From d780636075d978f68f321ba864d54900a765a8aa Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Mon, 23 Sep 2013 12:13:15 +0200 Subject: [PATCH] =?UTF-8?q?=C3=9Cbungsblatt=2011,=20Aufgabe=2031=20hinzuge?= =?UTF-8?q?f=C3=BCgt?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- documents/Numerik/Klausur2/Aufgabe5.tex | 2 +- documents/Numerik/Klausur2/Klausur2.pdf | Bin 286432 -> 286403 bytes documents/Numerik/UB11/Aufgabe31.tex | 58 ++++++++++++++++++++++++ documents/Numerik/UB11/Makefile | 8 ++++ documents/Numerik/UB11/UB11.tex | 47 +++++++++++++++++++ 5 files changed, 114 insertions(+), 1 deletion(-) create mode 100644 documents/Numerik/UB11/Aufgabe31.tex create mode 100644 documents/Numerik/UB11/Makefile create mode 100644 documents/Numerik/UB11/UB11.tex diff --git a/documents/Numerik/Klausur2/Aufgabe5.tex b/documents/Numerik/Klausur2/Aufgabe5.tex index a665023..256b512 100644 --- a/documents/Numerik/Klausur2/Aufgabe5.tex +++ b/documents/Numerik/Klausur2/Aufgabe5.tex @@ -8,7 +8,7 @@ Zunächst ist nach der Familie von Quadraturformeln gefragt, für die gilt: ($p \end{align} Nach Satz 29 sind in der Familie genau die QFs, für die gilt: \\ -Für alle Polynome $g(x)$ mit Grad $\le m-1$ gilt: +Für alle Polynome $g(x)$ mit Grad $\le m-1 = 0$ gilt: \begin{align} \int_0^1 M(x) \cdot g(x) \mathrm{d}x = 0 \label{a3} \end{align} diff --git a/documents/Numerik/Klausur2/Klausur2.pdf b/documents/Numerik/Klausur2/Klausur2.pdf index 0e098cba06636fe3a8628d240a2b4eee37b890a1..5d54ddedc9d3748961841dbecb5a4af39c20e538 100644 GIT binary patch delta 18632 zcmZshQ*fYN&~D=iCdot-+qUgYY}>Z=#Cd<|LfeHQ`NhwZ~AW6?p13) zyWL2G-AHxG9EpBEDS;zRsl)}bUp?2gvkEm;WcR-Edmza)$?_iZ_OSN9)Gu5NtWYKq zV0!s@&oMRTV?LnaP?HxV3ti=U-!IaO*N6(&5+g_H>3X^ixr~xH&efmxxf)~aN0|%J z_Fz?y*Cszww>s8ocBWwo)xVrot}?bpdT+80H2v-PuQP0#HGpnp%^CIWf9Jzh`s

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Da $c_1 = 0$ +ist, kann es jedoch keine Gauß-Quadraturformel sein. Also kann +die Ordnung höchstens $5$ sein. + +\paragraph*{Ordnung 5} + +Nutze Satz 29: + +\begin{align} + M(x) &= (x-c_1) (x-c_2) (x-c_3)\\ + &= x (x-c_2) (x-1)\\ + &= (x^2- x) (x-c_2)\\ + &= x^3 - (1+c_2)x^2 + c_2 x\\ + \int_0^1 M(x) \cdot g(x) \mathrm{d} x &\stackrel{!}{=} 0 +\end{align} + +Da wir Ordnung $5 = s + 2$ erreichen wollen, muss $g$ ein beliebiges +Polynom vom Grad $\leq 2-1 = 1$ sein. Also: +\begin{align} + g(x) &= ax + b\\ + M(x) \cdot g(x) &= ax^4 + (b-a-ac_2)x^3 + (ac_2-bc_2-b)x^2 + b c_2 x\\ + \int_0^1 M(x) g(x) \mathrm{d} x &= \frac{a}{5} + \frac{b-a-ac_2}{4} + \frac{ac_2 - bc_2-b}{3} + \frac{b c_2}{2}\\ + &= \frac{a c_2}{12}-\frac{a}{20}+\frac{b c_2}{6}-\frac{b}{12}\\ + 0 &\stackrel{!}{=}\frac{a c_2}{12}-\frac{a}{20}+\frac{b c_2}{6}-\frac{b}{12}\\ + \Leftrightarrow 0 &\stackrel{!