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\title{{\bf {\Huge  Analiz\u a Numeric\u a II}\\
{\Large Note de curs}\\
}}
\author{{\bf \LARGE Aurelian Nicola \\
UNIVERSITATEA "OVIDIUS" CONSTAN\c TA}}
%\makeindex
\renewcommand{\contentsname}{{ \bf CUPRINS}}
\def\chaptername{{\bf CAPITOLUL}}
\renewcommand{\listfigurename}{{\bf List\u{a} de Figuri}}
\renewcommand{\listtablename}{{\bf List\u{a} de Tabele}}
\renewcommand{\tablename}{{\bf TABELUL}}
\renewcommand{\figurename}{{ \bf FIGURA}}
\renewcommand{\bibname}{{ \bf Bibliografie}} 
%\date{20 februarie 2006}

\newtheorem{definition}{Defini\c tie}
\newtheorem{theorem}{Teorema}
\newtheorem{lemma}{Lema}
\newtheorem{remark}{Observa\c tie}
\newtheorem{proposition}{Propozi\c tia}
\newtheorem{corollary}{Corolar}
\newtheorem{example}{Exemplu}
\newtheorem{exercise}{Exerci\c{t}iu}
\def\R{I\!\!R}
\def\N{I\!\!N}
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\def\ra{\Rightarrow}
\def\be{\begin{equation}}
\def\ee{\end{equation}}

\begin{document}

\pagestyle{empty}

\vspace*{6cm}
\begin{center}
{\bf \Huge Exemple formule}\\

\end{center}

%salt la pagina noua
\newpage

\section {Capitolul I}
Aceasta este litera de la mult. nr. reale $\R$ (de ex.  $\R^n$)\\
Pentru scrierea cu diacritice. Dac\u a se dore\c ste scrierea cu literele \u a, \c s, \c t, \^ i \c si \^ a....\\
% \\ este pentru a forta saltul la linie noua
se continu\u a cu...
%pentru formula pe rand nenumerotata cu  indici superori si inferiori
S\u a se arate c\u a $\sum\limits_{i-1}^n x^4\leq 23,~\forall x\in[0,1].$ Dac\u a $x\in[0,\frac{1}{2}],~~{\rm atunci} S\geq 2$. \\
S\u a se arate c\u a
%pentru formula pe rand separat, dar nenumerotata
$$\int\limits_7^8 \displaystyle{\frac{3x}{\sqrt{x+4}}}{\rm dx}$$
poate fi calculat\u a prin diverse formule.\\
%pentru formula pe rand separat numerotata
\be
\lim\limits_{n\rightarrow\infty}\sqrt[n]{n}=1
\ee
% Tabel tip vector (o singura coloana)
%in acest mod se lasa un spatiu suplimentar de la inceputul liniei
\hspace*{2cm}
Acesta este un vector coloana.
%in acest mod se lasa un spatiu suplimentar intre cele doua randuri
\vspace*{2cm}
\begin{center}
\begin{tabular}{||c||}
$A_1$\\
$A_2$\\
$\dots$\\
$A_p$\\
\end{tabular}
\end{center}

%Tabel tip matrice (mai multe coloane)
Aceasta este o matrice.
\begin{center}
\begin{tabular}{||ccc||}
$A_1$ & $\dots$ & $A_N$\\
$A_2$ & $\dots$ & $A_N$\\
$\dots$ & $\dots$ & $\dots$\\
$A_p$ & $\dots$ & $A_N$\\
\end{tabular}
\end{center}

\subsection{Subcapitol 1}
\begin{equation}
\begin{tabular}{||c||}
$A_1$\\
$A_2$\\
$\dots$\\
$A_p$\\
\end{tabular}
\end{equation}

\subsection{Subcapitol 2}
\begin{equation}
A=\left[ 
\begin{array}{ll}
1 & x_{1} \\ 
1 & x_{2} \\ 
. & . \\ 
. & . \\ 
. & . \\ 
1 & x_{m}
\end{array}
\begin{array}{l}
x_{1}^{2} \\ 
x_{2}^{2} \\ 
. \\ 
. \\ 
. \\ 
x_{m}^{2}
\end{array}
\begin{array}{l}
. \\ 
. \\ 
. \\ 
. \\ 
. \\ 
.
\end{array}
\begin{array}{l}
. \\ 
. \\ 
. \\ 
. \\ 
. \\ 
.
\end{array}
\begin{array}{l}
. \\ 
. \\ 
. \\ 
. \\ 
. \\ 
.
\end{array}
\begin{array}{l}
x_{1}^{n} \\ 
x_{2}^{n} \\ 
. \\ 
. \\ 
. \\ 
x_{m}^{n}
\end{array}
\right] ,
\end{equation}
\section{Capitolul 2}
\begin{equation}
\left\{ 
\begin{array}{ll}
\Delta v=0\;\;\;\;, & \; {in}\;\;\;\Omega \\ 
\frac{\partial v}{\partial n}=0\;\;, &  {on}\;\;\;\Gamma \backslash
\Gamma _{1} \\ 
v=\delta, & {on} \;\;\;\Gamma_1, 
\end{array}
\right.
\end{equation}

