From ba27b18ccef59f2dfe1a8a2a5319e43f6ede8b62 Mon Sep 17 00:00:00 2001 From: NeevoLima Date: Thu, 19 Sep 2013 15:53:38 +0200 Subject: [PATCH 1/4] Schreibfehler in K2A5 Einfach bitte den Change ansehen. Nur ein kleiner Schreibfehler ist dort passiert. --- documents/Numerik/Klausur2/Aufgabe5.tex | 2 +- documents/Numerik/Klausur2/Klausur2.pdf | Bin 155804 -> 155787 bytes 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/documents/Numerik/Klausur2/Aufgabe5.tex b/documents/Numerik/Klausur2/Aufgabe5.tex index 68bf1f1..7aa460b 100644 --- a/documents/Numerik/Klausur2/Aufgabe5.tex +++ b/documents/Numerik/Klausur2/Aufgabe5.tex @@ -40,5 +40,5 @@ Dann gilt: $C$ ist invertierbar und $b = C^{-1} \cdot $. Es gibt genau eine symmetrische QF in der Familie. Begründung: \\ -Aus $c_1 = 0 $ folgt, dass $c_3 = 0$ ist. Außerdem muss $c_2 = \frac{1}{2} $ sein. Also sind die Knoten festgelegt. Da wir die Ordnung $\ge s = 3$ fordern, sind auch die Gewichte eindeutig. \\ +Aus $c_1 = 0 $ folgt, dass $c_3 = 1$ ist. Außerdem muss $c_2 = \frac{1}{2} $ sein. Also sind die Knoten festgelegt. Da wir die Ordnung $\ge s = 3$ fordern, sind auch die Gewichte eindeutig. \\ Es handelt sich um die aus der Vorlesung bekannte Simpsonregel. diff --git a/documents/Numerik/Klausur2/Klausur2.pdf b/documents/Numerik/Klausur2/Klausur2.pdf index c22c76d1fe302c860808925ed43642a76192a076..77e985b2f9264f3c3a305e711a21da9880fa2e8a 100644 GIT binary patch delta 33876 zcmZsBQ+FLI;tNU ziCjrcf}WXy9fo{vX?P8WJst@Sm@|=anmS$q0RWdvVx1?2&jlSv5ps#eKQtrUa;D#} zV_zw$^0w;vHYbr_?WzAnqH^SvvpW^XwNA$6)%(}A{qJ+^5p=(lseHm@rjuRyf1 z@hAbylD8r(g#pxk+5;WXJyz!(Hrl1~EqJkA*Z=u)MObIbyXh5^6q zG@S8eQ^K$dl=!MK3tQ`GeVinVwB8`pf1?>)G~`fC~J$;$iNn? z(-H%VShOMGJasrR+BQkjM5riq0eA&lsG5P3{b0a}sGv+lhp``_UF-CTfBr2G938Mtk@@H6bd~77DKT{)F$Qcr_VCYb$%;LRy*4k_F#p3#qeS@>w z>UJ;k%9$H|2QzQMOg~`vqMGc*r37DKe{DD&wl_d^e>txP9G1ZlME6!he;ZfSPF7BK zd|PQhT0S;Ar!_p!+8YTP=>ZuFOZ<`VV6%_+zYv;I6MLa~k@MLg-MdS!nOLGG( zApFEnSSa$uR3UH{s2zoo+t5dz4Bv78uC4mDg*@{ZY4Qt+-y1{O?g7r6A;KJ@`J&X*7X0OXsz#R{VOFPr;9}KJm3~H1EUq&SGnKycSbOaRjjn_|2+S{`7HC0z!cFRU}Z01fe$Uz?wsk%CVO-t9X1UP>{fSiEx zLOcw(@q@t~1*i2V5PGyIzW5nr`1`*OJ53k4DZ-T&@LOes)G@D$f)*?53&6V@erszY zqH}%D&4>i4hg)7A3+@jw z+v@(#_wNuGSP(Prfnp+{Qe8LEPxMWOOeo+qmg_}wDTgFVFFbnOCY0jcfa2H1bgvLq z$uP=P0UG#atgQ51BE7Fv(QocuX_p3I(mN!JieWofqg}_=TV5-hsVn7u%OE_)aY0Za zmwZ2s00S+L5LKNyrj)0CcRzKMVH`{@^ki1oSFn!R#gp`1fjurK?y1nCoCA%|w2qY) zu#Ed&&@9ye|xB68%rXFiIqHPZI*Rb);PvrOC%CE{R|j&79+Ebpr*zfo|bBNAG)hU61@TwcO_>LBoPt9<)9=W za1-zdC^v{~tO|;R)cUfuA3VNnd~>QF8po5ES^V0iR)_zZ|IRT#zn1*$=3W*|X+!}? 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From 416e1420afcc84bec1a2c0c29bea5f9d627117c8 Mon Sep 17 00:00:00 2001 From: NeevoLima Date: Thu, 19 Sep 2013 16:21:52 +0200 Subject: [PATCH 2/4] Added K3A5 MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Die Aufgabe 5 wurde noch nicht ausgefüllt und wurde mit TODO gefüllt. Habe die Lösung mal ergänzt. Teilaufgabe b ist vermutlich nichtmals im Ansatz die Musterlösung, da es keine Herleitung ist. Aber das liegt daran, dass ich es etwas unnötig fände es aus dem Skript abzuschreiben. Ich glaube auch, dass so eine Frage nicht in unserer Klausur dran kommt. --- documents/Numerik/Klausur3/Aufgabe5.tex | 17 ++++++++++++++++- documents/Numerik/Klausur3/Klausur3.pdf | Bin 134990 -> 138804 bytes 2 files changed, 16 insertions(+), 1 deletion(-) diff --git