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g3T19&b98cLVQmU!mtmw66ALpjH8=_-B}Gq03fP(shyVZp diff --git a/semester3/numcs/parts/02_quadrature/02_non-equidistant/00_gauss.tex b/semester3/numcs/parts/02_quadrature/02_non-equidistant/00_gauss.tex index 23fe3c0..7c06a66 100644 --- a/semester3/numcs/parts/02_quadrature/02_non-equidistant/00_gauss.tex +++ b/semester3/numcs/parts/02_quadrature/02_non-equidistant/00_gauss.tex @@ -8,3 +8,17 @@ welches hier $[0, 1]$ ist, mit $c_1, c_2, \ldots, c_s \in [0, 1]$ bezeichnet. % TODO: It would probably be a good idea to link the document together much better (and maybe create an index in the end?) Wir möchten unsere Gewichte $b_i$ und Knoten $c_i$ so bestimmen, dass die Quadraturordnung maximal ist. + +Wir definieren die Notation $\langle M, g \rangle = \int_{0}^{1} M(t) g(t) \dx t$ (also das Skalarprodukt). + +\begin{theorem}[]{Ordnung der Quadraturformel} + Die Ordnung ist $s + m$ genau dann, wenn $\langle M, g \rangle = 0$ für alle Polynome $g$ mit $\deg(g) \leq m - 1$ und + $M(t) = (t - c_1) \cdot (t - c_2) \cdot \ldots \cdot (t - c_s)$ für $s$. Also steht $M$ senkrecht zu allen $g$. +\end{theorem} + +\fancytheorem{Maximale Ordnung einer Quadraturformel} Die Ordnung einer Quadraturformel mit $s$ Knoten ist $\leq 2s$ + + +\fhlc{lime}{Orthogonale Polynome} + +Für $I = ]a, b[$ sei $w: I \rightarrow \R$ eine stetige Gewichtsfunktion mit $w(x) > 0 \smallhspace \forall x \in I$