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Diffstat (limited to 'tutorials/module_1/latex/arrays.tex')
| -rw-r--r-- | tutorials/module_1/latex/arrays.tex | 829 |
1 files changed, 829 insertions, 0 deletions
diff --git a/tutorials/module_1/latex/arrays.tex b/tutorials/module_1/latex/arrays.tex new file mode 100644 index 0000000..5d0f1b3 --- /dev/null +++ b/tutorials/module_1/latex/arrays.tex @@ -0,0 +1,829 @@ +\documentclass[11pt]{article} + + \usepackage[breakable]{tcolorbox} + \usepackage{parskip} % Stop auto-indenting (to mimic markdown behaviour) + + + % Basic figure setup, for now with no caption control since it's done + % automatically by Pandoc (which extracts  syntax from Markdown). + \usepackage{graphicx} + % Keep aspect ratio if custom image width or height is specified + \setkeys{Gin}{keepaspectratio} + % 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{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{iftex} + \ifPDFTeX + \usepackage[T1]{fontenc} + \IfFileExists{alphabeta.sty}{ + \usepackage{alphabeta} + }{ + \usepackage[mathletters]{ucs} + \usepackage[utf8]{inputenc} + } + \else + \usepackage{fontspec} + \usepackage{unicode-math} + \fi + + \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 old versions of grffile with XeLaTeX + \@ifpackagelater{grffile}{2019/11/01} + { + % Do nothing on new versions + } + { + \def\Gread@@xetex#1{% + \IfFileExists{"\Gin@base".bb}% + {\Gread@eps{\Gin@base.bb}}% + {\Gread@@xetex@aux#1}% + } + } + \makeatother + \usepackage[Export]{adjustbox} % Used to constrain images to a maximum size + \adjustboxset{max size={0.9\linewidth}{0.9\paperheight}} + + % 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{array} % table support for pandoc >= 2.11.3 + \usepackage{calc} % table minipage width calculation for pandoc >= 2.11.1 + \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{soul} % strikethrough (\st) support for pandoc >= 3.0.0 + \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} + + % common color for the border for error outputs. + \definecolor{outerrorbackground}{HTML}{FFDFDF} + + % 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}}}}} + \makeatletter + \newsavebox\pandoc@box + \newcommand*\pandocbounded[1]{% + \sbox\pandoc@box{#1}% + % scaling factors for width and height + \Gscale@div\@tempa\textheight{\dimexpr\ht\pandoc@box+\dp\pandoc@box\relax}% + \Gscale@div\@tempb\linewidth{\wd\pandoc@box}% + % select the smaller of both + \ifdim\@tempb\p@<\@tempa\p@ + \let\@tempa\@tempb + \fi + % scaling accordingly (\@tempa < 1) + \ifdim\@tempa\p@<\p@ + \scalebox{\@tempa}{\usebox\pandoc@box}% + % scaling not needed, use as it is + \else + \usebox{\pandoc@box}% + \fi + } + \makeatother + + % 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{arrays} + + + + + + + +% 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}} + +\@namedef{PY@tok@w}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.73,0.73}{##1}}} +\@namedef{PY@tok@c}{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.24,0.48,0.48}{##1}}} +\@namedef{PY@tok@cp}{\def\PY@tc##1{\textcolor[rgb]{0.61,0.40,0.00}{##1}}} +\@namedef{PY@tok@k}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@kp}{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@kt}{\def\PY@tc##1{\textcolor[rgb]{0.69,0.00,0.25}{##1}}} +\@namedef{PY@tok@o}{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\@namedef{PY@tok@ow}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.67,0.13,1.00}{##1}}} +\@namedef{PY@tok@nb}{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@nf}{\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}} +\@namedef{PY@tok@nc}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}} +\@namedef{PY@tok@nn}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}} +\@namedef{PY@tok@ne}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.80,0.25,0.22}{##1}}} +\@namedef{PY@tok@