\documentclass[10pt,a4paper]{article} % Packages \usepackage{fancyhdr} % For header and footer \usepackage{multicol} % Allows multicols in tables \usepackage{tabularx} % Intelligent column widths \usepackage{tabulary} % Used in header and footer \usepackage{hhline} % Border under tables \usepackage{graphicx} % For images \usepackage{xcolor} % For hex colours %\usepackage[utf8x]{inputenc} % For unicode character support \usepackage[T1]{fontenc} % Without this we get weird character replacements \usepackage{colortbl} % For coloured tables \usepackage{setspace} % For line height \usepackage{lastpage} % Needed for total page number \usepackage{seqsplit} % Splits long words. %\usepackage{opensans} % Can't make this work so far. Shame. Would be lovely. \usepackage[normalem]{ulem} % For underlining links % Most of the following are not required for the majority % of cheat sheets but are needed for some symbol support. \usepackage{amsmath} % Symbols \usepackage{MnSymbol} % Symbols \usepackage{wasysym} % Symbols %\usepackage[english,german,french,spanish,italian]{babel} % Languages % Document Info \author{Unobia} \pdfinfo{ /Title (metodi-numerici.pdf) /Creator (Cheatography) /Author (Unobia) /Subject (Metodi numerici Cheat Sheet) } % Lengths and widths \addtolength{\textwidth}{6cm} \addtolength{\textheight}{-1cm} \addtolength{\hoffset}{-3cm} \addtolength{\voffset}{-2cm} \setlength{\tabcolsep}{0.2cm} % Space between columns \setlength{\headsep}{-12pt} % Reduce space between header and content \setlength{\headheight}{85pt} % If less, LaTeX automatically increases it \renewcommand{\footrulewidth}{0pt} % Remove footer line \renewcommand{\headrulewidth}{0pt} % Remove header line \renewcommand{\seqinsert}{\ifmmode\allowbreak\else\-\fi} % Hyphens in seqsplit % This two commands together give roughly % the right line height in the tables \renewcommand{\arraystretch}{1.3} \onehalfspacing % Commands \newcommand{\SetRowColor}[1]{\noalign{\gdef\RowColorName{#1}}\rowcolor{\RowColorName}} % Shortcut for row colour \newcommand{\mymulticolumn}[3]{\multicolumn{#1}{>{\columncolor{\RowColorName}}#2}{#3}} % For coloured multi-cols \newcolumntype{x}[1]{>{\raggedright}p{#1}} % New column types for ragged-right paragraph columns \newcommand{\tn}{\tabularnewline} % Required as custom column type in use % Font and Colours \definecolor{HeadBackground}{HTML}{333333} \definecolor{FootBackground}{HTML}{666666} \definecolor{TextColor}{HTML}{333333} \definecolor{DarkBackground}{HTML}{637BA3} \definecolor{LightBackground}{HTML}{F5F6F9} \renewcommand{\familydefault}{\sfdefault} \color{TextColor} % Header and Footer \pagestyle{fancy} \fancyhead{} % Set header to blank \fancyfoot{} % Set footer to blank \fancyhead[L]{ \noindent \begin{multicols}{3} \begin{tabulary}{5.8cm}{C} \SetRowColor{DarkBackground} \vspace{-7pt} {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}} } \end{tabulary} \columnbreak \begin{tabulary}{11cm}{L} \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Metodi numerici Cheat Sheet}}}} \\ \normalsize{by \textcolor{DarkBackground}{Unobia} via \textcolor{DarkBackground}{\uline{cheatography.com/216860/cs/47508/}}} \end{tabulary} \end{multicols}} \fancyfoot[L]{ \footnotesize \noindent \begin{multicols}{3} \begin{tabulary}{5.8cm}{LL} \SetRowColor{FootBackground} \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}} \\ \vspace{-2pt}Unobia \\ \uline{cheatography.com/unobia} \\ \end{tabulary} \vfill \columnbreak \begin{tabulary}{5.8cm}{L} \SetRowColor{FootBackground} \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}} \\ \vspace{-2pt}Not Yet Published.\\ Updated 16th December, 2025.\\ Page {\thepage} of \pageref{LastPage}. \end{tabulary} \vfill \columnbreak \begin{tabulary}{5.8cm}{L} \SetRowColor{FootBackground} \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}} \\ \SetRowColor{white} \vspace{-5pt} %\includegraphics[width=48px,height=48px]{dave.jpeg} Measure your website readability!