From 97caaa59b8f57a88641f308948652f059bb53cec Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Mon, 10 Nov 2025 12:05:10 +0100 Subject: [PATCH] [NumCS] Start Chapter 6.1, fix some errors --- semester3/numcs/numcs-summary.pdf | Bin 1333509 -> 1338063 bytes semester3/numcs/numcs-summary.tex | 7 +++ .../numcs/parts/02_quadrature/03_adaptive.tex | 4 ++ .../numcs/parts/02_quadrature/04_in-rd.tex | 4 ++ .../parts/02_quadrature/05_monte-carlo.tex | 4 ++ .../06_reduction-of-variance.tex | 4 ++ semester3/numcs/parts/03_zeros/00_intro.tex | 48 ++++++++++++++++++ 7 files changed, 71 insertions(+) create mode 100644 semester3/numcs/parts/03_zeros/00_intro.tex diff --git a/semester3/numcs/numcs-summary.pdf b/semester3/numcs/numcs-summary.pdf index 077ae66f6e7cfb89c156aa87028a824aa560e418..49b81c146f91df1cd6efafdd76f4b50821691edb 100644 GIT binary patch delta 85592 zcmZs?b8se3&^4MovAMBrJh5%tww-MF#L33?#v9wVoosBI8~gs=?^fNaukJg4bxlq6 z%vAN%oIa;JtU+aVNu_Qa6^n<2g~Z9!7M`CUo>|_)(aPPLgd50{j0T_urKZZc{bfN7 zym|eBr}!jaeQGg&4E;ypYz`(C#*#*$Elv*~y_4HB({-=5abdzi7q5f>n5bE~blr-= z=Bj{`sjzLCeAa_b@uwUs%UnzRk2;%ze1)~GX%gd@w=;j$L=DqF3mh5RkfMg#bry7) zI_;vv3M7-2wODx1jiu>1P&z3w)O{gSc3~*KJ{wj}bGkpE+hFzKcAv^Qj#eLPp`2#xK%``N+#7|>0kYzk`j42ZRT;me=Y-l zLmOX-;_{(|RIHnYkr?oRLH4kn9f~Gg8R>8eVE_dMG%~)@z>Op#2ok8Cp8K##dOy zjni^%>EDZv+Hg7Drphxey9xmL--gy%36o;+IwH-=uFA|V^IC)e%dmwrUKYhqjNtjq z^vhg=ZlT;Yj+CwO@w=7?5-^EB3uDk^%Jud3U_#H%J+9^aGL7Z_L|ydRHw7#rXN_(f z>%UQMsu(Ju6|Y4cHbbFd4GY#iFYF2GBgnLf6#hPW z8^znEm+kVC0>jC$5&|UKXXl-~`e;IWKnX4W{#)-2x-o`T8%O98I9{=mcK`|o%NA~uEi^f{8AEzF;1|$X;o#SgB{MC7LX^P{saU#a>k*cF z4x$e{eeGYxpaqX8qzNTqP|@Jwb^aHblV8rW$o}&CZf@^{RD|UpwaN5urS;y(-SK1N zV^!5$+1trE^*75%XYByj@Ri4gq0g#GY>(v50{+W`A$M-8YeK(k zg#3JGj87rr!0J3+6K_F0ebn2({mtMBG)P5EVKsMNw?CuuSSWMfzF_w4M{VIP9L?R_ zT`f!;{>S>CSH_kOzKVef0CFbpiqM1f<&ytOqxZaPoi>?BRf%^Kw}#~g1>wSWoMj9G z-IO)qC|IUeB1`~qMt5&FMyaDMiCadU*%yaV(*G|WePUqDm z(SO^Bh`W0KX88?AfnKgtLaD@LHNJ6I>8Y`J#BQXhDtq=~G0(7^oKkjYe?S3Rxq#bT zIu4JWU9QY-p{3%6-Ikv$%)+{b#)6GyUe>ZW%HX@ z6cf-x&`oS|$|&o+!958bvFhl2PuRVS%T(X}C0G9DRpN6P@RVKBsFon(=A z{^;QlRj^o;Wk4{fZ-#cUcE?2!+j-NxZ1pgxhaEupGH^qoddVg2O+}5Qix*(Df z2&I_8X$mo853xp=Hyi`T;mAo-r5G;)NGOlcN(}2D%G1|tOK~;CN0|pt#C#(kk%kHx z`9ppp%#t8(oo}4>)Lnq&62#k}(H4K$!WQ0oX$63t7_VywiP)JTavx-Dn}zbvslX-D z3ZriluNVms`BU0xP5K)ww%UI-)&vfaS8F>%;Xq@jr$8wMT9j~SWU3IZ$Wv%rVaio? 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zwte&+Caz?L=r_v0!Shx)vkMYgLse{p46D*AoimU+5b148!