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875 lines
22 KiB
TeX
875 lines
22 KiB
TeX
\batchmode
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\documentclass[a4paper,twoside,11pt]{article}
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\usepackage{pstricks}
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\usepackage{fancybox}
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\usepackage{amsfonts}
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\usepackage{ifpdf}
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\usepackage[bookmarks=true, bookmarksnumbered=true, bookmarksopen=false, plainpages=false, pdfpagelabels, colorlinks, linkcolor=blue]{hyperref}
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\usepackage{ifthen}
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\usepackage{graphicx}
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\newtheorem{theorem}{Theorem}
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\newtheorem{corollary}{Corollary} \usepackage{rotating}
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\pdfoutput=0
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\setlength \textwidth{1.15\textwidth}
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\pagestyle{myheadings}
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\newcounter{subroutine}[subsection]
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\newcounter{example}[subroutine] \makeatletter%
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\providecommand{\subsubroutine}[2]{%
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\stepcounter{subroutine}%
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\subsection*{\flushleft #1---#2 \endflushleft}%
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\addcontentsline{toc}{subsubsection}{#1}%
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\markright{#1}}%%
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\providecommand{\examplename}{Example}%
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\providecommand{\syntaxname}{Syntax}
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\makeatother%
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\providecommand{\example}{\stepcounter{example}%
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\section*{Example~\arabic{example}}}
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%
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\providecommand{\precdata}{\hyperlink{precdata}{{\tt mld\_prec\_type}}}%
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\providecommand{\descdata}{\hyperlink{descdata}{{\tt psb\_desc\_type}}}%
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\providecommand{\spdata}{\hyperlink{spdata}{{\tt psb\_spmat\_type}}}%
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\providecommand{\Ref}[1]{\mbox{(\ref{#1})}}
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\usepackage[dvips]{pstcol}
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\pagecolor[gray]{.7}
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\usepackage[latin1]{inputenc}
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\makeatletter
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\makeatletter
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\count@=\the\catcode`\_ \catcode`\_=8
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\newenvironment{tex2html_wrap}{}{}%
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\catcode`\<=12\catcode`\_=\count@
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\newcommand{\providedcommand}[1]{\expandafter\providecommand\csname #1\endcsname}%
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\newcommand{\renewedcommand}[1]{\expandafter\providecommand\csname #1\endcsname{}%
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\expandafter\renewcommand\csname #1\endcsname}%
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\newcommand{\newedenvironment}[1]{\newenvironment{#1}{}{}\renewenvironment{#1}}%
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\let\newedcommand\renewedcommand
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\let\renewedenvironment\newedenvironment
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\makeatother
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\let\mathon=$
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\let\mathoff=$
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\ifx\AtBeginDocument\undefined \newcommand{\AtBeginDocument}[1]{}\fi
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\newbox\sizebox
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\setlength{\hoffset}{0pt}\setlength{\voffset}{0pt}
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\addtolength{\textheight}{\footskip}\setlength{\footskip}{0pt}
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\addtolength{\textheight}{\topmargin}\setlength{\topmargin}{0pt}
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\addtolength{\textheight}{\headheight}\setlength{\headheight}{0pt}
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\addtolength{\textheight}{\headsep}\setlength{\headsep}{0pt}
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\setlength{\textwidth}{349pt}
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\newwrite\lthtmlwrite
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\makeatletter
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\let\realnormalsize=\normalsize
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\global\topskip=2sp
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\def\preveqno{}\let\real@float=\@float \let\realend@float=\end@float
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\def\@float{\let\@savefreelist\@freelist\real@float}
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\def\liih@math{\ifmmode$\else\bad@math\fi}
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\def\end@float{\realend@float\global\let\@freelist\@savefreelist}
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\let\real@dbflt=\@dbflt \let\end@dblfloat=\end@float
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\let\@largefloatcheck=\relax
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\let\if@boxedmulticols=\iftrue
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\def\@dbflt{\let\@savefreelist\@freelist\real@dbflt}
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\def\adjustnormalsize{\def\normalsize{\mathsurround=0pt \realnormalsize
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\parindent=0pt\abovedisplayskip=0pt\belowdisplayskip=0pt}%
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\def\phantompar{\csname par\endcsname}\normalsize}%
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\def\lthtmltypeout#1{{\let\protect\string \immediate\write\lthtmlwrite{#1}}}%
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\newcommand\lthtmlhboxmathA{\adjustnormalsize\setbox\sizebox=\hbox\bgroup\kern.05em }%
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\newcommand\lthtmlhboxmathB{\adjustnormalsize\setbox\sizebox=\hbox to\hsize\bgroup\hfill }%
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\newcommand\lthtmlvboxmathA{\adjustnormalsize\setbox\sizebox=\vbox\bgroup %
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\let\ifinner=\iffalse \let\)\liih@math }%
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\newcommand\lthtmlboxmathZ{\@next\next\@currlist{}{\def\next{\voidb@x}}%
