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% Document Info
\author{Boko}
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  /Title (analysis-part1-2.pdf)
  /Creator (Cheatography)
  /Author (Boko)
  /Subject (Analysis Part1-2 Cheat Sheet)
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\noindent
\begin{multicols}{3}
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    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Analysis Part1-2 Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{Boko} via \textcolor{DarkBackground}{\uline{cheatography.com/55472/cs/15052/}}}
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\noindent
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}Boko \\
  \uline{cheatography.com/boko} \\
  \end{tabulary}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 9th March, 2018.\\
   Page {\thepage} of \pageref{LastPage}.
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  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
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\begin{multicols*}{2}

\begin{tabularx}{8.4cm}{x{2.584 cm} x{2.508 cm} x{2.508 cm} }
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\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{Systems of Linear Equations - Methods}}  \tn
% Row 0
\SetRowColor{LightBackground}
Elimination Methods & {\emph{Inverse Method}} & Iterative Methods \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Need scale system because system becomes more sensitive to round offs & solve multiple times for different constants & make unknowns the subject of equations \tn 
% Row Count 8 (+ 6)
% Row 2
\SetRowColor{LightBackground}
Maximum Coefficients on Main diagonal & Advantages & default all unknowns are 0 \tn 
% Row Count 11 (+ 3)
% Row 3
\SetRowColor{white}
{\emph{Gauss Elimination}} & calculate inverse once & Dominant Diagonal System DDS \tn 
% Row Count 14 (+ 3)
% Row 4
\SetRowColor{LightBackground}
{\bf{1}} forward elimination {\bf{2}} back substitution & iterate for dynamic cases & DDS ensures convergence \tn 
% Row Count 18 (+ 4)
% Row 5
\SetRowColor{white}
eliminate what is below main diagonal & Limitations & {\emph{Gauss Seidel }} \tn 
% Row Count 21 (+ 3)
% Row 6
\SetRowColor{LightBackground}
Issues & matrix has to have a solution & use updated values in equations \tn 
% Row Count 24 (+ 3)
% Row 7
\SetRowColor{white}
Zero at pivot - solution: switch rows & \seqsplit{under-determined} systems (\# equations\textless{}\#unknowns) & if system is converging \tn 
% Row Count 28 (+ 4)
% Row 8
\SetRowColor{LightBackground}
ill conditioned system - round off & do not have an inverse - infinite solutions & {\emph{Jacobi}} \tn 
% Row Count 32 (+ 4)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{8.4cm}{x{2.584 cm} x{2.508 cm} x{2.508 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{Systems of Linear Equations - Methods (cont)}}  \tn
% Row 9
\SetRowColor{LightBackground}
Limitations & Augmentation & update values at the end of each iteration \tn 
% Row Count 4 (+ 4)
% Row 10
\SetRowColor{white}
Lengthy- Cumbersome- Time consuming & {[}A:I{]} -\textgreater{} {[}I:A-{]} & help overcome divergence \tn 
% Row Count 7 (+ 3)
% Row 11
\SetRowColor{LightBackground}
2 distinct steps & equations have to be linearly independent & Relaxation \tn 
% Row Count 11 (+ 4)
% Row 12
\SetRowColor{white}
{\emph{Gauss Jordan}} &  & Xinew= \textasciitilde{}Xinew + (1-\textasciitilde{})Xiold \tn 
% Row Count 13 (+ 2)
% Row 13
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eliminate what is above and below the main diagonal &  & 0\textless{}\textasciitilde{}\textless{}2 \tn 
% Row Count 17 (+ 4)
% Row 14
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translate from coefficient matrix to identity matrix &  & \textasciitilde{}=0 diverging (initial conditions are most accurate) \tn 
% Row Count 21 (+ 4)
% Row 15
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Advantage: no need for back substitution &  & \textasciitilde{}=1 regular \tn 
% Row Count 25 (+ 4)
% Row 16
\SetRowColor{white}
 &  & \textasciitilde{}=2 converging \tn 
% Row Count 27 (+ 2)
% Row 17
\SetRowColor{LightBackground}
 &  & \textasciitilde{}\textless{}1 diverging or converging with fluctuations \tn 
% Row Count 31 (+ 4)
\end{tabularx}
\par\addvspace{1.3em}

