write your code in the cell below.
+
+    \begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
+\prompt{In}{incolor}{ }{\boxspacing}
+\begin{Verbatim}[commandchars=\\\{\}]
+
+\end{Verbatim}
+\end{tcolorbox}
+
+        \vspace{1cm}
+    \textbf{Exercise 2}
+
+After exchanging messages with the RSA protocol seen in class, Alice and
+Bob decide to meet and play their favorite game: flip a coin. They like
+this game very much because it does not take long to set it up and they
+have exactly the same chances of winning.
+
+Unfortunately, due to the COVID-19 pandemic they cannot meet in person,
+and despite being good friends they don't trust each other enough to
+play this game via Webex call. Luckily, Alice is an expert in
+cryptography and she knows how to play this game using the Chinese
+remainder theorem.
+
+The game plays out as follows:
+
+\begin{itemize}
+\item[(A1)] Alice picks two large prime numbers \(p\) and \(q\), she computes
+\(n=pq\) and sends \(n\) to Bob, keeping \(p\) and \(q\) secret.
+
+\item[(B1)] Bob picks a random number \(a\) with \(1<a<n\) and \(\gcd(a,n)=1\),
+computes \(b=a^2\mod n\) and sends \(b\) to Alice, keeping \(a\) secret.
+
+\item[(A2)] Alice computes two numbers \(x\) and \(y\) such that
+\(x^2\equiv b\pmod p\) and \(y^2\equiv b\pmod q\) and she uses the
+Chinese remainder theorem to compute a number \(z\) such that
+\(z\equiv x\pmod p\) and \(z\equiv y\pmod q\), so that
+\(z^2\equiv b\pmod n\). Then she sends \(z\) to Bob.
+
+Since \(n\) is the product of two primes, there are \(4\) possible
+square roots of \(b\) modulo \(n\), corresponding to the solutions of
+the four systems of congruences (one for each possible combination of
+\(\pm\)) \begin{align*}\begin{cases}
+z\equiv \pm x\pmod p\\
+z \equiv \pm y\pmod q
+\end{cases}\end{align*}
+
+One of those solutions is \(a\) and another is \(-a\), and Bob knows
+them. Alice is picking one of the \(4\) possible roots at random (she
+chooses between \(x\) and \(-x\) and between \(y\) and \(-y\)), so she
+has \(50\%\) chance of picking one that Bob already knows. This
+corresponds to Alice flipping a coin, and she wins if she picks
+\(\pm a\):
+
+\item[(B2)] If \(z\equiv\pm a\pmod n\), Bob declares to have lost. Otherwise,
+Bob claims to have won, and as proof he produces one prime factor of
+\(n\) by computing \(g=\gcd(n,a+z)\). \emph{(One can prove that in this
+situation \(g\) is always one of the two prime factors of \(n\).)}
+
+Since factoring a number without extra information is very hard, Alice
+will be convinced that she must have given Bob one of the square roots
+that he did not know, so she admits the loss.
+
+\end{itemize}
+
+Now to the actual exercise:
+
+\begin{enumerate}
+\def\labelenumi{(\alph{enumi})}
+\item
+  Write the code for the functions A1, B1 and B2 as indicated in the
+  cell below. The function A2 is already written.
+\item
+  Modify the functions B1, A2 and B2 to check that the opponent is not
+  cheating. More precisely:
+
+\begin{itemize}
+\tightlist
+\item
+  In B1, Bob should check that \(n\) is not a prime power. \emph{(This
+  is the only way Alice can try to cheat: if she sends Bob a number
+  \(n\) that is the product of more than two primes, than she has less
+  than \(50\%\) chance of winning!)}
+\item
+  In A2, Alice should check that \(b\) is a square modulo \(n\).
+\item
+  In B2, Bob should check that \(z^2\equiv a^2\pmod n\).
+\end{itemize}
+
+In case cheating is detected, a message should be printed saying that
+the person is cheating.