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\expandafter\box\next\egroup}%
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\newcommand\lthtmlmathtype[1]{\gdef\lthtmlmathenv{#1}}%
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\newcommand\lthtmllogmath{\dimen0\ht\sizebox \advance\dimen0\dp\sizebox
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\ifdim\dimen0>.95\vsize
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\lthtmltypeout{%
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*** image for \lthtmlmathenv\space is too tall at \the\dimen0, reducing to .95 vsize ***}%
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\ht\sizebox.95\vsize \dp\sizebox\z@ \fi
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\lthtmltypeout{l2hSize %
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:\lthtmlmathenv:\the\ht\sizebox::\the\dp\sizebox::\the\wd\sizebox.\preveqno}}%
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\newcommand\lthtmlfigureA[1]{\let\@savefreelist\@freelist
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\lthtmlmathtype{#1}\lthtmlvboxmathA}%
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\newcommand\lthtmlpictureA{\bgroup\catcode`\_=8 \lthtmlpictureB}%
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\newcommand\lthtmlpictureB[1]{\lthtmlmathtype{#1}\egroup
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\let\@savefreelist\@freelist \lthtmlhboxmathB}%
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\newcommand\lthtmlpictureZ[1]{\hfill\lthtmlfigureZ}%
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\newcommand\lthtmlfigureZ{\lthtmlboxmathZ\lthtmllogmath\copy\sizebox
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\global\let\@freelist\@savefreelist}%
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\newcommand\lthtmldisplayA{\bgroup\catcode`\_=8 \lthtmldisplayAi}%
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\newcommand\lthtmldisplayAi[1]{\lthtmlmathtype{#1}\egroup\lthtmlvboxmathA}%
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\newcommand\lthtmldisplayB[1]{\edef\preveqno{(\theequation)}%
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\lthtmldisplayA{#1}\let\@eqnnum\relax}%
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\newcommand\lthtmldisplayZ{\lthtmlboxmathZ\lthtmllogmath\lthtmlsetmath}%
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\newcommand\lthtmlinlinemathA{\bgroup\catcode`\_=8 \lthtmlinlinemathB}
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\newcommand\lthtmlinlinemathB[1]{\lthtmlmathtype{#1}\egroup\lthtmlhboxmathA
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\vrule height1.5ex width0pt }%
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\newcommand\lthtmlinlineA{\bgroup\catcode`\_=8 \lthtmlinlineB}%
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\newcommand\lthtmlinlineB[1]{\lthtmlmathtype{#1}\egroup\lthtmlhboxmathA}%
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\newcommand\lthtmlinlineZ{\egroup\expandafter\ifdim\dp\sizebox>0pt %
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\expandafter\centerinlinemath\fi\lthtmllogmath\lthtmlsetinline}
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\newcommand\lthtmlinlinemathZ{\egroup\expandafter\ifdim\dp\sizebox>0pt %
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\expandafter\centerinlinemath\fi\lthtmllogmath\lthtmlsetmath}
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\newcommand\lthtmlindisplaymathZ{\egroup %
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\centerinlinemath\lthtmllogmath\lthtmlsetmath}
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\def\lthtmlsetinline{\hbox{\vrule width.1em \vtop{\vbox{%
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\kern.1em\copy\sizebox}\ifdim\dp\sizebox>0pt\kern.1em\else\kern.3pt\fi
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\ifdim\hsize>\wd\sizebox \hrule depth1pt\fi}}}
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\def\lthtmlsetmath{\hbox{\vrule width.1em\kern-.05em\vtop{\vbox{%
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\kern.1em\kern0.8 pt\hbox{\hglue.17em\copy\sizebox\hglue0.8 pt}}\kern.3pt%
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\ifdim\dp\sizebox>0pt\kern.1em\fi \kern0.8 pt%
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\ifdim\hsize>\wd\sizebox \hrule depth1pt\fi}}}
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\def\centerinlinemath{%
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\dimen1=\ifdim\ht\sizebox<\dp\sizebox \dp\sizebox\else\ht\sizebox\fi
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\advance\dimen1by.5pt \vrule width0pt height\dimen1 depth\dimen1
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\dp\sizebox=\dimen1\ht\sizebox=\dimen1\relax}
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\def\lthtmlcheckvsize{\ifdim\ht\sizebox<\vsize
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\ifdim\wd\sizebox<\hsize\expandafter\hfill\fi \expandafter\vfill
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\else\expandafter\vss\fi}%
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\providecommand{\selectlanguage}[1]{}%
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\makeatletter \tracingstats = 1
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\begin{document}
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\pagestyle{empty}\thispagestyle{empty}\lthtmltypeout{}%
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\lthtmltypeout{latex2htmlLength hsize=\the\hsize}\lthtmltypeout{}%
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\lthtmltypeout{latex2htmlLength vsize=\the\vsize}\lthtmltypeout{}%
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\lthtmltypeout{latex2htmlLength hoffset=\the\hoffset}\lthtmltypeout{}%
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\lthtmltypeout{latex2htmlLength voffset=\the\voffset}\lthtmltypeout{}%
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\lthtmltypeout{latex2htmlLength topmargin=\the\topmargin}\lthtmltypeout{}%
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\lthtmltypeout{latex2htmlLength topskip=\the\topskip}\lthtmltypeout{}%
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\lthtmltypeout{latex2htmlLength headheight=\the\headheight}\lthtmltypeout{}%
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\lthtmltypeout{latex2htmlLength headsep=\the\headsep}\lthtmltypeout{}%
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\lthtmltypeout{latex2htmlLength parskip=\the\parskip}\lthtmltypeout{}%
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\lthtmltypeout{latex2htmlLength oddsidemargin=\the\oddsidemargin}\lthtmltypeout{}%
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\makeatletter
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\if@twoside\lthtmltypeout{latex2htmlLength evensidemargin=\the\evensidemargin}%
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\else\lthtmltypeout{latex2htmlLength evensidemargin=\the\oddsidemargin}\fi%
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\lthtmltypeout{}%
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\makeatother
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\setcounter{page}{1}
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\onecolumn
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% !!! IMAGES START HERE !!!
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\newlength{\centeroffset}%