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\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{Systems of Linear Equations - Methods (cont)}}  \tn
% Row 18
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 &  & \textasciitilde{}\textgreater{}1 converging without fluctuations \tn 
% Row Count 3 (+ 3)
% Row 19
\SetRowColor{white}
 &  & as system grows , \textasciitilde{} is close to 1 \tn 
% Row Count 6 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{4 cm} x{4 cm} }
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\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Roots of Non linear Equations -{}-{}-Numerical Methods}}  \tn
% Row 0
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Bracketed Methods & Open Methods \tn 
% Row Count 1 (+ 1)
% Row 1
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2 initial guesses bracket the root & initial guesses do {\bf{not}} have to bracket root \tn 
% Row Count 4 (+ 3)
% Row 2
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to check that intial guesses bracket root: f(xl)*f(xu)\textless{}0 & Newton Raphson \tn 
% Row Count 7 (+ 3)
% Row 3
\SetRowColor{white}
Bisection Method & Takes into account {\bf{1}} initial guess {\bf{2}} function behavior {\bf{3}} rate of change \tn 
% Row Count 12 (+ 5)
% Row 4
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Xm= Xl+Xu / 2 & Xi+1= Xi- (f(xi)/ f'(xi)) \tn 
% Row Count 14 (+ 2)
% Row 5
\SetRowColor{white}
Limitations: & pitfalls \tn 
% Row Count 15 (+ 1)
% Row 6
\SetRowColor{LightBackground}
1 miss roots & diverge due to infliction point \tn 
% Row Count 17 (+ 2)
% Row 7
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2 inefficient (time consuming) & converge to local min/max \tn 
% Row Count 19 (+ 2)
% Row 8
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3 if even \# of roots between initial guesses are missed & jumping roots- converge to a different root \tn 
% Row Count 22 (+ 3)
% Row 9
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4 disregard function behavior; function of initial guesses & if xi is close is zero, it will offshute \tn 
% Row Count 25 (+ 3)
% Row 10
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False Position & Limitation: differentiation \tn 
% Row Count 27 (+ 2)
% Row 11
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Xr= Xu - (f(xu) * \seqsplit{(xl-xu))/(f(xl)-f(xu))} & Secant Method \tn 
% Row Count 29 (+ 2)
% Row 12
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in some cases, bisection may converge faster & xi+1 = xi - ((f(xi)* (xi-i \seqsplit{-xi))/(f(xi-1)-f(xi))} \tn 
% Row Count 32 (+ 3)
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\begin{tabularx}{8.4cm}{x{4 cm} x{4 cm} }
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\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Roots of Non linear Equations -{}-{}-Numerical Methods (cont)}}  \tn
% Row 13
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 & Modified Secant \tn 
% Row Count 1 (+ 1)
% Row 14
\SetRowColor{white}
 & 1 initial guess \tn 
% Row Count 2 (+ 1)
% Row 15
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 & xi-1= xi + oxi \tn 
% Row Count 3 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{3.76 cm} x{4.24 cm} }
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\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Roots of Non-linear Equations}}  \tn
% Row 0
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Analytical Solution & Graphical Solution \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
cannot solve complex equations & Visual Preceptions \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Roots of an equation & Miss roots due to choice of window \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{2}{x{8.4cm}}{find the value of independent variable when the dependent variable is zero.} \tn 
% Row Count 8 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{3.84 cm} x{4.16 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Systems of Linear Equations}}  \tn
% Row 0
\SetRowColor{LightBackground}
Graphical Solution & \# equations = \# unknowns \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Visual perception  - accuracy & 1 solution \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Time consuming & \# equations \textless{}  \# unknowns \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
impractical beyond 3D & infinite solutions \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
 & \# equations \textgreater{} \# unknowns \tn 
% Row Count 10 (+ 2)
% Row 5
\SetRowColor{white}
 & 1 solution (redundant equation) \tn 
% Row Count 12 (+ 2)
% Row 6
\SetRowColor{LightBackground}
 & no solution - do not intersect \tn 
% Row Count 14 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{5.04 cm} x{2.96 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Systems of Linear Equations - Cranmer's Rule}}  \tn
% Row 0
\SetRowColor{LightBackground}
D = determinant of coefficients & Limitations \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Dn = determinant of coefficients with n column replaced with B matrix & Time consuming \tn 
% Row Count 5 (+ 3)
% Row 2
\SetRowColor{LightBackground}
Singular System D=0 & if D=0 \tn 
% Row Count 6 (+ 1)
% Row 3
\SetRowColor{white}
1 no solution & ill- conditioned system \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
2 infinite solutions & D is close to 0 \tn 
% Row Count 10 (+ 2)
% Row 5
\SetRowColor{white}
 & instruction is a region \tn 
% Row Count 12 (+ 2)
% Row 6
\SetRowColor{LightBackground}
 & sensitive to round offs \tn 
% Row Count 14 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}


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

\end{document}