nv}{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\@namedef{PY@tok@no}{\def\PY@tc##1{\textcolor[rgb]{0.53,0.00,0.00}{##1}}} +\@namedef{PY@tok@nl}{\def\PY@tc##1{\textcolor[rgb]{0.46,0.46,0.00}{##1}}} +\@namedef{PY@tok@ni}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.44,0.44,0.44}{##1}}} +\@namedef{PY@tok@na}{\def\PY@tc##1{\textcolor[rgb]{0.41,0.47,0.13}{##1}}} +\@namedef{PY@tok@nt}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@nd}{\def\PY@tc##1{\textcolor[rgb]{0.67,0.13,1.00}{##1}}} +\@namedef{PY@tok@s}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\@namedef{PY@tok@sd}{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\@namedef{PY@tok@si}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.64,0.35,0.47}{##1}}} +\@namedef{PY@tok@se}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.67,0.36,0.12}{##1}}} +\@namedef{PY@tok@sr}{\def\PY@tc##1{\textcolor[rgb]{0.64,0.35,0.47}{##1}}} +\@namedef{PY@tok@ss}{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\@namedef{PY@tok@sx}{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@m}{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\@namedef{PY@tok@gh}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,0.50}{##1}}} +\@namedef{PY@tok@gu}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.50,0.00,0.50}{##1}}} +\@namedef{PY@tok@gd}{\def\PY@tc##1{\textcolor[rgb]{0.63,0.00,0.00}{##1}}} +\@namedef{PY@tok@gi}{\def\PY@tc##1{\textcolor[rgb]{0.00,0.52,0.00}{##1}}} +\@namedef{PY@tok@gr}{\def\PY@tc##1{\textcolor[rgb]{0.89,0.00,0.00}{##1}}} +\@namedef{PY@tok@ge}{\let\PY@it=\textit} +\@namedef{PY@tok@gs}{\let\PY@bf=\textbf} +\@namedef{PY@tok@gp}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,0.50}{##1}}} +\@namedef{PY@tok@go}{\def\PY@tc##1{\textcolor[rgb]{0.44,0.44,0.44}{##1}}} +\@namedef{PY@tok@gt}{\def\PY@tc##1{\textcolor[rgb]{0.00,0.27,0.87}{##1}}} +\@namedef{PY@tok@err}{\def\PY@bc##1{{\setlength{\fboxsep}{\string -\fboxrule}\fcolorbox[rgb]{1.00,0.00,0.00}{1,1,1}{\strut ##1}}}} +\@namedef{PY@tok@kc}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@kd}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@kn}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@kr}{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@bp}{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}} +\@namedef{PY@tok@fm}{\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}} +\@namedef{PY@tok@vc}{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\@namedef{PY@tok@vg}{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\@namedef{PY@tok@vi}{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\@namedef{PY@tok@vm}{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}} +\@namedef{PY@tok@sa}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\@namedef{PY@tok@sb}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\@namedef{PY@tok@sc}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\@namedef{PY@tok@dl}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\@namedef{PY@tok@s2}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\@namedef{PY@tok@sh}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\@namedef{PY@tok@s1}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}} +\@namedef{PY@tok@mb}{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\@namedef{PY@tok@mf}{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\@namedef{PY@tok@mh}{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\@namedef{PY@tok@mi}{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\@namedef{PY@tok@il}{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\@namedef{PY@tok@mo}{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}} +\@namedef{PY@tok@ch}{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.24,0.48,0.48}{##1}}} +\@namedef{PY@tok@cm}{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.24,0.48,0.48}{##1}}} +\@namedef{PY@tok@cpf}{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.24,0.48,0.48}{##1}}} +\@namedef{PY@tok@c1}{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.24,0.48,0.48}{##1}}} +\@namedef{PY@tok@cs}{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.24,0.48,0.48}{##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 + + + + + \hypertarget{matrixarrays}{% +\section{matrixArrays}\label{matrixarrays}} + +In computer programming, an array is a structure for storing and +retrieving data. We often talk about an array as if it were a grid in +space, with each cell storing one element of the data. For instance, if +each element of the data were a number, we might visualize a +``one-dimensional'' array like a list: + +\begin{longtable}[]{@{}llll@{}} +\toprule +1 & 5 & 2 & 0 \\ +\midrule +\endhead +\bottomrule +\end{longtable} + +A two-dimensional array would be like a table: + +\begin{longtable}[]{@{}llll@{}} +\toprule +1 & 5 & 2 & 0 \\ +\midrule +\endhead +8 & 3 & 6 & 1 \\ +1 & 7 & 2 & 9 \\ +\bottomrule +\end{longtable} + +A three-dimensional array would be like a set of tables, perhaps stacked +as though they were printed on separate pages. If we visualize the +position of each element as a position in space. Then we can represent +the value of the element as a property. In other words, if we were to +analyze the stress concentration of an aluminum block, the property +would be stress. + +\begin{itemize} +\tightlist +\item + From + \href{https://numpy.org/doc/2.2/user/absolute_beginners.html}{Numpy + documentation} + +\end{itemize} + +If the load on this block changes over time, then we may want to add a +4th dimension i.e.~additional sets of 3-D arrays for each time +increment. As you can see - the more dimensions we add, the more +complicated of a problem we have to solve. It is possible to increase +the number of dimensions to the n-th order. This course we will not be +going beyond dimensional analysis. + +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} + +\hypertarget{numpy---the-pythons-array-library}{% +\section{Numpy - the python's array +library}\label{numpy---the-pythons-array-library}} + +In this tutorial we will be introducing arrays and we will be using the +numpy library. Arrays, lists, vectors, matrices, sets - You might've +heard of them before, they all store data. In programming, an array is a +variable that can hold more than one value at a time. We will be using +the Numpy python library to create arrays. Since we already have +installed Numpy previously, we can start using the package. + +Before importing our first package, let's as ourselves \emph{what is a +package?} A package can be thought of as pre-written python code that we +can re-use. This means the for every script that we write in python we +need to tell it to use a certain package. We call this importing a +package. + +\hypertarget{importing-numpy}{% +\subsection{Importing Numpy}\label{importing-numpy}} + +When using packages in python, we need to let it know what package we +will be using. This is called importing. To import numpy we need to +declare it a the start of a script as follows: + +\begin{Shaded} +\begin{Highlighting}[] +\ImportTok{import}\NormalTok{ numpy }\ImportTok{as}\NormalTok{ np} +\end{Highlighting} +\end{Shaded} + +\begin{itemize} +\tightlist +\item + \texttt{import} - calls for a library to use, in our case it is Numpy. +\item + \texttt{as} - gives the library an alias in your script. It's common + convention in Python programming to make the code shorter and more + readable. We will be using \emph{np} as it's a standard using in many + projects. +\end{itemize} + +\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} + +\hypertarget{creating-arrays}{% +\section{Creating arrays}\label{creating-arrays}} + +Now that we have imported the library we can create a one dimensional +array or \emph{vector} with three