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\begingroup
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\renewcommand{\thepage}{toc}
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\endgroup
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\stepcounter{section}
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{\newpage\clearpage
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\lthtmldisplayA{displaymath238}%
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\begin{displaymath}
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Ax=b,
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\end{displaymath}%
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\lthtmldisplayZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline265}%
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$A$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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\stepcounter{section}
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\stepcounter{section}
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\stepcounter{section}
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\stepcounter{subsection}
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\stepcounter{subsection}
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\stepcounter{subsection}
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\stepcounter{subsection}
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\stepcounter{section}
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\stepcounter{subsection}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline586}%
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$A=(a_{ij}) \in \Re^{n \times n}$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline588}%
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$G=(W,E)$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline592}%
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$W=\{1, 2, \ldots, n\}$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline594}%
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$E=\{(i,j) : a_{ij} \neq 0\}$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline596}%
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$G$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline598}%
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$\delta > 0$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline600}%
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$\delta$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline602}%
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$W$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline606}%
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$m$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline608}%
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$W_i^0 \subset W$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline610}%
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$\cup_{i=1}^m W_i^0 = W$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline616}%
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$W_i^\delta \supset W_i^{\delta-1}$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline618}%
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$W_i^{\delta-1}$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline620}%
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$n_i^\delta$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline622}%
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$W_i^\delta$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline624}%
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$R_i^{\delta} \in
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\Re^{n_i^\delta \times n}$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline626}%
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$v \in \Re^n$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline628}%
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$v_i^{\delta} \in \Re^{n_i^\delta}$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline630}%
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$v$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline634}%
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$R_i^{\delta}$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline636}%
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$\Re^{n_i^\delta}$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline638}%
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$\Re^n$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline640}%
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$A_i^\delta=R_i^\delta A (R_i^\delta)^T \in
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\Re^{n_i^\delta \times n_i^\delta}$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline644}%
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$W_i^{\delta}$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmldisplayA{displaymath564}%
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\begin{displaymath}
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M_{AS}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
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(A_i^\delta)^{-1} R_i^{\delta},
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\end{displaymath}%
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\lthtmldisplayZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline646}%
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$A_i^\delta$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline652}%
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$v_i = R_i^{\delta} v$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline654}%
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$i=1,\ldots,m$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline656}%
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$A_i^\delta w_i = v_i$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline660}%
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$w_i$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline662}%
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$w = \sum_{i=1}^m (R_i^{\delta})^T w_i$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline664}%
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$p$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline668}%
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$p,t$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline672}%
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$(\tilde{R}_i^0)^T \in \Re^{n_i^\delta \times n}$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline674}%
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$\tilde{R}_i^0$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline676}%
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$R_i^\delta$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline678}%
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$W_i^\delta \backslash W_i^0$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
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{\newpage\clearpage
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\lthtmldisplayA{displaymath565}%
|
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\begin{displaymath}
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M_{RAS}^{-1}= \sum_{i=1}^m (\tilde{R}_i^0)^T
|
|
(A_i^\delta)^{-1} R_i^{\delta}.
