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\def\lea{\mathrel{\raise .4ex\hbox{\rlap{$<$}\lower 1.2ex\hbox{$\sim$}}}}

\makeatother
\newenvironment{tex2html_wrap}{}{}
\newwrite\lthtmlwrite
\def\lthtmltypeout#1{{\let\protect\string\immediate\write\lthtmlwrite{#1}}}%
\newbox\sizebox
\begin{document}
\pagestyle{empty}
\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}\delta F & = & 0 \\ 
F & = & \int_{\Omega} \frac{1}{2} \left( \nabla \varphi \right)^2
d^3 \underline{x} \\ 
\varphi (\underline{x}) & = & \mbox{known}\ f(x)\ \mbox{on}\
\partial\Omega
\end{eqnarray*}
}

\stepcounter{subsection}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}\varphi (\underline{r}) & = & \varphi_0+ \left( \varphi_1 - \varphi_0 \right)
\frac{\left( \underline{r} - \underline{r}_0 \right) \wedge \left(
\underline{r}_2 - \underline{r}_0 \right) \vert_z}{\left( \underline{r}_1 
- \underline{r}_0 \right) \wedge \left( \underline{r}_2 -
\underline{r}_0 \right) \vert_z} \\ 
&& \\ 
&& \phantom{\varphi_0} + \left( \varphi_2 - \varphi_0 \right)
\frac{\left( \underline{r} - \underline{r}_0 \right) \wedge \left(
\underline{r}_1 - \underline{r}_0 \right) \vert_z}{\left( \underline{r}_2 
- \underline{r}_0 \right) \wedge \left( \underline{r}_1 -
\underline{r}_0 \right) \vert_z}
\end{eqnarray*}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}F & = & \int_{\Delta} \frac{1}{2} \left( \nabla\varphi \right)^2 d^2 x \\ 
& = & \frac{1}{4} \left\lbrack \varphi_0\, \varphi_1\, \varphi_2
\right\rbrack \left\lbrack \begin{array}{ccc}
c_1 + c_2 & -c_2 & -c_1 \\ 
-c_2 & c_2 + c_0 & -c_0 \\ 
-c_1 & -c_0 & c_0 + c_1 \\  \end{array} \right\rbrack \quad
\left\lbrack \begin{array}{c}
\varphi_0 \\ 
\varphi_1 \\ 
\varphi_2 \\  \end{array} \right\rbrack \\ 
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 1025
$c_j = \cot\theta_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1025: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 1037
$S_ii = \displaystyle{\sum}_{\mbox{angles}\, \theta} \cot\, \theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1037: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 1039
$\theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1039: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 1059
$S_{ij} = - \left( \cot \theta_1 + \cot \theta_2
\right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1059: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}F & = & \frac{1}{4} \varphi^T {\twelvebxti S} \varphi \\ 
& = & \frac{1}{4} \sum_{ij} \varphi_j S_{ji} \varphi_i 
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\varphi_i & \rightarrow & \varphi_i + \eta_i \hspace{1em} \hbox{be
variation} \\ 
F & = & \frac{1}{4} \sum_{ij} \varphi_j S_{ji} \varphi_i \\ 
& + & \frac{1}{4} \left( \sum_{ij} \bigg( \eta_j S_{ji} \varphi_i
\bigg) + \bigg( \varphi_j S_{ji} \eta_i \bigg) \right)
\end{eqnarray*}
}

\stepcounter{subsection}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 1083
$\theta_1 +
\theta_2 \leq 180^\circ$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1083: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
\setcounter{equation}{0}
{\newpage
\clearpage
\samepage \begin{equation}F(\underline{x}) = \frac{1}{2} \underline{x}^T A\underline{x} -
\underline{x}^T b
\label{pp1.eqn}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}A\underline{x} = \underline{b}
\label{pp2.eqn}
\end{equation}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 1111
$\underline{x}^{(k+1)} = \underline{x}^{(k)} + \alpha
\underline{r}^{(k)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1111: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 1113
$\alpha$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1113: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 1117
$\alpha$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1117: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 318
F (\alpha) & = & \alpha^2 \left\lbrack \frac{1}{2}\,
\underline{r}^{(k)T} A \underline{r}^{(k)} \right\rbrack \\ 
& - & \alpha \underline{r}^{(k)T} \cdot \underline{r}^{(k)} \\ 
& + & \mbox{Terms Independent of}\ \alpha
\end{eqnarray*}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 334
F (\alpha) & = & \frac{1}{2} \underline{r}^{(k)T} A
\underline{r}^{(k)} \left( \alpha - \alpha_k \right)^2 \\ 
& + & \mbox{Terms Independent of}\ \alpha
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 1127
$\alpha_k = \underline{r}^{(k)T}\cdot
\underline{r}^{(k)} / \underline{r}^{(k)T} A
\underline{r}^{(k)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1127: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}% latex2html id marker 1123
\underline{x}^{(k+1)} = \underline{x}^{(k)} + \alpha_k
\underline{r}^{(k)}
\end{displaymath}
}

