\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{Kamva}
\pdfinfo{
  /Title (numerical-methods-262.pdf)
  /Creator (Cheatography)
  /Author (Kamva)
  /Subject (Numerical Methods 262 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}{C21CFF}
\definecolor{LightBackground}{HTML}{FBF0FF}
\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{Numerical Methods 262 Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{Kamva} via \textcolor{DarkBackground}{\uline{cheatography.com/169429/cs/35481/}}}
\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}Kamva \\
  \uline{cheatography.com/kamva} \\
  \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 14th November, 2022.\\
   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*}{4}

\begin{tabularx}{3.833cm}{p{0.3433 cm} p{0.3433 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Curve fitting: Higher order Polynomials}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 0 (+ 0)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{The formula: ax + by + cz = 1 \newline `A= {[}-1 -1 -3;1 0 0;2 1 2;2 3 6{]};` \newline `b = {[}1;1;1;1{]};` \newline `x = A\textbackslash{}b;`     \newline NOTE: A{\bf{x}} = b...{\bf{backslash(\textbackslash{})}} to solve for {\bf{a}},{\bf{b}},{\bf{c}}}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.3433 cm} p{0.3433 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Curve fitting: linearization}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 0 (+ 0)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{{\bf{Exponential linearized function}} \newline The formula is {\emph{y = be\textasciicircum{}mx\textasciicircum{}}}: \newline `x = {[}1 2 3 4{]}; y = {[}6 3 2 1{]};   ` {\emph{Given}} \newline `a = polyfit(x,log(y),1)`  \newline `\textgreater{}\textgreater{}a = -0.5781 2.3411; ` \newline `m = a(1) ` \newline `\textgreater{}\textgreater{}m = -0.5781;` \newline `b = exp(a(2));` \newline `\textgreater{}\textgreater{}b = 10.3923;` \newline The final product: {\emph{y = 10.39e\textasciicircum{}-0.5781x\textasciicircum{}}}}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.3433 cm} p{0.3433 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Eigenvalues \& Eigenvectors}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 0 (+ 0)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{`A={[}2 -1 0;-1 2 -1;0 -1 2{]}; a=eig(A);` \newline `{[}V,D{]} = eig(A)` {\emph{V - eigenvectors}} \newline `V(:,1)`:{\emph{get 1st column}} \newline `v1 = V(:,1);` \newline `v1 = v1/v1(1);`: normalise eigenvector}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.3433 cm} p{0.3433 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{ODEs using ode45}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 0 (+ 0)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{`a = 0.1; b = 0.002; c = 0.0025; d = 0.2;` \newline `f = @(t,x) {[}-a{\emph{x(1)+b}}x(1){\emph{x(2);` \newline `d}}x(2)-c{\emph{x(1)}}x(2){]};` \newline `X0 = {[}20; 100{]};` \newline `Tspan = linspace(0,150,1000);` \newline `{[}T,X{]} = ode45(f,Tspan,X0);` \newline `plot(T,X,'LineWidth',2);`}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.3433 cm} p{0.3433 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{ODEs using ode45}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 0 (+ 0)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{`a = 0.1; b = 0.002; c = 0.0025; d = 0.2;` \newline `f = @(t,x) {[}-a{\emph{x(1)+b}}x(1){\emph{x(2);` \newline `d}}x(2)-c{\emph{x(1)}}x(2){]};` \newline `X0 = {[}20; 100{]};` \newline `Tspan = linspace(0,150,1000);` \newline `{[}T,X{]} = ode45(f,Tspan,X0);` \newline `plot(T,X,'LineWidth',2);`}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{1.7165 cm} x{1.7165 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Gauss Integration}}  \tn
% Row 0
\SetRowColor{LightBackground}
Gauss 2-pt rule & Gauss 3-pt rule \tn 
% Row Count 1 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{`f=@(x) 1/2{\emph{sqrt(16+(x+3).\textasciicircum{}4)./(x+3)\textasciicircum{}2;`  \newline `G2=f(-1/sqrt(3))+f(1/sqrt(3));` \newline `\textgreater{}\textgreater{}G2= 1.1289;` \newline and \newline `G3=5/9}}f(-sqrt(3/5))+8/9{\emph{f(0)+5/9}}f(sqrt(3/5));` \newline `\textgreater{}\textgreater{}G3= 1.1319;`}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.3433 cm} p{0.3433 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{ODEs: Boundary-value problems (BVPs)}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{Finite Difference Method} \tn 