a/documents/Numerik/Klausur3/Aufgabe5.tex b/documents/Numerik/Klausur3/Aufgabe5.tex index 75caec4..95a8de3 100644 --- a/documents/Numerik/Klausur3/Aufgabe5.tex +++ b/documents/Numerik/Klausur3/Aufgabe5.tex @@ -1,2 +1,17 @@ \section*{Aufgabe 5} -TODO +\subsection*{Teilaufgabe a} +Eine Quadraturformel $(b_i, c_i)_{i=1, \dots, s}$ hat die Ordnung +$p$, falls sie exakte Lösungen für alle Polynome vom Grad $\leq p -1$ +liefert. + +\subsection*{Teilaufgabe b} +Für die ersten 3. Ordnungsbedingungen gilt: + +\begin{align*} + 1 = \sum_{i = 0}^{s} b_i \\ + \frac{1}{2} = \sum_{i = 0}^{s} b_i * c_i \\ + \frac{1}{3} = \sum_{i = 0}^{s} b_i * c_i^2 +\end{align*} + +\subsection*{Teilaufgabe c} +Da die Ordnung 4 gewünscht ist müssen nach VL die Knoten der QF symmetrisch sein. Damit folgt sofort $c_2 = \frac{1}{2}$. Sind die Knoten gewählt, so sind die Gewichte eindeutig bestimmt. Die Berechnung erfolgt mit den Lagrangepolynomen. 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text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Habe meinen Autornamen ausgeschrieben. Aufgabe 5 c war noch nicht vollständig, da noch nicht geklärt wurde, dass die QF die Ordnung 4 hat. --- documents/Numerik/Klausur3/Aufgabe5.tex | 2 +- documents/Numerik/Klausur3/Klausur3.pdf | Bin 138804 -> 138971 bytes documents/Numerik/Klausur3/Klausur3.tex | 2 +- 3 files changed, 2 insertions(+), 2 deletions(-) diff --git a/documents/Numerik/Klausur3/Aufgabe5.tex b/documents/Numerik/Klausur3/Aufgabe5.tex index 95a8de3..7c00166 100644 --- a/documents/Numerik/Klausur3/Aufgabe5.tex +++ b/documents/Numerik/Klausur3/Aufgabe5.tex @@ -14,4 +14,4 @@ Für die ersten 3. Ordnungsbedingungen gilt: \end{align*} \subsection*{Teilaufgabe c} -Da die Ordnung 4 gewünscht ist müssen nach VL die Knoten der QF symmetrisch sein. Damit folgt sofort $c_2 = \frac{1}{2}$. Sind die Knoten gewählt, so sind die Gewichte eindeutig bestimmt. Die Berechnung erfolgt mit den Lagrangepolynomen. Es gilt $b_0 = b_2 = \frac{1}{6}, b_1 = \frac{4}{6}$. \ No newline at end of file +Da die Ordnung 4 gewünscht ist müssen nach VL die Knoten der QF symmetrisch sein. Damit folgt sofort $c_2 = \frac{1}{2}$. Sind die Knoten gewählt, so sind die Gewichte eindeutig bestimmt. Die Berechnung erfolgt mit den Lagrangepolynomen. Es gilt $b_0 = b_2 = \frac{1}{6}, b_1 = \frac{4}{6}$. Entweder man setzt alles in die 4. 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Thu, 19 Sep 2013 16:32:11 +0200 Subject: [PATCH 4/4] added K3A4 MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Lösung zu Aufgabe 4 Klausur 3 hinzugefügt. --- documents/Numerik/Klausur3/Aufgabe4.tex | 5 ++++- documents/Numerik/Klausur3/Klausur3.pdf | Bin 138971 -> 140158 bytes 2 files changed, 4 insertions(+), 1 deletion(-) diff --git a/documents/Numerik/Klausur3/Aufgabe4.tex b/documents/Numerik/Klausur3/Aufgabe4.tex index ef2b2e3..d098a24 100644 --- a/documents/Numerik/Klausur3/Aufgabe4.tex +++ b/documents/Numerik/Klausur3/Aufgabe4.tex @@ -1,2 +1,5 @@ \section*{Aufgabe 4} -TODO + +\begin{align*} + I = \int_0^1 \frac{1}{1+4x} dx = \int_0^{0.5} \frac{1}{1+4x} dx +\int_{0.5}^{1} \frac{1}{1+4x} dx = \\ (0.5 - 0) * \frac{f(0) + f(0.5)}{2} + (1 - 0.5) * \frac{f(1) + f(0.5)}{2} = \frac{7}{15} +\end{align*} diff --git a/documents/Numerik/Klausur3/Klausur3.pdf b/documents/Numerik/Klausur3/Klausur3.pdf index 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