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    \Large Izpit iz Numeri"cnih metod\vspace*{3mm} \\
    \normalsize 14. februar 2011 
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\begin{enumerate}

\item Re"si robni problem:
\begin{eqnarray*}
    \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial
y^2}=x^2+y^2-1,\quad \text{na obmo"cju}\quad \mathcal{D}&=&[0,1]\times[0,1]\\
	\text{na robu obmo"cja}\quad u(x,y)|_{\partial{\mathcal{D}}}&=&x-y
\end{eqnarray*}
Druge parcialne odvode nadomesti"s z drugimi razlikami.
Koliko je pribli"zna vrednost $u(\frac12,\frac12)$, "ce izbere"s $h=\frac12$? 

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Re"sitev: $$\frac{u_{0,1/2}+u_{1,1/2}-4u_{1/2,1/2}+u_{1/2,0}+u_{1/2,1}}{\frac1{2^2}}=1-x_{\frac12}^2-y_{\frac12}^2$$
$u_{0,\frac12}=-\frac12, u_{1,\frac12}=\frac12, u_{\frac12,0}=\frac12, u_{\frac12,1}=-\frac12, x_{\frac12}=
\frac12, y_{\frac12}=\frac12$
$$-16 u_{\frac12,\frac12}=(1-\frac14-\frac14+\frac12-\frac12-\frac12+\frac12),\quad u_{\frac12,\frac12}=\frac1{32}$$
\item Dolo"ci parametra $a$ in $b$ tako, da se bo graf funkcije 
$$
	y=a+bx,\quad\text{"cim bolj prilegal  podatkom}\quad \{(0,1),(1,\frac{1}{3}),(2,\frac{1}{8})\}.
$$
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Re"sitev: Oredolo"cen sistem $$1=0 a+b,\quad 1/3=a+b,\quad  1/8=2 a+b, \rightarrow a=-7/16,\quad b=-133/144$$


\item S pomo"cjo Newtonove iteracije, dolo"ci $\sqrt{15}$ na pet decimalnih mest natan"cno. 

Re"sitev $f(x)=x^2-15,\quad x_0=4$,  $x=x-(x^2-15)/(2*x)$, $x=[4,3.8750,3.8730]$.

\end{enumerate}
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