% Row Count 1 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{`y0=1;y1=2; \% N = 1/h;h = 0.2`  \newline `i = {[}1:N-1{]}';` \newline `xi = i{\emph{h;    \% Interior nodes`  \newline `A=(h\textasciicircum{}2-2)}}eye(N-1); \% Diagonal part of A ` \newline `A=A+diag(1-0.5{\emph{h}}xi(2:N-1),-1);\% Add sub- and superdiagonal` \newline `A=A+diag(1+0.5{\emph{h}}xi(1:N-2),+1);\% Add sub- and superdiagonal` \newline `b = 2{\emph{h\textasciicircum{}2}}xi;\% RHS refined formula` \newline `b(1)=b(1)-(1-0.5{\emph{h}}xi(1)){\emph{y0;\% Add boundary contributions` \newline `b(N-1)=b(N-1)-(1+0.5}}h{\emph{xi(N-1))}}y1;\% Add boundary contributions` \newline `y = sparse(A)\textbackslash{}b;   \% Solve`}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Finite Difference: Laplace\&Poisson PDEs}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{3.833cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/kamva_1668461848_Capture4.PNG}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{You then use backslash after you have the equations}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.3433 cm} p{0.3433 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Multiple Integrals}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 0 (+ 0)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{`f = @(x,y) x.\textasciicircum{}2-y.\textasciicircum{}2-1;`           \newline `a = 0; b = 3; g = 0; p = 1;`            \newline `integral2(f,a,b,g,p);`               \newline `\textgreater{}\textgreater{}ans = 5.000000000001309;` \newline or \newline `f = @(x,y) y;`                        \newline `a = 0; b = pi/4;`                       \newline `g = @(x) sin(x); p = @(x) cos(x);` \newline `m = integral2(f,a,b,g,p);` \newline `\textgreater{}\textgreater{}m = 0.250000000000013;`}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.3433 cm} p{0.3433 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Curve fitting\& Interpolation}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{Least Squares: Backslash \& Polyfit} \tn 
% Row Count 1 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{{\bf{y = a0 + a1x + a2x\textasciicircum{}2\textasciicircum{}}}: \newline `x = {[}1; 2; 3; 4{]}; y = {[}6; 3; 2; 1{]};` \newline `A = {[}ones(4,1) x x.ˆ2{]}; a = A\textbackslash{}y` \newline `a = 9.5000 -4.1000 0.5000` \newline `a2 = polyfit(x,y,2)` \newline `xx = linspace(0,5);` \newline `y2 = polyval(a2,xx);` \newline `plot(x,y,'r.',xx,y2,'b-')`}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.3433 cm} p{0.3433 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Error,Residual,Norms,Condition number}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 0 (+ 0)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{`x = {[}1; -1; 3; -5{]};` \newline `norm(x,inf)`     ||x||inf(subscript) \newline ||A||2 = (||A{\bf{x}}||2)/(||{\bf{x}}||2) \newline \textasciitilde{} cond(A) = (||A||)(||A\textasciicircum{}-1\textasciicircum{}||) \newline \textasciitilde{}`cond(A)` \textgreater{}\textgreater{} 1:ill-conditioned\&close singular matrix \newline \textasciitilde{}`cond(A)` ≈ 1:well-conditioned \newline \textasciitilde{} decimals digits that trustworthy = 10\textasciicircum{}-a\textasciicircum{} = 10\textasciicircum{}c\textasciicircum{} x 10\textasciicircum{}-d\textasciicircum{} \newline a = d-c \newline \textasciitilde{}r = b - A{\bf{x}} \newline det (A) = 0; No A\textasciicircum{}-1\textasciicircum{}; Singular matrix \newline test singularity: `cond,condest,rank`}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{x{2.12846 cm} p{1.30454 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Built-in Functions}}  \tn
% Row 0
\SetRowColor{LightBackground}
Integral & Trapz \tn 
% Row Count 1 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{`f=@(x)sqrt(1+x.\textasciicircum{}4)./x.\textasciicircum{}2;`  \newline `L=integral(f,1,2);`                 \newline `\textgreater{}\textgreater{}L=1.132090393305918;`    \newline or \newline `y={[}0 0.3 0.5 1 1.5 2 2.5 3 4 5{]};`                 \newline `v={[}0 0.4 0.5 0.56 0.6 0.63 0.66 0.68 0.71 0.72{]};`  \newline `Q=10*trapz(y,v);` \newline `\textgreater{}\textgreater{}Q=30.8000;`}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Euler's Method}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{3.833cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/kamva_1668460132_Capture17.PNG}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{`f=@(x,y) -x{\emph{y;                     \% Define rhs`  \newline `N=100;                               \% Number of steps`  \newline `a=1; b=2;                           \% Interval` \newline `h=(b-a)/N;                           \% step size`  \newline `x(1)=1; y(1)=4;                   \% Initial values`  \newline `for i=1:N` \newline `y(i+1)=y(i)+h}}f(x(i),y(i));     \% Euler` \newline `x(i+1)=x(i)+h;` \newline `end`}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.3433 cm} p{0.3433 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Runge-Kutta Methods}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 0 (+ 0)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{`f = @(x,y) -x{\emph{y;                   \%Define rhs`  \newline `N = 100;                             \% Number of steps ` \newline ....             \newline `for i = 1:N` \newline     `K1 = f(x(i),y(i));` \newline     `K2 = f(x(i)+0.5}}h,y(i)+0.5{\emph{h}}K1);` \newline     `K3 = f(x(i)+0.5{\emph{h,y(i)+0.5}}h{\emph{K2);` \newline     `K4 = f(x(i+1),y(i)+h}}K3);`            \newline     `x(i+1) = x(i)+h;` \newline     `y(i+1) = y(i)+(1/6){\emph{h}}(K1+2K2+2K3+K4);` \newline `end`}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.3433 cm} p{0.3433 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Runge-Kutta Methods}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 0 (+ 0)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{`f = @(x,y) -x{\emph{y;                   \%Define rhs`  \newline `N = 100;                             \% Number of steps ` \newline .... \newline `for i = 1:N` \newline     `K1 = f(x(i),y(i));` \newline     `K2 = f(x(i)+0.5}}h,y(i)+0.5{\emph{h}}K1);` \newline     `K3 = f(x(i)+0.5{\emph{h,y(i)+0.5}}h{\emph{K2);` \newline     `K4 = f(x(i+1),y(i)+h}}K3);`            \newline     `x(i+1) = x(i)+h;` \newline     `y(i+1) = y(i)+(1/6){\emph{h}}(K1+2K2+2K3+K4);` \newline `end`}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Numerical Differentiation}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{3.833cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/kamva_1668457102_Capture10.PNG}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{`x = 0;`                                        \newline `f = @(x) sqrt(1-2{\emph{sin(x));`                     \newline `h = {[}0.002 0.001 0.0005 0.00025{]};`                 \newline `D2f = @(h) (f(x)-2}}f(x+h)+f(x+2{\emph{h))./h.\textasciicircum{}2;`   \newline `D2f(h) = -1.0040 -1.0020 -1.0010 -1.0005; `}}error decreases by factor 2 thus n = 1}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.3433 cm} p{0.3433 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Composite Simpson's Rule}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{Remember: N even} \tn 
% Row Count 1 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{`f = @(x) sqrt(1+x.\textasciicircum{}4)./x.\textasciicircum{}2;`                         \newline `a = 1; b = 2;`                                                  \newline `N = 100; h = (b-a)/N;` \newline `x = a+{[}0:N{]}{\emph{h;`                                              \newline `fx = f(x);` \newline `L = h/3}}(fx(1)+4{\emph{sum(fx(2:2:N))+2}}sum(fx(3:2:N-1))+fx(N+1))` \newline or \newline `w = ones(size(x)); w(2:2:N) = 4; w(3:2:N-1) = 2;` \newline `L = h/3{\emph{sum(w.}}f(x))` \newline `\textgreater{}\textgreater{}L = 1.132090394695845;`}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.3433 cm} p{0.3433 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Composite Trapezium Rule}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{} \tn 
% Row Count 0 (+ 0)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{`f = @(x) sqrt(1+x.\textasciicircum{}4)./x.\textasciicircum{}2;`              \newline `a = 1; b = 2;`                                  \newline `N = 100; h = (b-a)/N;`                        \newline `x = a+{[}0:N{]}{\emph{h;` \newline `fx = f(x);` \newline `L = h3}}( 0.5{\emph{fx(1) + sum(fx(2:N)) + 0.5}}fx(N+1) )` \newline or  \newline `w = ones(size(x)); w(1) = 0.5; w(N+1) = 0.5;`  \newline `L = h{\emph{sum(w.}}f(x));` \newline `\textgreater{}\textgreater{}L = 1.132101672788808;`}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Numerical Partial Differentiation}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{3.833cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/kamva_1668457464_Capture11.PNG}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{`x = 0; y = 0;`                                 \newline `f = @(x,y) sqrt(1+2{\emph{x-3}}y.\textasciicircum{}2);`           \newline `h = {[}0.2 0.1 0.05 0.025{]};`                       \newline `Lapf = @(h) (f(x+h,y)+f(x-h,y)+f(x,y+h)+f(x,y-h)-4*f(x,y))./h.\textasciicircum{}2; ` \newline `Lapf(h) = -4.1505 -4.0356 -4.0088 -4.0022;`}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Modified Euler}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{3.833cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/kamva_1668460466_Capture18.PNG}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{`f = @(x,y) -x{\emph{y;    \% Define rhs`  \newline `N = 100;           \% Number of steps`  \newline `a = 1; b = 2;      \% Interval` \newline `h= (b-a)/N;         \% step size` \newline `x(1)=1;y(1) = 4;  \% Initial values`  \newline `for i = 1:N7` \newline `yeu = y(i)+h}}f(x(i),y(i)); \% Euler` \newline `x(i+1) = x(i)+h;` \newline `y(i+1) = y(i)+0.5{\emph{h}}(f(x(i),y(i))+f(x(i+1),yeu));` \newline `end`}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{p{0.3433 cm} p{0.3433 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{3.833cm}}{\bf\textcolor{white}{Spline}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{Extrapolate data set} \tn 
% Row Count 1 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{3.833cm}}{`x = {[}22 35 48 61 74{]};`                      \newline `y = {[}320 490 540 500 480{]};`                 \newline `yy = spline(x,y,42);`  {\emph{not-a-knot}}        \newline `\textgreater{}\textgreater{}yy =  531.1971;`                         \newline `xx = linspace(20,80,1000);` \newline *clamped conditions: `{[}0 y 0{]} `               \newline `ss = spline(x,y,xx);`                     \newline `plot(x,y,'r.',xx,ss,'b-');`}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}