}{=} 5 a c_2 - 3a + 10 b c_2 - 5 b\\ + \Leftrightarrow -5 a c_2 - 10 b c_2&\stackrel{!}{=} - 3a - 5 b\\ + \Leftrightarrow 5 a c_2 + 10 b c_2&\stackrel{!}{=} 3a + 5 b\\ + \Leftrightarrow c_2(5 a + 10 b)&\stackrel{!}{=} 3a + 5 b\\ + \Leftrightarrow c_2 &\stackrel{!}{=} \frac{3a + 5 b}{5 a + 10 b} +\end{align} + +Offensichtlich gibt es kein $c_2$, dass diese Bedingung für jedes $a,b \in \mathbb{R}$ +erfüllt. Daher kann es keine Quadraturformel der Ordnung $5$ mit den Knoten +$0$ und $1$ geben. + +\paragraph*{Ordnung 4} +Die Simpson-Regel erfüllt offensichtlich alle Bedinungen und hat +Ordnung 5: + +\begin{align} + c_2 &= \nicefrac{1}{2}\\ + b_1 &= \nicefrac{1}{6}\\ + b_2 &= \nicefrac{4}{6}\\ + b_3 &= \nicefrac{1}{6} +\end{align} diff --git a/documents/Numerik/UB11/Makefile b/documents/Numerik/UB11/Makefile new file mode 100644 index 0000000..82e184d --- /dev/null +++ b/documents/Numerik/UB11/Makefile @@ -0,0 +1,8 @@ +SOURCE = UB11 +make: + pdflatex -shell-escape $(SOURCE).tex -output-format=pdf + pdflatex -shell-escape $(SOURCE).tex -output-format=pdf + make clean + +clean: + rm -rf $(TARGET) *.class *.html *.log *.aux *.out *.pyg diff --git a/documents/Numerik/UB11/UB11.tex b/documents/Numerik/UB11/UB11.tex new file mode 100644 index 0000000..625eee1 --- /dev/null +++ b/documents/Numerik/UB11/UB11.tex @@ -0,0 +1,47 @@ +\documentclass[a4paper]{scrartcl} +\usepackage{amssymb, amsmath} % needed for math +\usepackage[utf8]{inputenc} % this is needed for umlauts +\usepackage[ngerman]{babel} % this is needed for umlauts +\usepackage[T1]{fontenc} % this is needed for correct output of umlauts in pdf +\usepackage{pdfpages} % Signatureinbingung und includepdf +\usepackage{geometry} % [margin=2.5cm]layout +\usepackage[pdftex]{hyperref} % links im text +\usepackage{color} +\usepackage{framed} +\usepackage{enumerate} % for advanced numbering of lists +\usepackage{marvosym} % checkedbox +\usepackage{wasysym} +\usepackage{braket} % for \Set{} +\usepackage{pifont}% http://ctan.org/pkg/pifont +\usepackage{gauss} +\usepackage{algorithm,algpseudocode} +\usepackage{parskip} +\usepackage{lastpage} +\usepackage{gauss} +\usepackage{units} +\usepackage{amsthm} +\allowdisplaybreaks + +\newcommand{\cmark}{\ding{51}}% +\newcommand{\xmark}{\ding{55}}% + +\title{Numerik Übungsblatt 11 - Musterlösung} +\makeatletter +\AtBeginDocument{ + \hypersetup{ + pdfauthor = {Martin Thoma, Peter, Felix}, + pdfkeywords = {Numerik, KIT, Übungsblatt}, + pdftitle = {\@title} + } + \pagestyle{fancy} + \lhead{\@title} + \rhead{Seite \thepage{} von \pageref{LastPage}} +} +\makeatother + +\usepackage{fancyhdr} +\fancyfoot[C]{} + +\begin{document} + \input{Aufgabe31} +\end{document}