From 14b3575baffeb18fc81e06a75cc85633150bc85e Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Fri, 20 Sep 2013 18:24:17 +0200 Subject: [PATCH] misc --- documents/Numerik/Klausur1/Aufgabe3.tex | 11 +---------- documents/Numerik/Klausur1/Klausur1.pdf | Bin 271037 -> 270338 bytes documents/Numerik/Klausur3/Aufgabe2.tex | 6 +++--- documents/Numerik/Klausur3/Aufgabe3.tex | 5 +++-- documents/Numerik/Klausur3/Aufgabe4.tex | 5 ++++- documents/Numerik/Klausur3/Aufgabe5.tex | 6 +++++- documents/Numerik/Klausur3/Klausur3.pdf | Bin 157630 -> 159444 bytes documents/Numerik/Klausur4/Aufgabe2.tex | 4 ++-- documents/Numerik/Klausur4/Klausur4.pdf | Bin 151446 -> 151446 bytes documents/Numerik/Klausur5/Aufgabe2.tex | 6 +++++- documents/Numerik/Klausur5/Klausur5.pdf | Bin 147543 -> 148350 bytes documents/Numerik/Klausur6/Aufgabe2.tex | 10 ++++++---- documents/Numerik/Klausur6/Klausur6.pdf | Bin 168825 -> 169202 bytes documents/Numerik/Klausur6/aufgabe2.py | 13 ++++++++++++- 14 files changed, 41 insertions(+), 25 deletions(-) diff --git a/documents/Numerik/Klausur1/Aufgabe3.tex b/documents/Numerik/Klausur1/Aufgabe3.tex index 40ae087..5dac9d6 100644 --- a/documents/Numerik/Klausur1/Aufgabe3.tex +++ b/documents/Numerik/Klausur1/Aufgabe3.tex @@ -10,7 +10,7 @@ zweiten Spalte nach $y$. \subsection*{Lösungsvorschlag 1 (Numerische Lösung)} Eine Iteration des Newton-Verfahren ist durch \begin{align} -x_{k+1}&=x_{k}-f'(x_k)^{-1}\cdot f(x_k) +x_{k+1}&=x_{k}\underbrace{-f'(x_k)^{-1}\cdot f(x_k)}_{\Delta x} \end{align} gegeben (vgl. Skript, S. 35). @@ -167,12 +167,3 @@ also ausführlich: \end{pmatrix}\\ P &= I_2 \end{align} - -TODO: Eigentlich sollten sich ab hier die Lösungsvorschläge gleichen\dots - -Es folgt: -\begin{align} --f ( \nicefrac{-1}{3}, 0) &= \begin{pmatrix} -2\\ -\frac{26}{27}\end{pmatrix}\\ -c &= \begin{pmatrix} 2\\ \nicefrac{82}{27} \end{pmatrix}\\ %TODO: Was ist c? -(x_1, y_1) &= \begin{pmatrix} \nicefrac{5}{3}\\ \nicefrac{82}{27}\end{pmatrix} -\end{align} diff --git a/documents/Numerik/Klausur1/Klausur1.pdf b/documents/Numerik/Klausur1/Klausur1.pdf index 6ca2a8a93a9db233771b2260ee58a20abdce5072..8deaaa6da2350afd2ea83aff1b240949e8aa0b58 100644 GIT binary patch delta 42576 zcmZUZV~j3L5T(bqZQJ(Tv2EKn-mz^PcWm3XZQQYq{dPCI*<|}ir<1CmCtXiYot_b- zpnjwTEkJRIyVmU8E{RYs34B7Ca5fgzJQ<>WF_pL(<#Q>p^ZP5ojocKvMj9S9^*p9Z zl0`V{5G!l=@uacR_~Fm-&HZsU;RFVmSVCabQF&HzMTH&vcIpMnz(9Y7F>NlT1;Co6ub=7#giE8G1TyoRzXCc(z^Yc1`1MO?GRsUz##t?Z0qKLVZzHLBWa z3Q(d!3BzR{Ybq#zRQXe-B$3KvQAQcb^H=E`JOIaCL`$QpL(pIv*JQRY7Ewe%i>XoC 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r_{kj}}{r_{jj}} \end{align} @@ -9,8 +9,8 @@ \begin{algorithm} \begin{algorithmic} \For{$d \in \Set{1, \dots n}$} - \State berechne d-te Zeile von R - \State berechne d-te Spalte von L + \State berechne d-te Zeile von $R$ + \State berechne d-te Spalte von $L$ \EndFor \end{algorithmic} \end{algorithm} diff --git a/documents/Numerik/Klausur3/Aufgabe3.tex b/documents/Numerik/Klausur3/Aufgabe3.tex index 3e939f0..164f656 100644 --- a/documents/Numerik/Klausur3/Aufgabe3.tex +++ b/documents/Numerik/Klausur3/Aufgabe3.tex @@ -3,11 +3,11 @@ \begin{enumerate} \item Selbstabbildung: \\ - Sei $x \in D := [1.75 , 2]$. + Sei $x \in D := [1.75 , 2] = [\frac{7}{4}, \frac{8}{4}]$. Dann: \begin{align} - F(x) = 1 + \frac{1}{x} + \frac{1}{x^2} \le 1 + \frac{1}{1.75} + \frac{1}{1.75^2} = 1 + \frac{44}{49} \le 2 + F(x) = 1 + \frac{1}{x} + \frac{1}{x^2} \le 1 + \frac{1}{1.75} + \frac{1}{1.75^2} = 1 + \frac{44}{49} \le 2 %TODO: schöner formulieren \end{align} und: \\ \begin{align} @@ -19,6 +19,7 @@ %\textbf{Behauptung:} $F(x)$ ist auf $A$ eine Kontraktion. %\textbf{Beweis:} %z.Z.: $\exists L \in [0, 1): \forall x,y \in A: || F(x) - F(y) || \leq L \cdot || x - y||$ + Hier ist der Mittelwertsatz der Differentialrechnung von Nutzen.\\ $F$ ist Lipschitz-stetig auf $D$ und für alle $x \in D$ gilt: \\ \begin{align} |F'(x)| = |-\frac{1}{x^2}-2 \cdot \frac{1}{x^3}| \le \frac{240}{343} =: \theta < 1 diff --git a/documents/Numerik/Klausur3/Aufgabe4.tex b/documents/Numerik/Klausur3/Aufgabe4.tex index d098a24..ff6e764 100644 --- a/documents/Numerik/Klausur3/Aufgabe4.tex +++ b/documents/Numerik/Klausur3/Aufgabe4.tex @@ -1,5 +1,8 @@ \section*{Aufgabe 4} \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} + 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/Aufgabe5.tex b/documents/Numerik/Klausur3/Aufgabe5.tex index 7c00166..20a970a 100644 --- a/documents/Numerik/Klausur3/Aufgabe5.tex +++ b/documents/Numerik/Klausur3/Aufgabe5.tex @@ -14,4 +14,8 @@ 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 $b_0 = b_2 = \frac{1}{6}, b_1 = \frac{4}{6}$. Entweder man setzt alles in die 4. Ordnungsbedingung ein oder aber argumentiert, dass es sich hierbei um die Simpson-Regel handelt und diese die Ordnung 4 hat. \ No newline at end of file +Wähle die Simpson-Regel, also $c_1=0, c_2 = \frac{1}{2}, c_3 = 1$ und +$b_1 = b_3 = \frac{1}{6}$ und $b_2 = \frac{4}{6}$. + +Überprüfe nun Ordnungsbedingungen 1-4 $\Rightarrow$ Simpson-Regel hat Ordnung 4 +Überprüfe Ordnungsbedingung 5 $\Rightarrow$ Simpson-Regel hat nicht Ordnung 5. %TODO ausführlicher diff --git a/documents/Numerik/Klausur3/Klausur3.pdf b/documents/Numerik/Klausur3/Klausur3.pdf index b443df25a3a5decf9812eb56976dae4dfa2cc207..588819b540697b75fc87af7bd651d226280d4c2c 100644 GIT binary patch delta 65810 zcmV)9K*hhl&k5As39x?!12!}_lW-I$f3;cLa^pl0efL-BQ7+Ih_e-dKSZZ0SmP^4V zMNuSqaAGGoaUAf4zA& zIlrXE9RK#AphP%Ze5xfM3duCl&Of&*PA9)iXmE~po#eb=j5wuu*GcZ+w`iO^n>tx8 zmhkT8c6J3^tn`SDDz7i7Vl~33e-ipsH)u>6VzekKp^0LuQTWzB0!f=lQlhDbl3L_~ zQWm5VeTqn>qR>B$1WP2~fNmw;Yt3D0fBG5j$j1zD5AkVqroM*2$)#Oh>Zx)m2O*f8HKH_^O2c 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zc`H}&MnJEar~R3?PD(A_vBR6#e-+HtlDYc|=B=YmlXuaW9?aYd^`3hfa)ZZ=6eyh?YXZIC&w~F5u?