elements. + +\begin{Shaded} +\begin{Highlighting}[] +\NormalTok{x }\OperatorTok{=}\NormalTok{ np.array([}\DecValTok{1}\NormalTok{,}\DecValTok{2}\NormalTok{,}\DecValTok{3}\NormalTok{])} +\end{Highlighting} +\end{Shaded} + +To create a \emph{matrix} we can nest the arrays to create a two +dimensional array. This is done as follows. + +\begin{Shaded} +\begin{Highlighting}[] +\NormalTok{matrix }\OperatorTok{=}\NormalTok{ np.array([[}\DecValTok{1}\NormalTok{,}\DecValTok{2}\NormalTok{,}\DecValTok{3}\NormalTok{],} +\NormalTok{ [}\DecValTok{4}\NormalTok{,}\DecValTok{5}\NormalTok{,}\DecValTok{6}\NormalTok{],} +\NormalTok{ [}\DecValTok{7}\NormalTok{,}\DecValTok{8}\NormalTok{,}\DecValTok{9}\NormalTok{]])} +\end{Highlighting} +\end{Shaded} + +\emph{Note: for every array we nest, we get a new dimension in our data +structure.} + + \hypertarget{display-arrays}{% +\section{Display arrays}\label{display-arrays}} + +Using command print("") Accessing particular elements of an array +\ldots.. + + \hypertarget{practice-problem}{% +\section{Practice Problem}\label{practice-problem}} + +Problem statement + + \begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder] +\prompt{In}{incolor}{1}{\boxspacing} +\begin{Verbatim}[commandchars=\\\{\}] +\PY{k+kn}{import} \PY{n+nn}{numpy} \PY{k}{as} \PY{n+nn}{np} + +\PY{n}{x} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{array}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{7}\PY{p}{,} \PY{l+m+mi}{10} \PY{p}{,}\PY{l+m+mi}{12}\PY{p}{]}\PY{p}{)} + +\PY{n+nb}{print}\PY{p}{(}\PY{n}{x}\PY{p}{)} + +\PY{n+nb}{print}\PY{p}{(}\PY{n}{x}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)} +\end{Verbatim} +\end{tcolorbox} + + \begin{Verbatim}[commandchars=\\\{\}] +[ 7 10 12] +10 + \end{Verbatim} + + \hypertarget{numpy-array-creation-functions}{% +\subsection{Numpy array creation +functions}\label{numpy-array-creation-functions}} + +Numpy comes with some built-in function that we can use to create arrays +quickly. Here are a couple of functions that are commonly used in +python. + +\hypertarget{np.arange}{% +\subsubsection{np.arange}\label{np.arange}} + +The \texttt{np.arange()} function returns an array with evenly spaced +values within a specified range. It is similar to the built-in +\texttt{range()} function in Python but returns a Numpy array instead of +a list. The parameters for this function are the start value +(inclusive), the stop value (exclusive), and the step size. If the step +size is not provided, it defaults to 1. + +\begin{Shaded} +\begin{Highlighting}[] +\OperatorTok{\textgreater{}\textgreater{}\textgreater{}}\NormalTok{ np.arange(}\DecValTok{4}\NormalTok{)} +\NormalTok{array([}\FloatTok{0.}\NormalTok{ , }\FloatTok{1.}\NormalTok{, }\FloatTok{2.}\NormalTok{, }\FloatTok{3.}\NormalTok{ ])} +\end{Highlighting} +\end{Shaded} + +In this example, \texttt{np.arange(4)} generates an array starting from +0 and ending before 4, with a step size of 1. + +\hypertarget{np.linspace}{% +\subsubsection{np.linspace}\label{np.linspace}} + +The \texttt{np.linspace()} function returns an array of evenly spaced +values over a specified range. Unlike \texttt{np.arange()}, which uses a +step size to define the spacing between elements, \texttt{np.linspace()} +uses the number of values you want to generate and calculates the +spacing automatically. It accepts three parameters: the start value, the +stop value, and the number of samples. + +\begin{Shaded} +\begin{Highlighting}[] +\OperatorTok{\textgreater{}\textgreater{}\textgreater{}}\NormalTok{ np.linspace(}\FloatTok{1.}\NormalTok{, }\FloatTok{4.}\NormalTok{, }\DecValTok{6}\NormalTok{)} +\NormalTok{array([}\FloatTok{1.}\NormalTok{ , }\FloatTok{1.6}\NormalTok{, }\FloatTok{2.2}\NormalTok{, }\FloatTok{2.8}\NormalTok{, }\FloatTok{3.4}\NormalTok{, }\FloatTok{4.