*aP;GYauGBt^4ovYqMB z*8=HXkGQ4qP5^sc`9Fm_levT<{9sb@|c zkr;ct=vri0xELG_)K}|7wE&*tQ~$9{z-@ z%c2+=#{>&lNjz5Q>~7*1Mtxz}IY5I|uq1X@up+Y~NV$O-q~HF=-;Cx$2?I*8U%)UJ z5W$Mnm!W>k8FSAZ+Unsp_BcTg@hSy~n1fRyEU+oJ4r%iVSJZl$@fB}&h98l&9 z+;Y+}n_(6+d%&?QCI%lV(hGk@PQc?h+5P7B*kKg>dpFNxy5C`BgitdrqDa)0g9t2UhBJUEFFB4O&W6GWr(lEvVG*lcHg`i=7T;?uN z53q@c;#GpM2^=y)P!_a;)cxv(^Ut#VZ{SytmSPlG*AgEN3#;Nbb)oZiM^zo% z)gc|~kl{6{x?~=Y1;v$3Mg0rOL}eR5^1(9#2!R0_V8huvbXzs5|F&9k3@daWn&S*W z^YYhxExjbjTH6=gZV|Jkh3LP{H`{5E*bCHz8H-t)q-Z_Tq15;)>+~MvIfuY$H>7TJ z033ndsaWbIGpTBqKm>TA9Au^ylDZa^*;a-B zqw2#>fe5&F3(v*i}EAY@%-#2X#`Bg!M_E-h{(P2P% zJ@iQK-%0RuQ}XSA9L;e(>gnBSfWMjiLoecK+yxB?q}&1McN$_}i&obUO_)hM50jPs zi74s(HhbkQF&`Z9uozL55OMwu?Oh`H98LW57j75v&!eme^3OzSx6$NQ=s@2__{ARe z8e~_lc4$|IdB|Op#DS`!iqMDb+tMbS4~W>aC{? zSBQ&sIGpnkPOcSzHvqI)K#T|$Mqw │ +% └ ┘ + \newsection \subsection{Adaptive Quadratur} Der lokale Fehler einer zusammengesetzten Quadraturformel auf dem Gitter $\mathcal{M} := \{ a = x_0 < x_1 < \dots < x_m = b \}$ ist (für $f \in C^2([a, b])$): diff --git a/semester3/numcs/parts/02_quadrature/04_in-rd.tex b/semester3/numcs/parts/02_quadrature/04_in-rd.tex index 4810a2c..020d123 100644 --- a/semester3/numcs/parts/02_quadrature/04_in-rd.tex +++ b/semester3/numcs/parts/02_quadrature/04_in-rd.tex @@ -1,3 +1,7 @@ +% ┌ ┐ +% │ AUTHOR: Janis Hutz │ +% └ ┘ + \newsectionNoPB \subsection{Quadratur in $\R^d$ und dünne Gitter} Eine einfache Option wäre natürlich, zwei eindimensionale Quadraturformeln aneinander zu hängen. diff --git a/semester3/numcs/parts/02_quadrature/05_monte-carlo.tex b/semester3/numcs/parts/02_quadrature/05_monte-carlo.tex index e4700af..eead1fa 100644 --- a/semester3/numcs/parts/02_quadrature/05_monte-carlo.tex +++ b/semester3/numcs/parts/02_quadrature/05_monte-carlo.tex @@ -1,3 +1,7 @@ +% ┌ ┐ +% │ AUTHOR: Janis Hutz │ +% └ ┘ + \newsection \newcommand{\tsigma}{\tilde{\sigma}} \subsection{Monte-Carlo Quadratur} diff --git a/semester3/numcs/parts/02_quadrature/06_reduction-of-variance.tex b/semester3/numcs/parts/02_quadrature/06_reduction-of-variance.tex index ccba90f..49f0fe6 100644 --- a/semester3/numcs/parts/02_quadrature/06_reduction-of-variance.tex +++ b/semester3/numcs/parts/02_quadrature/06_reduction-of-variance.tex @@ -1,3 +1,7 @@ +% ┌ ┐ +% │ AUTHOR: Janis Hutz │ +% └ ┘ + \newsection \subsection{Methoden zur Reduktion der Varianz} % NOTE: Mostly from TA slides, as the script is quite convoluted there diff --git a/semester3/numcs/parts/03_zeros/00_intro.tex