|
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\end{displaymath}%
|
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\lthtmldisplayZ
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\lthtmlcheckvsize\clearpage}
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|
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{\newpage\clearpage
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\lthtmldisplayA{displaymath566}%
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\begin{displaymath} M_{ASH}^{-1}= \sum_{i=1}^m (R_i^{\delta})^T
|
|
(A_i^\delta)^{-1} \tilde{R}_i^0.
|
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\end{displaymath}%
|
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\lthtmldisplayZ
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\lthtmlcheckvsize\clearpage}
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|
|
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{\newpage\clearpage
|
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\lthtmlinlinemathA{tex2html_wrap_inline680}%
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$\delta=0$%
|
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\lthtmlinlinemathZ
|
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\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
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\lthtmlinlinemathA{tex2html_wrap_inline686}%
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$A_C$%
|
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\lthtmlinlinemathZ
|
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\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
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\lthtmlinlinemathA{tex2html_wrap_inline692}%
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$W_C$%
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\lthtmlinlinemathZ
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\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
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\lthtmlinlinemathA{tex2html_wrap_inline694}%
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$n_C$%
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\lthtmlinlinemathZ
|
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\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline696}%
|
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$R_C \in \Re^{n_C \times n}$%
|
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\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
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\lthtmldisplayA{displaymath567}%
|
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\begin{displaymath}
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A_C=R_C A R_C^T
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\end{displaymath}%
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\lthtmldisplayZ
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\lthtmlcheckvsize\clearpage}
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|
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline700}%
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$M_{1L}$%
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\lthtmlinlinemathZ
|
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\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
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\lthtmldisplayA{displaymath568}%
|
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\begin{displaymath}
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M_{C}^{-1}= R_C^T A_C^{-1} R_C,
|
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\end{displaymath}%
|
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\lthtmldisplayZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
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{\newpage\clearpage
|
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\lthtmlinlinemathA{tex2html_wrap_inline704}%
|
|
$M_{C}^{-1}$%
|
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\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline710}%
|
|
$M_{C}$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmldisplayA{displaymath569}%
|
|
\begin{displaymath}
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|
M_{2LA}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
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|
\end{displaymath}%
|
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\lthtmldisplayZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline714}%
|
|
$M_{2L-A}^{-1}$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline720}%
|
|
$M_{1L}^{-1}$%
|
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\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmldisplayA{displaymath570}%
|
|
\begin{displaymath}
|
|
\begin{array}{l}
|
|
w = M_{1L}^{-1} v, \\
|
|
z = w + M_{C}^{-1} (v-Aw);
|
|
\end{array}
|
|
\end{displaymath}%
|
|
\lthtmldisplayZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmldisplayA{displaymath571}%
|
|
\begin{displaymath}
|
|
M_{2LH-PRE}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1}.