\stepcounter{subsection}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}A & = & P^{-1} \Lambda P \\ 
\Lambda & = & \mbox{diag.}\ \left( \lambda_1\,, \lambda_2\,, \ldots
\lambda_n\, \right) \\ 
& = & \left\lbrack \begin{array}{cccc}
\lambda_1 & 0 & 0 & \ldots \\ 
0 & \lambda_2 & 0 & \ldots \\ 
0 & 0 & \lambda_3 & \ldots \end{array} \right\rbrack \\ 
&& \lambda_1 \geq \lambda_2 \geq \ldots
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\underline{\tilde{x}}^{(k)} & = & P\underline{x}^{(k)} \\ 
\underline{\tilde{b}} & = & P\underline{b} \\ 
\underline{\tilde{r}}^{(k)} & = & P\underline{r}^{(k)} \\ 
\end{eqnarray*}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}&& \frac{1}{2} \underline{x}^T A \underline{x} - \underline{b}^T
\cdot \underline{x} \\ 
&& \\ 
& = & \frac{1}{2} \underbrace{\normalsize\underline{x}^T P^{-1}}_{x^T P^T}
\Lambda P\underline{x} - \underbrace{\underline{b}^T
P^{-1}}_{\normalsize(P\underline{b})^T} P\underline{x} \\ 
&& \\ 
& = & \frac{1}{2} ( P\underline{x} )^T \Lambda P\underline{x} - (
P\underline{b} )^T P\underline{x} \\ 
&& \\ 
& = & \frac{1}{2} \underline{\tilde{x}}^T \Lambda
\underline{\tilde{x}} - \tilde{b}^T \underline{\tilde{x}}
\end{eqnarray*}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 1163
$\alpha$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1163: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}% latex2html id marker 1153
\alpha^{(k)} = \frac{\underline{\tilde{r}}^{(k-1)T} \cdot
\underline{\tilde{r}}^{(k-1)}}{\underline{\tilde{r}}^{(k-1)T}
\Lambda \underline{\tilde{r}}^{(k-1)}} \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}% latex2html id marker 1154
\underline{\tilde{x}}^{(k)} = \underline{\tilde{x}}^{(k-1)} 
+ \alpha_k \underline{\tilde{r}}^{(k-1)} \end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 464
\underline{\tilde{r}}^{(k)} & = & \underline{\tilde{b}} - \Lambda\,
\underline{\tilde{x}}^{(k)} \\ 
& = & \underline{\tilde{b}} - \Lambda\, \underline{\tilde{x}}^{(k-1)}
- \alpha_k \Lambda \underline{\tilde{r}}^{(k-1)} \\ 
& = & \left( I - \alpha_k \Lambda \right) \tilde{r}^{(k-1)}
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{displaymath}% latex2html id marker 1155
\tilde{r}^{(k)} \big|_i = \left( 1 - \alpha_k \lambda_i
\right) \tilde{r}^{(k-1)} \big|_i \end{displaymath}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}% latex2html id marker 1167
\alpha_k = \frac{\sum_i \left\lbrack \left( \tilde{r}^{(k-1)}
\right) \big|_i \right\rbrack^2}{\sum_i \lambda_i \left\lbrack
\tilde{r}^{(k-1)} \big|_i \right\rbrack^2}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 507
\alpha_k & \sim & \frac{1}{\lambda_1} \ \mbox{and} \\ 
\tilde{r}^{(k)}\big|_1 & = & \left( 1 - \frac{\lambda_1}{\lambda_1}
\right) \tilde{r}^{(k-1)}\big|_1 \\ 
& \sim & 0
\end{eqnarray*}
}

\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}% latex2html id marker 1215
\underline{x}^{(k)} = \underline{x}^{(k-1)} + \alpha_k
\underline{p}^{(k)} \end{displaymath}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}\underline{r}^{(0)} & = & \underline{b} - A\underline{x}^{(0)} \\ 
k=1 \hspace{5em} p^{(1)} & = & \underline{r}^{(0)} \\ 
k>1 \hspace{5em} p^{(k)} & = & \underline{r}^{(k-1)} + \beta_k
\underline{p}^{(k-1)}
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 1233
$\alpha_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1233: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 1235
$\underline{x}^{(k-1)} + \alpha_k \underline{p}^{(k)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1235: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 597
\alpha_k & = & \underline{p}^{(k)T} \underline{p}^{(k)} /
\underline{p}^{(k)T} A\underline{p}^{(k)} \\ 
\underline{x}^{(k)} & = & \underline{x}^{(k-1)} + \alpha_k
\underline{p}^k \\ 
\underline{r}^{(k)} & = & b - A x^{(\underline{k})}
\end{eqnarray*}
}

\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}k & = & \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \\ 
& \propto & \frac{1}{h}
\end{eqnarray*}
}

\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}x_n^{(k)} & \mbox{requires} & x_{n-1}^{(k)}\ \ \ldots \\ 
&& \\ 
x_{n-1}^{(k)} & \mbox{requires} & x_{n-2}^{(k)}\ \ \ldots \\ 
\vdots\hspace{1em} \\ 
x_2^{(k)} & \mbox{requires} & x_1^{(k)}, \mbox{etc.} \\ 
\end{eqnarray*}
}

\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}\frac{1}{1-\rho} & = & \frac{2N^2}{\pi^2} \hspace{1em} \mbox{Jacobi} \\ 
& = & \frac{N^2}{\pi^2} \hspace{1.5em} \mbox{Gauss\ Seidel} \\ 
& = & \frac{N}{\pi} \hspace{2em} \mbox{SOR} \\ 
\end{eqnarray*}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}\omega_{opt} & = & 2 - \frac{\mbox{constant}}{N} \\ 
& = & 2 - \mbox{constant}\ \cdot\ h \\ 
\end{eqnarray*}
}

\stepcounter{subsection}

\end{document}
