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\author{ {\fontspec{Trebuchet MS}Marcin Chrz\k{a}szcz, Danny van Dyk} (UZH)}
\institute{UZH}
\title[Linear equation systems I]{Linear equation systems I}
\date{21 September 2016}


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			\flushright\fontspec{Trebuchet MS}\bfseries \Huge {Linear equation \\systems I}
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		  %\includegraphics[width=\textwidth]{SHiP-2}
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	\quad
	\vspace{3em}
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\flushright \vspace{-1.8em} {\fontspec{Trebuchet MS} \large Marcin Chrząszcz, Danny van Dyk\\\vspace{-0.1em}\small \href{mailto:mchrzasz@cern.ch}{mchrzasz@cern.ch}, \href{mailto:dany.van.dyk@gmail.com}{danny.van.dyk@gmail.com}}

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\vspace{1em}
%		\footnotesize\textcolor{gray}{With N. Serra, B. Storaci\\Thanks to the theory support from M. Shaposhnikov, D. Gorbunov}\normalsize\\
\vspace{0.5em}
	\textcolor{normal text.fg!50!Comment}{Numerical Methods, \\ 10 October, 2016}
\end{center}
\end{frame}
}


\begin{frame}\frametitle{Linear eq. system}

\ARROW This and the next lecture will focus on a well known problem. Solve the following equation system:
\begin{align*}
A \cdot x =b,
\end{align*} 
\ARROWR $A = a_{ij} 	\in \mathbb{R}^{n\times n}$ and $\det(A) \neq 0$\\
\ARROWR $b=b_i \in  \mathbb{R}^n$.\\
\ARROW The problem: Find the $x$ vector.



\end{frame}

\begin{frame}\frametitle{Error digression}
\begin{small}
\ARROW There is enormous amount of ways to solve the linear equation system.\\
\ARROW The choice of one over the other of them should be gathered by the {\it condition} of the matrix $A$ denoted at $cond(A)$.
\ARROW If the $cond(A)$ is small we say that the problem is well conditioned, otherwise we say it's ill conditioned.\\
\ARROW The {\it condition} relation is defined as:
\begin{align*}
cond(A) = \Vert A \Vert \cdot \Vert A^{-1} \Vert
\end{align*}
\ARROW Now there are many definitions of different norms... The most popular one (so-called ''column norm''):
\begin{align*}
\Vert A \vert_1 = \max_{1 \leq j \leq n} \sum_{i=1}^n \vert a_{i,j} \vert,
\end{align*}
where $n$ -is the dimension of $A$, $i,j$ are columns and rows numbers.



\end{small}
\end{frame}


\begin{frame}\frametitle{More norms}
\begin{small}
\ARROW A different norm is a spectral norm:
\begin{align*}
\Vert A \Vert_2 &= \sqrt{\rho(A^T A)}\\
\rho(M) &= \max \lbrace \vert\lambda_i \vert: \det{M- \lambda I} =0,~i=1,...n \rbrace
\end{align*}



\end{small}
\end{frame}


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