|
|
\end{displaymath}%
|
|
\lthtmldisplayZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmldisplayA{displaymath572}%
|
|
\begin{displaymath}
|
|
\begin{array}{l}
|
|
w = M_{C}^{-1} v , \\
|
|
z = w + M_{1L}^{-1} (v-Aw) ,
|
|
\end{array}
|
|
\end{displaymath}%
|
|
\lthtmldisplayZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmldisplayA{displaymath573}%
|
|
\begin{displaymath}
|
|
M_{2LH-POST}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
|
|
\end{displaymath}%
|
|
\lthtmldisplayZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline734}%
|
|
$M$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline738}%
|
|
$w=M^{-1}v$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline740}%
|
|
$nlev$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline742}%
|
|
$l$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline744}%
|
|
$A_l$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline746}%
|
|
$M_l$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline748}%
|
|
$A_1=A$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlpictureA{tex2html_wrap860}%
|
|
\framebox{
|
|
\begin{minipage}{.85\textwidth} {\small
|
|
\begin{tabbing}
|
|
\quad \=\quad \=\quad \=\quad \\[-1mm]
|
|
$v_1 = v$; \\[2mm]
|
|
\textbf{for $l=2, nlev$\ do}\\[1mm]
|
|
\> ! transfer $v_{l-1}$\ to the next coarser level\\
|
|
\> $v_l = R_lv_{l-1}$\ \\[1mm]
|
|
\textbf{endfor} \\[2mm]
|
|
! apply the coarsest-level correction\\[1mm]
|
|
$y_{nlev} = A_{nlev}^{-1} v_{nlev}$\\[2mm]
|
|
\textbf{for $l=nlev -1 , 1, -1$\ do}\\[1mm]
|
|
\> ! transfer $y_{l+1}$\ to the next finer level\\
|
|
\> $y_l = R_{l+1}^T y_{l+1}$;\\[1mm]
|
|
\> ! compute the residual at the current level\\
|
|
\> $r_l = v_l-A_l^{-1} y_l$;\\[1mm]
|
|
\> ! apply the basic Schwarz preconditioner to the residual\\
|
|
\> $r_l = M_l^{-1} r_l$\\[1mm]
|
|
\> ! update $y_l$\\
|
|
\> $y_l = y_l+r_l$\\
|
|
\textbf{endfor} \\[1mm]
|
|
$w = y_1$;
|
|
\end{tabbing}
|
|
}
|
|
\end{minipage}
|
|
}%
|
|
\lthtmlpictureZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
\stepcounter{subsection}
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline776}%
|
|
$R_C$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline784}%
|
|
$R_C^T$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline798}%
|
|
$N_r$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline802}%
|
|
$r \in W$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmldisplayA{displaymath574}%
|
|
\begin{displaymath} N_r = \left\{s \in W: |a_{rs}| > \theta \sqrt{|a_{rr}a_{ss}|} \right\}
|
|
\cup \left\{ r \right\} ,
|
|
\end{displaymath}%
|
|
\lthtmldisplayZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline804}%
|
|
$\theta \in [0,1]$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline806}%
|
|
$i$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline808}%
|
|
$W_i^0$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline812}%
|
|
$P_C=R_C^T$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline814}%
|
|
$P \in \Re^{n \times n_C}$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmldisplayA{displaymath531}%
|
|
\begin{displaymath}
|
|
P=(p_{ij}), \quad p_{ij}=
|
|
\left\{ \begin{array}{ll}
|
|
1 & \quad \mbox{if} \; i \in V^j_C \\
|
|
0 & \quad \mbox{otherwise}
|
|
\end{array} \right. .
|
|
\end{displaymath}%
|
|
\lthtmldisplayZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline816}%
|
|
$P_C$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline818}%
|
|
$P$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline820}%
|
|
$S \in \Re^{n \times n}$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmldisplayA{displaymath542}%
|
|
\begin{displaymath}
|
|
P_C = S P,
|
|
\end{displaymath}%
|
|
\lthtmldisplayZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline822}%
|
|
$S$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmldisplayA{displaymath546}%
|
|
\begin{displaymath}
|
|
S = I - \omega D^{-1} A ,
|
|
\end{displaymath}%
|
|
\lthtmldisplayZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline824}%
|
|
$\omega$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline826}%
|
|
$D^{-1}A$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
\stepcounter{section}
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline983}%
|
|
$\theta = 0$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
\stepcounter{subsection}
|
|
\stepcounter{section}
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline1324}%
|
|
$w$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
\stepcounter{subsection}
|
|
\stepcounter{subsection}
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline1328}%
|
|
$\ge 0$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline1342}%
|
|
$t$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline1346}%
|
|
$\in [0, 1]$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline1348}%
|
|
$\theta$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline1352}%
|
|
$>0$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
\stepcounter{subsection}
|
|
\stepcounter{subsection}
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline1374}%
|
|
$y = op(M^{-1})\, x$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline1378}%
|
|
$op$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline1382}%
|
|
$x$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline1384}%
|
|
$y$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline1386}%
|
|
$op(M^{-1}) = M^{-1}$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline1388}%
|
|
$op(M^{-1}) = M^{-T}$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline1390}%
|
|
$M^{-1})$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
{\newpage\clearpage
|
|
\lthtmlinlinemathA{tex2html_wrap_inline1392}%
|
|
$op(M^{-1}) = M^{-C}$%
|
|
\lthtmlinlinemathZ
|
|
\lthtmlcheckvsize\clearpage}
|
|
|
|
\stepcounter{subsection}
|
|
\stepcounter{subsection}
|
|
\stepcounter{section}
|
|
\appendix
|
|
\stepcounter{section}
|
|
|
|
\end{document}
|