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\Large \centerline{\textbf{IZPIT IZ NUMERI"CNIH METOD}}
\centerline{\textbf{12. september 2013}}

\normalsize\9\9
\begin{enumerate}
  \9\9\item Dane so tri to"cke v ravnini
    $$
      A(-2,1), B(1,-1) \ \textrm{in} \ C(3,2).
    $$
    Skozi te tri to"cke interpolirajte polinom druge stopnje. Z Newtonovo metodo dolo"cite teme tako dobljene parabole.
  \9\9\item Dan je sistem $Ax = b$, kjer je
    $$
      A = \left[ \begin{array}{cc} -3 & 1 \\ 2 & -4 \end{array} \right]
      \quad \textrm{in} \quad b = \left[ \begin{array}{c} -5 \\ 0 \end{array} \right].
    $$
    Ali Gauss-Seidlova iteracija konvergira? Odgovor utemeljite! Naredite tri korake Gauss-Seidlove iteracije
    z za"cetnim pribli"zkom $x_0 = \left[ \begin{array}{c} 0 \\ 0 \end{array} \right]$.
  \9\9\item Dolo"cite ute"zi kvadraturne formule
    $$
      \int_0^2 \frac{f(x)}{\sqrt{x}} \, \dd x \approx \omega_1 f(\textstyle{\frac{1}{4}}) + \omega_2 f(1),
    $$
    da bo formula to"cna za $f(x) = 1$ in $f(x) = x$. Z uporabo te formule nato izra"cuanjte vrednost integrala
    $$
      \int_0^2 \frac{\ee^x}{\sqrt{x}} \, \dd x.
    $$
\end{enumerate}

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