\\ www.readability-score.com \end{tabulary} \end{multicols}} \begin{document} \raggedright \raggedcolumns % Set font size to small. Switch to any value % from this page to resize cheat sheet text: % www.emerson.emory.edu/services/latex/latex_169.html \footnotesize % Small font. \begin{multicols*}{2} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Sistemi lineari}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{} \tn % Row Count 0 (+ 0) \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Jacobi}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{n = size(A,1); \newline D = spdiags(diag(A),0,n,n); \newline N = A-D; \newline for k = 1:kmax \newline x = D\textbackslash{}(b-N{\emph{x); \newline res\_rel = norm(b-A}}x)/norm(b); \newline if res\_rel \textless{}= toll \newline break \newline end \newline end} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Gauss-Seidel}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{M = tril(A); \newline N = A-M; \newline for k = 1:kmax \newline x = M\textbackslash{}(b-N{\emph{x); \newline res\_rel = norm(b-A}}x)/norm(b); \newline if res\_rel \textless{}= toll \newline break \newline end \newline end} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Gradiente}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{r = b-A{\emph{x; \newline for k = 1:kmax \newline z = A}}r; \newline alpha = (r'{\emph{r)/(r'}}z); \newline x = x + alpha{\emph{r; \newline r = r - alpha}}z; \newline res\_rel = norm(r)/norm(b); \newline if res\_rel \textless{}= tol \newline break \newline end \newline end} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Quadratura}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{} \tn % Row Count 0 (+ 0) \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Trapezi composita}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{x = linspace(a,b,n+1); \newline y = f(x); \newline h = (b-a)/n; \newline Itrap = (h/2){\emph{(y(1) + 2}}sum(y(2:end-1)) + y(end));} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Simpson composita}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{x = linspace(a,b,2{\emph{n+1); \newline y = f(x); \newline h = (b-a)/n; \newline Isimpson = (h/6)}}(y(1) + 4{\emph{sum(y(2:2:end-1)) + 2}}sum(y(3:2:end-2)) + y(end));} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{PDE}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{} \tn % Row Count 0 (+ 0) \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Differenze finite DD}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{N = length(x) - 1; \newline h = x(2) - x(1); \newline xM = (x(1:end-1)+x(2:end))/2; \newline muVec = mu(xM); \newline \newline d0 = 1/h\textasciicircum{}2{\emph{(muVec(1:end-1) + muVec(2:end)); \newline d1 = -1/h\textasciicircum{}2}}muVec(2:end); \newline d\_1 = -1/h\textasciicircum{}2*muVec(1:end-1); \newline \newline A = spdiags({[}d\_1, d0, d1{]}, {[}-1, 0, 1{]}, N-1, N-1); \newline b = f(x(2:end-1)); \newline b(1) = b(1) + (1/h\textasciicircum{}2)*muVec(1) *BC(1); \newline b(end) = b(end) + (1/h\textasciicircum{}2)*muVec(end)*BC(2); \newline \newline u = A\textbackslash{}b; \newline u = {[}BC(1); u; BC(2){]};} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Differenze finite ND}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{N = length(x) - 1; \newline h = x(2) - x(1); \newline xM = (x(1:end-1)+x(2:end))/2; \newline \newline d0 = mu/h\textasciicircum{}2*({[}1; 2{\emph{ones(N-1,1){]}); \newline d1 = -mu/h\textasciicircum{}2}}ones(N,1); \newline d\_1 = -mu/h\textasciicircum{}2{\emph{ones(N,1); \newline \newline A = spdiags({[}d\_1, d0, d1{]}, {[}-1, 0, 1{]}, N, N); \newline b = f(x(1:end-1)); \newline b(1) = 