+\end{enumerate}
+
+\newpage
+
+    \begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
+\prompt{In}{incolor}{1}{\boxspacing}
+\begin{Verbatim}[commandchars=\\\{\}]
+\PY{c+c1}{\PYZsh{} Alice needs this to compute the square roots}
+\PY{k+kn}{from} \PY{n+nn}{sage}\PY{n+nn}{.}\PY{n+nn}{rings}\PY{n+nn}{.}\PY{n+nn}{finite\PYZus{}rings}\PY{n+nn}{.}\PY{n+nn}{integer\PYZus{}mod} \PY{k}{import} \PY{n}{square\PYZus{}root\PYZus{}mod\PYZus{}prime}
+
+\PY{k}{def} \PY{n+nf}{A1}\PY{p}{(}\PY{p}{)}\PY{p}{:}
+    \PY{c+c1}{\PYZsh{} This function must return two distinct primes and their product.}
+    
+\PY{k}{def} \PY{n+nf}{B1}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{:}
+    \PY{c+c1}{\PYZsh{} This function must return a random integer a}
+    \PY{c+c1}{\PYZsh{} with 1\PYZlt{}a\PYZlt{}n and gcd(a,n)=1.}
+
+\PY{k}{def} \PY{n+nf}{A2}\PY{p}{(}\PY{n}{b}\PY{p}{,} \PY{n}{p}\PY{p}{,} \PY{n}{q}\PY{p}{)}\PY{p}{:}
+    \PY{n}{x} \PY{o}{=} \PY{n}{ZZ}\PY{p}{(}\PY{n}{square\PYZus{}root\PYZus{}mod\PYZus{}prime}\PY{p}{(}\PY{n}{Integers}\PY{p}{(}\PY{n}{p}\PY{p}{)}\PY{p}{(}\PY{n}{b}\PY{p}{)}\PY{p}{,} \PY{n}{p}\PY{p}{)}\PY{p}{)}
+    \PY{n}{y} \PY{o}{=} \PY{n}{ZZ}\PY{p}{(}\PY{n}{square\PYZus{}root\PYZus{}mod\PYZus{}prime}\PY{p}{(}\PY{n}{Integers}\PY{p}{(}\PY{n}{q}\PY{p}{)}\PY{p}{(}\PY{n}{b}\PY{p}{)}\PY{p}{,} \PY{n}{q}\PY{p}{)}\PY{p}{)}
+    \PY{k}{return} \PY{n}{crt}\PY{p}{(}\PY{n}{x}\PY{p}{,} \PY{n}{y}\PY{p}{,} \PY{n}{p}\PY{p}{,} \PY{n}{q}\PY{p}{)}
+
+\PY{k}{def} \PY{n+nf}{B2}\PY{p}{(}\PY{n}{a}\PY{p}{,} \PY{n}{z}\PY{p}{,} \PY{n}{n}\PY{p}{)}\PY{p}{:}
+    \PY{c+c1}{\PYZsh{} This function must print out one of two messages:}
+    \PY{c+c1}{\PYZsh{} \PYZdq{}Bob has lost\PYZdq{} if z is congruent to a or \PYZhy{}a modulo n.}
+    \PY{c+c1}{\PYZsh{} \PYZdq{}Bob has won, proof: \PYZdq{} followed by a prime factor of n otherwise.}
+    \PY{c+c1}{\PYZsh{} In this case the prime must be calculated as explained above.}
+
+
+\PY{c+c1}{\PYZsh{} This is how the game plays out:}
+\PY{n}{p}\PY{p}{,} \PY{n}{q}\PY{p}{,} \PY{n}{n} \PY{o}{=} \PY{n}{A1}\PY{p}{(}\PY{p}{)}
+\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Alice picked n =}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n}{n}\PY{p}{)}
+\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{[[ Alice}\PY{l+s+s2}{\PYZsq{}}\PY{l+s+s2}{s secret:}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n}{p}\PY{p}{,} \PY{n}{q}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{]]}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
+\PY{n}{a} \PY{o}{=} \PY{n}{B1}\PY{p}{(}\PY{n}{n}\PY{p}{)}
+\PY{n}{b} \PY{o}{=} \PY{n}{a}\PY{o}{\PYZca{}}\PY{l+m+mi}{2} \PY{o}{\PYZpc{}} \PY{n}{n}
+\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Bob picked b =}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n}{b}\PY{p}{)}
+\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{[[ Bob}\PY{l+s+s2}{\PYZsq{}}\PY{l+s+s2}{s secret:}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n}{a}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{]]}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
+\PY{n}{z} \PY{o}{=} \PY{n}{A2}\PY{p}{(}\PY{n}{b}\PY{p}{,} \PY{n}{p}\PY{p}{,} \PY{n}{q}\PY{p}{)}
+\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Alice picked z =}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n}{z}\PY{p}{)}
+\PY{n}{B2}\PY{p}{(}\PY{n}{a}\PY{p}{,} \PY{n}{z}\PY{p}{,} \PY{n}{n}\PY{p}{)}
+\end{Verbatim}
+\end{tcolorbox}
+
+    \vspace{1cm}
+    \textbf{Grading}
+
+This homework assignment is worth \(20\%\) of your final grade. Exercise
+1 is worth 4 points (one for each part) and Exercise 2 is worth 12
+points (8 points for part (a) and 4 points for part (b)), for a total of
+\textbf{16 points}.
+
+    
+    
+\end{document}