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131 lines
13 KiB
TeX
131 lines
13 KiB
TeX
\documentclass[leqno]{article}
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%% Created with wxMaxima 19.07.0
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\setlength{\parskip}{\medskipamount}
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\setlength{\parindent}{0pt}
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\usepackage{iftex}
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\ifPDFTeX
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% PDFLaTeX or LaTeX
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\DeclareUnicodeCharacter{00B5}{\ensuremath{\mu}}
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\else
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% XeLaTeX or LuaLaTeX
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\usepackage{fontspec}
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\fi
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\usepackage{graphicx}
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\usepackage{color}
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\usepackage{amsmath}
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\usepackage{grffile}
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\usepackage{ifthen}
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\newsavebox{\picturebox}
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\newlength{\pictureboxwidth}
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\newlength{\pictureboxheight}
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\newcommand{\includeimage}[1]{
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\savebox{\picturebox}{\includegraphics{#1}}
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\settoheight{\pictureboxheight}{\usebox{\picturebox}}
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\settowidth{\pictureboxwidth}{\usebox{\picturebox}}
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\ifthenelse{\lengthtest{\pictureboxwidth > .95\linewidth}}
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{
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\includegraphics[width=.95\linewidth,height=.80\textheight,keepaspectratio]{#1}
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}
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{
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\ifthenelse{\lengthtest{\pictureboxheight>.80\textheight}}
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{
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\includegraphics[width=.95\linewidth,height=.80\textheight,keepaspectratio]{#1}
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}
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{
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\includegraphics{#1}
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}
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}
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}
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\newlength{\thislabelwidth}
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\DeclareMathOperator{\abs}{abs}
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\usepackage{animate} % This package is required because the wxMaxima configuration option
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% "Export animations to TeX" was enabled when this file was generated.
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\definecolor{labelcolor}{RGB}{100,0,0}
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\begin{document}
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\noindent
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%%%%%%%%%%%%%%%
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%%% INPUT:
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\begin{minipage}[t]{4em}\color{red}\bfseries
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(\% i1)
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\end{minipage}
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\begin{minipage}[t]{\textwidth}\color{blue}
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A: matrix([1 -l -s*l\_s,1-l -s*l\_s - Nsl\_S,M*(l\_s + Nl\_S)],[-s*l\_S,-Nsl\_S,M*Nl\_S],[-s*l\_D,-NSl\_D,1-delta + M*Nl\_D + g\_D]);
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\end{minipage}
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%%% OUTPUT:
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\[\displaystyle \tag{A}
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\begin{pmatrix}-{l_s} s-l+1 & -{l_s} s-l-{{\mathit{Nsl}}_S}+1 & M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \\
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-{l_S} s & -{{\mathit{Nsl}}_S} & M\, {{\mathit{Nl}}_S}\\
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-{l_D} s & -{{\mathit{NSl}}_D} & {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\end{pmatrix}\mbox{}
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\]
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%%%%%%%%%%%%%%%
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\noindent
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%%%%%%%%%%%%%%%
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%%% INPUT:
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\begin{minipage}[t]{4em}\color{red}\bfseries
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(\% i2)
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\end{minipage}
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\begin{minipage}[t]{\textwidth}\color{blue}
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invert(A);
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\end{minipage}
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%%% OUTPUT:
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\[\displaystyle \tag{\% o2}
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\begin{pmatrix}\frac{M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) } & \frac{-\left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) -M\, {{\mathit{NSl}}_D} \left( {l_s}+{{\mathit{Nl}}_S}\right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) } & \frac{M\, {{\mathit{Nl}}_S} \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +M\, {{\mathit{Nsl}}_S} \left( {l_s}+{{\mathit{Nl}}_S}\right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }\\
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\frac{\left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s}{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) } & \frac{\left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \left( -{l_s} s-l+1\right) +M\, {l_D} \left( {l_s}+{{\mathit{Nl}}_S}\right) s}{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) } & \frac{-M\, {{\mathit{Nl}}_S} \left( -{l_s} s-l+1\right) -M\, {l_S} \left( {l_s}+{{\mathit{Nl}}_S}\right) s}{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }\\
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\frac{{{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s}{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) } & \frac{{{\mathit{NSl}}_D} \left( -{l_s} s-l+1\right) -{l_D} s\, \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) } & \frac{{l_S} s\, \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) -{{\mathit{Nsl}}_S} \left( -{l_s} s-l+1\right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }\end{pmatrix}\mbox{}
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\]
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%%%%%%%%%%%%%%%
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\noindent
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%%%%%%%%%%%%%%%
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%%% INPUT:
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\begin{minipage}[t]{4em}\color{red}\bfseries
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(\% i3)
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\end{minipage}
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\begin{minipage}[t]{\textwidth}\color{blue}
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b: matrix([b1],[b2],[b3]);
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\end{minipage}
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%%% OUTPUT:
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\[\displaystyle \tag{b}
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\begin{pmatrix}\mathit{b1}\\
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\mathit{b2}\\
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\mathit{b3}\end{pmatrix}\mbox{}
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\]
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%%%%%%%%%%%%%%%
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\noindent
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%%%%%%%%%%%%%%%
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%%% INPUT:
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\begin{minipage}[t]{4em}\color{red}\bfseries
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(\% i8)
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\end{minipage}
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\begin{minipage}[t]{\textwidth}\color{blue}
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x : invert(A).b;
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\end{minipage}
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%%% OUTPUT:
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\[\displaystyle \tag{x}
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\begin{pmatrix}\frac{\mathit{b2}\, \left( -\left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) -M\, {{\mathit{NSl}}_D} \left( {l_s}+{{\mathit{Nl}}_S}\right) \right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }+\frac{\mathit{b3}\, \left( M\, {{\mathit{Nl}}_S} \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +M\, {{\mathit{Nsl}}_S} \left( {l_s}+{{\mathit{Nl}}_S}\right) \right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }+\frac{\mathit{b1}\, \left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }\\
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\frac{\mathit{b2}\, \left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \left( -{l_s} s-l+1\right) +M\, {l_D} \left( {l_s}+{{\mathit{Nl}}_S}\right) s\right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }+\frac{\mathit{b3}\, \left( -M\, {{\mathit{Nl}}_S} \left( -{l_s} s-l+1\right) -M\, {l_S} \left( {l_s}+{{\mathit{Nl}}_S}\right) s\right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }+\frac{\mathit{b1}\, \left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }\\
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\frac{\mathit{b3}\, \left( {l_S} s\, \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) -{{\mathit{Nsl}}_S} \left( -{l_s} s-l+1\right) \right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }+\frac{\mathit{b2}\, \left( {{\mathit{NSl}}_D} \left( -{l_s} s-l+1\right) -{l_D} s\, \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) \right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }+\frac{\mathit{b1}\, \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }\end{pmatrix}\mbox{}
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\]
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%%%%%%%%%%%%%%%
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\end{document}
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