1/2}}b(1)-BC(1)/h; \newline b(end) = b(end) + 1/h\textasciicircum{}2*mu*BC(2); \newline \newline u = A\textbackslash{}b; \newline u = {[}u; BC(2){]};} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Elementi finiti DD}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{N = length(x) - 1; \newline h = diff(x); \newline xM = x(1:N) + 0.5{\emph{h; \newline muVec = mu(xM)./h; \newline \newline d = muVec(1:N-1) + muVec(2:N); \newline d\_1 = -muVec(2:N); \newline d1 = -muVec(1:N-1); \newline \newline A = spdiags({[}d\_1 d d1{]}, {[}-1 0 1{]}, N-1, N-1); \newline \newline b = f(x(2:N)).*(h(1:N-1) + h(2:N))}}0.5; \newline b(1) = b(1) + muVec(1){\emph{BC(1); \newline b(N-1) = b(N-1) + muVec(N-1)}}BC(2); \newline \newline uDof = A\textbackslash{}b; \newline u = {[}BC(1); uDof; BC(2){]};} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Elementi finiti ND}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{N = length(x) - 1; \newline h = diff(x); \newline xM = x(1:N) + 0.5{\emph{h; \newline muVec = mu(xM)./h; \newline \newline d = {[}muVec(1); muVec(1:N-1) + muVec(2:N){]}; \newline d\_1 = -{[}muVec(1); muVec(2:N){]}; \newline d1 = -{[}muVec(1); muVec(1:N-1){]}; \newline \newline A = spdiags({[}d\_1 d d1{]}, {[}-1 0 1{]}, N, N); \newline \newline b = f(x(2:N)).*(h(1:N-1) + h(2:N))}}0.5; \newline b = {[}0.5*f(x(1)){\emph{h(1) - BC(1); b{]}; \newline b(end) = b(end) + muVec(end)}}BC(2); \newline \newline uDof = A\textbackslash{}b; \newline u = {[}uDof; BC(2){]};} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Equazioni non lineari}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{} \tn % Row Count 0 (+ 0) \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Newton}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{for k=1:kmax \newline x = x0 - m*f(x0)/df(x0); \newline errx(k) = abs(x-x0)/abs(x); \newline val\_f = f(x); \newline if val\_f\textless{}=tolf \newline break \newline end \newline if errx(k)\textless{}=tolr \newline break \newline end \newline x0 = x; \newline end} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Punto fisso}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{x\_vec(1) = x0; \newline for k=1:kmax \newline x = phi(x0); \newline x\_vec(k+1) = x; \newline errx(k) = abs(x-x0)/x; \newline if errx(k)\textless{}=tolr \newline break \newline end \newline val\_f = abs(phi(x)-phi(x0)); \newline if val\_f\textless{}=tolf \newline break \newline end \newline x0 = x; \newline end} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{Attenzione in versione 2014 l'aggiunta di elementi in un vettore} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{ODE}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{} \tn % Row Count 0 (+ 0) \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Eulero esplicito}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{t = t0:dt:T; \newline u = zeros(length(t),1); \newline u(1) = u0; \newline for k = 1:length(t) - 1 \newline u(k+1) = u(k) + dt*f(t(k),u(k)); \newline end \newline t = t';} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Eulero implicito}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{t = t0:dt:T; \newline u = zeros(length(t),1); \newline u(1) = u0; \newline for k = length(t)-1 \newline u(k+1) = 1/(1 - dt{\emph{lambda)}}u(k); \newline end \newline t = t';} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{8.4cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Eulero sistemi}} \tn \SetRowColor{LightBackground} \mymulticolumn{1}{x{8.4cm}}{t = t0:dt:T; \newline dim\_u = size(u0,1); \newline u = zeros(length(t),dim\_u); \newline u(1,:) = u0; \newline for k = 1:length(t) - 1 \newline u(k+1,:) = u(k,:) + dt*f(t(k),u(k,:))'; \newline end \newline t = t';} \tn \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} % That's all folks \end{multicols*} \end{document}