<commit_before>\section{Przypadki specjalne}

\subsection{Liniowy model}
  \begin{frame}{Liniowy model}
    \begin{block}{Minimalizacja $\chi^{2}$ (chisquare minimization, least squares fitting)}
      \begin{displaymath}
        min!F(\vec{x}) = \sum_{k=1}^{k} \left[ \frac{Y_{k} - T_{k}(\vec{x})}{\sigma_{k}} \right]^{2} = \sum_{k=1}^{k} f_{k}^{2}(\vec{x})
      \end{displaymath}
      gdzie: $Y_{k}$ -- wartości zamierzone, $\sigma_{k}$ -- błąd,\\
      $T_{k}$ -- wartości teoretyczne; $k \geq n$, zwykle $k \gg n$ \\
      \begin{tabular}{|c|} \hline
        min$F(\vec{x})$ \\ \hline
      \end{tabular}
      $\Rightarrow$ najlepsze estymaty parametrów $\vec{x}$
      (modelu teoretycznego)
    \end{block}

  \end{frame}

  \begin{frame}{Liniowy model}
    \begin{eqnarray}
      (*)\ \frac{\partial^2 F}{\partial x_{i} \partial x_{j}} =
      \frac{\partial}{\partial x_{i}} \frac{\partial}{\partial x_{j}}
      \sum_{k} f_{k}^2 = \frac{\partial}{\partial x_{i}}
      \sum_{k} 2 \cdot f_{k} \cdot \frac{\partial f_{k}}{\partial x_{j}} =
      \nonumber \\
    \end{eqnarray}
  \end{frame}<commit_msg>fixed first part of eq in 29th frame<commit_after>\section{Przypadki specjalne}

\subsection{Liniowy model}
  \begin{frame}{Liniowy model}
    \begin{block}{Minimalizacja $\chi^{2}$ (chisquare minimization, least squares fitting)}
      \begin{displaymath}
        min!F(\vec{x}) = \sum_{k=1}^{k} \left[ \frac{Y_{k} - T_{k}(\vec{x})}{\sigma_{k}} \right]^{2} = \sum_{k=1}^{k} f_{k}^{2}(\vec{x})
      \end{displaymath}
      gdzie: $Y_{k}$ -- wartości zamierzone, $\sigma_{k}$ -- błąd,\\
      $T_{k}$ -- wartości teoretyczne; $k \geq n$, zwykle $k \gg n$ \\
      \begin{tabular}{|c|} \hline
        min$F(\vec{x})$ \\ \hline
      \end{tabular}
      $\Rightarrow$ najlepsze estymaty parametrów $\vec{x}$
      (modelu teoretycznego)
    \end{block}

  \end{frame}

  \begin{frame}{Liniowy model}
    \begin{displaymath}
      (*)\ \frac{\partial^2 F}{\partial x_{i} \partial x_{j}} =
      \frac{\partial}{\partial x_{i}} \frac{\partial}{\partial x_{j}}
      \sum_{k} f_{k}^2 = \frac{\partial}{\partial x_{i}}
      \sum_{k} 2 \cdot f_{k} \cdot \frac{\partial f_{k}}{\partial x_{j}} =
    \end{displaymath}
    \begin{displaymath}
      \sum_{k} 2 \cdot \frac{\partial f_{k}}{\partial x_{i}} \cdot
      \frac{\partial f_{k}}{\partial x_{j}} +
      \sum_{k} 2 \cdot f_{k} \cdot
      \frac{\partial^2 f_{k}}{\partial x_{i} \partial x_{j}}
    \end{displaymath}

  \end{frame}
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