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a/semester3/numcs/parts/02_quadrature/00_introduction.tex +++ b/semester3/numcs/parts/02_quadrature/00_introduction.tex @@ -91,6 +91,8 @@ Wir erhalten nun eine Quadraturformel, wenn wir $p$ als Approximation von $f$ ve \begin{align*} w_j = \int_{a}^{b} l_j(x), \smallhspace j = 0, 1, \ldots, n \end{align*} +Diese Gewichte werden für die Trapez- und Simpson-Regeln verwendet, genau genommen, im Falle der Trapezregel haben wir $w_2$ und für die Simpsonregel $w_3$, +also müssen wir die entsprechenden Lagrange-Polynome integrieren \drmvspace Durch die Konstruktion der Formel ist sie exakt für alle Polynome aus $\mathcal{P}_{n + 1}$ und der Fehler ist: diff --git a/semester3/numcs/parts/02_quadrature/04_in-rd.tex b/semester3/numcs/parts/02_quadrature/04_in-rd.tex index 9db59d7..0452ed1 100644 --- a/semester3/numcs/parts/02_quadrature/04_in-rd.tex +++ b/semester3/numcs/parts/02_quadrature/04_in-rd.tex @@ -1,2 +1,22 @@ \newsection \subsection{Quadratur in $\R^d$ und dünne Gitter} +Eine einfache Option wäre natürlich, zwei eindimensionale Quadraturformeln aneinander zu hängen. +Für zweidimensionale Funktionen sieht dies so aus: +\rmvspace +\begin{align*} + I = \int_{j_1}^{n_1} \sum_{j_2}^{n_2} \omega_{j_1}^1 \omega_{j_2}^2 f(c_{j_1}^1, c_{j_2}^2) +\end{align*} + +\drmvspace +und für beliebige $d$ haben wir +\rmvspace +\begin{align*} + \left( w_{j_k}^k, c_{j_k}^k \right)_{1 \leq j_k \leq n_k} \smallhspace k = 1, \ldots, d +\end{align*} + +\drmvspace +Which has the same form as above, but with $d$ sums and $d$ times a $w_{j_k}$ and a $d$-dimensional function $f$ + +\begin{recall}[]{Tensor-Produkt} + \TODO Write this section +\end{recall}