\begin{eqnarray}
f_{i}(\omega ;b;x) &=&(1-\omega )x+\omega f_{i}(b;x), \\
F\left( \omega ;b;x\right) &=&(f_{1}\circ ...\circ f_{m})(\omega ;b;x), \\
\varphi _{j}(\alpha ;y) &=&(1-\alpha )y+\alpha \varphi _{j}(y), \\
\Phi (\alpha ;y) &=&(\varphi _{1}\circ ...\circ \varphi _{n})(\alpha ;y).
\end{eqnarray}

\begin{equation}
\left| \left\langle x^{k},a_{i}\right\rangle -b_{i}\right| =\left|
\left\langle Ax^{k}-b,e_{i}\right\rangle \right| , ~i=1,...,m,
\end{equation}

\be
A=\left[ 
\begin{array}{c}
A_{1} \\ 
A_{2} \\ 
. \\ 
. \\ 
. \\ 
A_{p}
\end{array}
\right] ,b=\left[ 
\begin{array}{c}
b_{1} \\ 
b_{2} \\ 
. \\ 
. \\ 
. \\ 
b_{p}
\end{array}
\right] ,
\ee


\begin{center}
\begin{tabular}{||c|c|c||} \hline \hline
\multicolumn{3}{||c||}{ \bf Table 5.}\\
\hline
{\rm n=16} & {\rm n=24} & {\rm n=32} \\
\hline
{\rm nc ~~nr} & {\rm nc ~~nr} & {\rm nc ~~nr} \\
\hline
0.0016 ~~5 $\cdot 10^{-5}$ & 19.80  ~~0.47 & 32.0  ~~0.59 \\
0.0019  ~~8 $\cdot 10^{-5}$ & 18.70  ~~0.50 & 25.6  ~~0.65 \\
0.0010  ~~6 $\cdot 10^{-5}$ & 5.65  ~~0.23 & 14.5  ~~0.45 \\
0.0009  ~~5 $\cdot 10^{-5}$ & 17.10  ~~0.71 & 26.9  ~~0.84 \\
0.0008  ~~3 $\cdot 10^{-5}$ & 7.00  ~~0.20 & 21.7  ~~0.48 \\
\hline
\end{tabular}
\end{center}

\be
\label{eq55}
%cu tagul \label se declara care este eticheta ecuatiei pentru a fi folosita ulterior in referintele incrutisate
 A = \left[
\begin{array}{cccc}
\Delta_h & E \\
S & 0 
\end{array}\right]
\ee
from (\ref{eq55})  we get (also using the symmetry of $\Delta_h$)
\be
\label{eq55}
A^t A = \left[
\begin{array}{cccc}
\Delta_h^2 + \tilde{I_1} & C \\
C^t & I_2 
\end{array}\right]
\ee
where 
\be
\label{eq56}
I_2 = E^t E = \left[
\begin{array}{cccc}
1  &   &  & 0 \\
   & 1 &  &   \\
   & \dots & \dots & \\
0  &   &   & 1
\end{array}\right]
\in {\cal M}_{n-3}~~,
\ee

% pentru referinte
Din ecua\c tia (\ref{eq56}), ob\c tinem c\u a....

\be
\label{eq57}
\tilde{I_1}=\left[
\begin{array}{cccccc}
1 & & & & & \\
& \ldots & & & & \\
& & 1 & & & \\
& & & 0 & & \\
& & & & \ldots & \\
& & & & & 0 
\end{array}\right]
\in {\cal M}_{(n-1)^2}
\ee
\be
\label{eq58}
C = \Delta_h E.
\ee
Pentru tabel\\
\begin{table}[htbp]
\begin{tabular}{|c|c|c|c|c|}
\hline
an&1&2&3&4\\ \hline
v\^ anz\u ari&23&27&30&34\\
\hline
\end{tabular}
\end{table}
\\
Pentru inserare de figuri.
\begin{figure}[!htbp]
	\centering
	\includegraphics[width=6.8cm,height=7cm]{LSS1.png}
	\caption{N(A) \c si LSS(A,b)}
	\label{fig:lss}
\end{figure} 

\end{document}













