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Zm!Oca3xY^{f$J0%WM~BibxrkX{{zZ1|LFh# literal 0 HcmV?d00001 diff --git a/semester3/numcs/numcs-summary.tex b/semester3/numcs/numcs-summary.tex index 1fd23ed..ec13881 100644 --- a/semester3/numcs/numcs-summary.tex +++ b/semester3/numcs/numcs-summary.tex @@ -4,6 +4,10 @@ \input{\dir/include.tex} \load{recommended} +\renewcommand{\authorTitle}{Robin Bacher, Janis Hutz\\\url{https://github.com/janishutz/eth-summaries}} +\renewcommand{\authorHeaders}{Robin Bacher, Janis Hutz} +\setLang{de} + \setup{Numerical Methods for Computer Science} \begin{document} @@ -13,7 +17,30 @@ % ╭────────────────────────────────────────────────╮ % │ Title page │ % ╰────────────────────────────────────────────────╯ -\textit{Denken vor Rechnen} +\vspace{2cm} +\begin{Huge} + \begin{center} + TITLE PAGE COMING SOON + \end{center} +\end{Huge} + + +\vspace{4cm} +\begin{center} + \begin{Large} + ``\textit{Denken vor Rechnen}'' + \end{Large} + + \hspace{3cm} - Vasile Grudinaru, 2025 +\end{center} + +\vspace{3cm} +\begin{center} + HS2025, ETHZ\\[0.2cm] + \begin{Large} + Summary of the Script and Lectures + \end{Large}\\[0.2cm] +\end{center} % ──────────────────────────────────────────────────────────────────── diff --git a/semester3/numcs/parts/introduction/rounding-errors.tex b/semester3/numcs/parts/introduction/rounding-errors.tex index e69de29..6645fb0 100644 --- a/semester3/numcs/parts/introduction/rounding-errors.tex +++ b/semester3/numcs/parts/introduction/rounding-errors.tex @@ -0,0 +1,22 @@ +\subsection{Rundungsfehler} + +\begin{definition}[]{Absoluter \& Relativer Fehler} + \begin{multicols}{2} + \begin{itemize} + \item \bi{Absoluter Fehler}: $||\tilde{x} - x||$ + \item \bi{Relativer Fehler}: $\displaystyle \frac{||\tilde{x} - x||}{||x||}$ für $||x|| \neq 0$ + \end{itemize} + \end{multicols} + wobei $\tilde{x}$ eine Approximation an $x \in \R$ ist +\end{definition} + +Rundungsfehler entstehen durch die (verhältnismässig) geringe Präzision die man mit der Darstellung von Zahlen auf Computern erreichen kann. +Zusätzlich kommt hinzu, dass durch Unterläufe (in diesem Kurs ist dies eine Zahl die zwischen $0$ und der kleinsten darstellbaren, positiven Zahl liegt) Präzision verloren gehen kann. + +Überläufe hingegen sind konventionell definiert, also eine Zahl, die zu gross ist und nicht mehr dargestellt werden kann. + + +\begin{remark}[]{Auslöschung} + Bei der Subtraktion von zwei ähnlich grossen Zahlen kann es zu einer Addition der Fehler der beiden Zahlen kommen, was dann den relativen Fehler um einen sehr grossen Faktor vergrössert. + Die Subtraktion selbst hat einen vernachlässigbaren Fehler +\end{remark}