}\NormalTok{ ])} +\end{Highlighting} +\end{Shaded} + +In this example, \texttt{np.linspace(1.,\ 4.,\ 6)} generates 6 evenly +spaced values between 1. and 4., including both endpoints. + +Try this and see what happens: + +\begin{Shaded} +\begin{Highlighting}[] +\NormalTok{x }\OperatorTok{=}\NormalTok{ np.linspace(}\DecValTok{0}\NormalTok{,}\DecValTok{100}\NormalTok{,}\DecValTok{101}\NormalTok{)} +\NormalTok{y }\OperatorTok{=}\NormalTok{ np.sin(x)} +\end{Highlighting} +\end{Shaded} + +\hypertarget{other-useful-functions}{% +\subsubsection{Other useful functions}\label{other-useful-functions}} + +\begin{itemize} +\tightlist +\item + \texttt{np.zeros()} +\item + \texttt{np.ones()} +\item + \texttt{np.eye()} +\end{itemize} + + \hypertarget{practice-problem}{% +\subsection{Practice problem}\label{practice-problem}} + +Problem statement below + + \begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder] +\prompt{In}{incolor}{2}{\boxspacing} +\begin{Verbatim}[commandchars=\\\{\}] +\PY{n}{y}\PY{o}{=}\PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{l+m+mi}{10}\PY{p}{,}\PY{l+m+mi}{20}\PY{p}{,}\PY{l+m+mi}{5}\PY{p}{)} +\PY{n+nb}{print}\PY{p}{(}\PY{n}{y}\PY{p}{)} +\end{Verbatim} +\end{tcolorbox} + + \begin{Verbatim}[commandchars=\\\{\}] +[10. 12.5 15. 17.5 20. ] + \end{Verbatim} + + \hypertarget{working-with-arrays}{% +\subsection{Working with Arrays}\label{working-with-arrays}} + +Now that we have been introduced to some ways to create arrays using the +Numpy functions let's start using them. + +\hypertarget{indexing}{% +\subsubsection{Indexing}\label{indexing}} + +Indexing in Python allows you to access specific elements within an +array based on their position. This means you can directly retrieve and +manipulate individual items as needed. + +Python uses \textbf{zero-based indexing}, meaning the first element is +at position \textbf{0} rather than \textbf{1}. This approach is common +in many programming languages. For example, in a list with five +elements, the first element is at index \texttt{0}, followed by elements +at indices \texttt{1}, \texttt{2}, \texttt{3}, and \texttt{4}. + +Here's an example of data from a rocket test stand where thrust was +recorded as a function of time. + +\begin{Shaded} +\begin{Highlighting}[] +\NormalTok{thrust\_lbf }\OperatorTok{=}\NormalTok{ np.array(}\FloatTok{0.603355}\NormalTok{, }\FloatTok{2.019083}\NormalTok{, }\FloatTok{2.808092}\NormalTok{, }\FloatTok{4.054973}\NormalTok{, }\FloatTok{1.136618}\NormalTok{, }\FloatTok{0.943668}\NormalTok{)} + +\OperatorTok{\textgreater{}\textgreater{}\textgreater{}}\NormalTok{ thrust\_lbs[}\DecValTok{3}\NormalTok{]} +\end{Highlighting} +\end{Shaded} + +Due to the nature of zero-based indexing. If we want to call the value +\texttt{4.054973} that will be the 3rd index. + +\hypertarget{operations-on-arrays}{% +\subsubsection{Operations on arrays}\label{operations-on-arrays}} + +\begin{itemize} +\tightlist +\item + Arithmetic operations (\texttt{+}, \texttt{-}, \texttt{*}, \texttt{/}, + \texttt{**}) +\item + \texttt{np.add()}, \texttt{np.subtract()}, \texttt{np.multiply()}, + \texttt{np.divide()} +\item + \texttt{np.dot()} for dot product +\item + \texttt{np.matmul()} for matrix multiplication +\item + \texttt{np.linalg.inv()}, \texttt{np.linalg.det()} for linear algebra +\end{itemize} + +\hypertarget{statistics}{% +\paragraph{Statistics}\label{statistics}} + +\begin{itemize} +\tightlist +\item + \texttt{np.mean()}, \texttt{np.median()}, \texttt{np.std()}, + \texttt{np.var()} +\item + \texttt{np.min()}, \texttt{np.max()}, \texttt{np.argmin()}, + \texttt{np.argmax()} +\item + Summation along axes: \texttt{np.sum(arr,\ axis=0)} +\end{itemize} + +\hypertarget{combining-arrays}{% +\paragraph{Combining arrays}\label{combining-arrays}} + +\begin{itemize} +\tightlist +\item + Concatenation: \texttt{np.concatenate((arr1,\ arr2),\ axis=0)} +\item + Stacking: \texttt{np.vstack()}, \texttt{np.hstack()} +\item + Splitting: \texttt{np.split()} +\end{itemize} + + \hypertarget{exercise}{% +\section{Exercise}\label{exercise}} + +Let's solve a statics problem given the following problem + +A simply supported bridge of length L=20L = 20L=20 m is subjected to +three point loads: + +\begin{itemize} +\tightlist +\item + \(P1=1010 kN\) at \(x=5m\) +\item + \(P2=15 kN\) at \(x=10m\) +\item + \(P3=20 kN\) at \(x=15m\) +\end{itemize} + +The bridge is supported by two reaction forces at points AAA (left +support) and BBB (right support). We assume the bridge is in static +equilibrium, meaning the sum of forces and sum of moments about any +point must be zero. + +\hypertarget{equilibrium-equations}{% +\paragraph{Equilibrium Equations:}\label{equilibrium-equations}} + +\begin{enumerate} +\def\labelenumi{\arabic{enumi}.} +\item + \textbf{Sum of Forces in the Vertical Direction}: + + \(R_A + R_B - P_1 - P_2 - P_3 = 0\) +\item + \textbf{Sum of Moments About Point A}: + + \(5 P_1 + 10 P_2 + 15 P_3 - 20 R_B = 0\) +\item + \textbf{Sum of Moments About Point B}: + + \(20 R_A - 15 P_3 - 10 P_2 - 5 P_1 = 0\) +\end{enumerate} + +\hypertarget{system-of-equations}{% +\paragraph{System of Equations:}\label{system-of-equations}} + +\[ +\begin{cases} +R_A + R_B - 10 - 15 - 20 = 0 \\ +5 \cdot 10 + 10 \cdot 15 + 15 \cdot 20 - 20 R_B = 0 \\ +20 R_A - 5 \cdot 10 - 10 \cdot 15 - 15 \cdot 20 = 0 +\end{cases} +\] + + \hypertarget{solution}{% +\subsubsection{Solution}\label{solution}} + + \begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder] +\prompt{In}{incolor}{3}{\boxspacing} +\begin{Verbatim}[commandchars=\\\{\}] +\PY{k+kn}{import} \PY{n+nn}{numpy} \PY{k}{as} \PY{n+nn}{np} + +\PY{c+c1}{\PYZsh{} Define the coefficient matrix A} +\PY{n}{A} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{array}\PY{p}{(}\PY{p}{[} + \PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{]}\PY{p}{,} + \PY{p}{[}\PY{l+m+mi}{0}\PY{p}{,} \PY{o}{\PYZhy{}}\PY{l+m+mi}{20}\PY{p}{]}\PY{p}{,} + \PY{p}{[}\PY{l+m+mi}{20}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{]} +\PY{p}{]}\PY{p}{)} + +\PY{c+c1}{\PYZsh{} Define the right\PYZhy{}hand side vector b} +\PY{n}{b} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{array}\PY{p}{(}\PY{p}{[} + \PY{l+m+mi}{45}\PY{p}{,} + \PY{l+m+mi}{5}\PY{o}{*}\PY{l+m+mi}{10} \PY{o}{+} \PY{l+m+mi}{10}\PY{o}{*}\PY{l+m+mi}{15} \PY{o}{+} \PY{l+m+mi}{15}\PY{o}{*}\PY{l+m+mi}{20}\PY{p}{,} + \PY{l+m+mi}{5}\PY{o}{*}\PY{l+m+mi}{10} \PY{o}{+} \PY{l+m+mi}{10}\PY{o}{*}\PY{l+m+mi}{15} \PY{o}{+} \PY{l+m+mi}{15}\PY{o}{*}\PY{l+m+mi}{20} +\PY{p}{]}\PY{p}{)} + +\PY{c+c1}{\PYZsh{} Solve the system of equations Ax = b} +\PY{c+c1}{\PYZsh{} Using least squares to handle potential overdetermination} +\PY{n}{x} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linalg}\PY{o}{.}\PY{n}{lstsq}\PY{p}{(}\PY{n}{A}\PY{p}{,} \PY{n}{b}\PY{p}{,} \PY{n}{rcond}\PY{o}{=}\PY{k+kc}{None}\PY{p}{)}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]} + +\PY{c+c1}{\PYZsh{} Display the results} +\PY{n+nb}{print}\PY{p}{(}\PY{l+s+sa}{f}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Reaction force at A (R\PYZus{}A): }\PY{l+s+si}{\PYZob{}}\PY{n}{x}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{l+s+si}{:}\PY{l+s+s2}{.2f}\PY{l+s+si}{\PYZcb{}}\PY{l+s+s2}{ kN}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)} +\PY{n+nb}{print}\PY{p}{(}\PY{l+s+sa}{f}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Reaction force at B (R\PYZus{}B): }\PY{l+s+si}{\PYZob{}}\PY{n}{x}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{l+s+si}{:}\PY{l+s+s2}{.2f}\PY{l+s+si}{\PYZcb{}}\PY{l+s+s2}{ kN}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)} +\end{Verbatim} +\end{tcolorbox} + + \begin{Verbatim}[commandchars=\\\{\}] +Reaction force at A (R\_A): 25.11 kN +Reaction force at B (R\_B): -24.89 kN + \end{Verbatim} + + + % Add a bibliography block to the postdoc + + + +\end{document} |