~bQ6@zyKeet+Jr z%CpHjRa;rF7~B0BcZf2&Lvs_!6A zw=!wT({agYH^}3^2R_|mq9sqqO>NqQkJ>>VdVzFCZ;;;6>z_UOY;-wWyzpMX6Pw>P zTyAGn^nm2v(d6RrY(ALI4@7<;@E#rA=*{PT)mI1fQ>)HsbwEF%>I{S0=qyJv zWXN(oU}I`*WpmF0{q~p^I&iN^T|TQmh<(DO-eoR+hh2{jKY7v(5Y0t&0Q#I{N)tCjmSh;BYbd z)XSM*`386fh<>NWmvr-FaygW*X2TbE^J`S;$QmzhhvWQX2eC{o{{t!_^hpY3Ze(+G za%Ev{3T19&Z(?c+mvlh_Us5?TGBiO!IYBrxGeR*lH8epmMn*g#RK>*b_Q{}YSS=e^ew9KsRg;TTTf6wcsWLGtnM2}fje Z_6zk7B6^oqLjoTPFfuX0$ definiert ist. Bei der Iteration kommt man aber schnell in +einen Bereich, der nicht erlaubt ist (das erlaubte Intervall ist klein; +Rechenungenauigkeit) $F_1$ ist auf $[0,1]$ eine Kontraktion mit Kontraktionszahl $\theta = \frac{1}{2}$: \begin{align} @@ -56,7 +57,8 @@ $F_1$ ist auf $[0,1]$ eine Kontraktion mit Kontraktionszahl $\theta = \frac{1}{2 \Leftrightarrow \underbrace{e^{-(x+y)}}_{\leq 1} \cdot \|e^{x} - e^{y}\| &\leq \|x-y\| \end{align} -TODO: Beweis ist noch nicht fertig +TODO: Beweis ist noch nicht fertig. Mittelwertsatz der Differentialrechnung +anwenden. $F_2$ ist auf $(0,1]$ eine Kontraktion mit Kontraktionszahl $\theta$: \begin{align} diff --git a/documents/Numerik/Klausur6/Klausur6.pdf b/documents/Numerik/Klausur6/Klausur6.pdf index d39d86d8f1a6c2a2bb2be5c2c47f592094e9d7c0..ca2b7cdaf6ff5f150df499c959bcbb5b3aa43b37 100644 GIT binary patch delta 37977 zcmV(&K;ggnrV8?@3a~8(12Hr;lOY!s<8;iV4Omf`GQf5+zF z_uD(6^@5envED3}%orxicfLCKu$F&Xa9cjr;Ssg?aQnl}o77D2U@J*-VX&O!r9`tcIB>+OcU+!xoz?8=BFDP)uKc7I4hVWLTWx#k2}y<1od7W z?>DPoU_ZRXJz7!nc2{F@U%!>2e_)I_rCM3ESPDiIk78~YPL{p`c%nGgiZiOL5^-b1 zK@2xUXcNS|X#z5vY6~D9zQJeV5jA{-8Ah~V#ZpRQ1rHDMu#zV$6_HB9E7}#m#}zg) zIMK?TR@_lab7I2bX_b#lTw#baG`@)1LJ~#gnIt_}rirq8w4YQr3kK4le_1c;u|^b2@Eav;c;rPEU_qSd^qoXj&1wR!rn?m{v#^5u zFm(Y|-{@(73ZI_nKm5Kg=3 zc$iS7kDosSksK7@IPwZFg_r>*`QN-0Vn$lPGrs| zbZieXXAhctn;}Mv1nVnEYSJq+h~%OM-qx`-2^GO>s1nK*Rt&*B55Bq+#vERR+rCSP zzZ}AqM7)NT-(+XL$#Z)Y2B@cd@R%nk&Y+GG&+~-j{du01 zR$Z(t^4ub(Dhu{Je;>+WoEXNCL!k`mg{3trH;OPQz!ZoVe@cvtR4H^{qyp8-u0#zb 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7a15118..16eb9c3 100644 --- a/documents/Numerik/Klausur6/aufgabe2.py +++ b/documents/Numerik/Klausur6/aufgabe2.py @@ -1,4 +1,4 @@ -from math import exp, log +"""from math import exp, log def iterate(x): #return x - (2.0*x - exp(-x))/(2.0+exp(-x)) #Newton @@ -9,3 +9,14 @@ x = 0.9 for i in range(10): print (i, x) x = iterate(x) +""" +from math import log +f = lambda x: -log(2*x) + +x = 0.35173371124919582602 +#x = 0.3517337112491958260249093009299510651715 +for i in range(200): + print("x 0.35173371124919582602") + print("x 0.3517337112491958260249093009299510651715") + print("%i %.30f" % (i, x)) + x = f(x)