b/semester3/numcs/parts/03_zeros/00_intro.tex new file mode 100644 index 0000000..8a78aaf --- /dev/null +++ b/semester3/numcs/parts/03_zeros/00_intro.tex @@ -0,0 +1,48 @@ +% ┌ ┐ +% │ AUTHOR: Janis Hutz │ +% └ ┘ + +\subsection{Iterative Verfahren} +\inlinedef Ein iteratives Verfahren ist ein Algorithmus $\phi_F$, der die Folge $x^{(0)}, x^{(1)}, \ldots$ von approximativen Lösungen $x^{(j)}$ generiert. +Die Definition ist dabei rekursiv: $x^{(k)} := \phi_F(x^{(k - 1)})$, sofern $x^{(0)}$ und $\phi$ gegeben sind. + +\setLabelNumber{all}{5} +\fancydef{Konvergenz} $\phi_F$ zur Lösung $F(x^*) = 0$ konvergiert, wenn $x^{(k)} \rightarrow x^*$, mit $x^*$ der gesuchte Wert. + +\setLabelNumber{all}{8} +\fancydef{Norm} + +\innumpy haben wir \texttt{numpy.linalg.norm}, welches zwei Argumente nimmt. Dabei ist das erste Argument der Vektor und das Zweite die Art der Norm. +Ohne zweites Argument wird die Euklidische Norm $||x||_2$, mit Argument $1$ wird die $1$-Norm $||x||_1 := |x_1| + \ldots + |x_n|$ +und mit mit \texttt{inf} als Argument wird die $\infty$-Norm, bzw die Max-Norm $||x||_\infty := \max\{ |x_1|, \ldots, |x_n| \}$ berechnet. + +\stepLabelNumber{all} +\inlinedef Zwei Normen $||\cdot||_1$ und $||\cdot||_2$ sind äquivalent auf $\cV$, falls es Konstanten $\underline{C}$ und $\overline{C}$ gibt so dass +\rmvspace +\begin{align*} + \underline{C} \cdot ||v||_1 \leq ||v||_2 \leq \overline{C} \cdot ||v||_1 \mediumhspace \forall v \in \cV, \text{ mit } \cV \text{ ein linearer Raum} +\end{align*} + +\drmvspace +\inlinetheorem Falls $\dim(\cV) < \infty$, dann sind alle Normen auf $\cV$ äquivalent + +\stepLabelNumber{all} +\fancydef{Lineare Konvergenz} $x^{(k)}$ konvergiert linear gegen $x^*$, falls es ein $L < 1$ gibt, so dass +\rmvspace +\begin{align*} + ||x^{(k + 1)} - x^*|| \leq L||x^{(k)} - x^*|| \smallhspace \forall k \geq k_0, \smallhspace L \text{ gennant Konvergenzrate } +\end{align*} + +\drmvspace\stepLabelNumber{all} +\fancydef{Konvergenzordnung} $p$ für das Verfahren, falls es ein $C > 0$ gibt, so dass +\rmvspace +\begin{align*} + ||x^{(k + 1)} - x^*|| \leq C||x^{(k)} - x^*||^p \smallhspace \forall k \in \N \text{ mit } C < 1 \text{ für } p = 1 +\end{align*} + +\drmvspace +Wir nehmen dabei an, dass $||x^{(0)} - x^*|| < 1$, damit wir eine konvergente Folge haben. + +\numberingOff +\inlineremark Eine höhere Konvergenzordnung ist in Lin-Log-Skala an einer gekrümmten Konvergenzkurve erkennbar. +\numberingOn