diff --git a/.gitignore b/.gitignore index 4d2488a..5268a3b 100644 --- a/.gitignore +++ b/.gitignore @@ -2,3 +2,7 @@ *.log *.out *.synctex.gz +*.asy +cviceni_NUM_A4-*.pdf +cviceni_NUM_kindle-*.pdf +*.pre \ No newline at end of file diff --git a/Makefile b/Makefile new file mode 100644 index 0000000..95ff028 --- /dev/null +++ b/Makefile @@ -0,0 +1,7 @@ +all: cviceni_NUM.tex cviceni_NUM_A4.tex cviceni_NUM_kindle.tex + pdflatex cviceni_NUM_kindle.tex + asy cviceni_NUM_kindle*.asy + pdflatex cviceni_NUM_kindle.tex + pdflatex cviceni_NUM_A4.tex + asy cviceni_NUM_A4*.asy + pdflatex cviceni_NUM_A4.tex \ No newline at end of file diff --git a/cviceni_NUM.tex b/cviceni_NUM.tex index a3e49d1..9b77a38 100644 --- a/cviceni_NUM.tex +++ b/cviceni_NUM.tex @@ -1,4 +1,4 @@ -%!TEX root = cvičení NUM - kindle.tex +%!TEX root = cviceni_NUM_kindle.tex % % Cvičení z NUM % @@ -40,7 +40,8 @@ \usepackage{amsthm} \usepackage{latexsym} -\usepackage{graphicx} +\usepackage[pdftex]{graphicx} +\usepackage{asymptote} \newtheorem{definition}{Definice} @@ -51,6 +52,8 @@ \newcommand{\inte}[2]{\int\limits_0^1 #1\,\mathrm{d}#2} \newcommand{\R}{\mathbb{R}} +\newcommand{\bbA}{\mathbb{A}} +\newcommand{\bbF}{\mathbb{F}} \newcommand{\pa}{\partial} \title{Zápisky z cvičení NUM 2011} @@ -230,7 +233,8 @@ Hledáme přesnost aproximace $$ \psi = L_h(P_h y ) - P_h(L_y).$$ - $P_h$ je projekce funkcí definovaných na $(a, b)$ do funkcí definovaných na síti $\overline{\omega_h}$. + $P_h$ je projekce funkcí definovaných na $(a, b)$ do funkcí definovaných na síti + $\overline{\omega_h}$. \begin{align*} \psi &= -(p(P_h y)_{\overline{x}})_x + qP_hy - f - P_h(-(py')' + qy - f) \\ \psi_j &= -(p(P_h y)_{\overline{x}})_x - P_h(-(py')') \quad \rightarrow O(h) @@ -241,10 +245,42 @@ $$ p_{i \pm \frac{1}{2}} = p\left(a + \left(i \pm \frac{1}{2}\right)h \right) \quad \rightarrow O(h^2)$$ \begin{enumerate} - \item ~\\ \includegraphics{mexson1.pdf} + \item ~\\ \vspace{-1cm} + \begin{asy} + size(7cm); + draw((0,0.5)--(1,0.5)); // vodorovná čára + draw((0.1,0.48)--(0.1,0.52)); // první svislítko + label("$i-1$",(0.1,0.48),S); + draw((0.5,0.48)--(0.5,0.52)); // druhé svislítko + label("$i$",(0.5,0.48),S); + label("$p_i$",(0.5,0.52),N); + draw((0.9,0.48)--(0.9,0.52)); // třetí svislítko + label("$i+1$",(0.9,0.48),S); + label("$p_{i+1}$",(0.9,0.52),N); + \end{asy} $$ (pu_{\overline{x}})_x = \frac{(pu_{\overline{x}})_{i+1} - (pu_{\overline{x}})_i} {h} = \frac{1}{h^2} (p_{i+1}(u_{i+1} - u_i) - p_i(u_i - u_{i-1})) $$ - \item ~\\ \includegraphics{mexson2.pdf} + \item ~\\ \vspace{-1cm} + \begin{asy} + size(12cm); + draw((0,0.5)--(1,0.5)); // vodorovná čára + draw((0.1,0.49)--(0.1,0.51)); // první svislítko + label("$i-1$",(0.1,0.49),S); + label("$p_{i-1}$",(0.1,0.56),N); + label("$u_{i-1}$",(0.1,0.51),N); + draw((0.3,0.49)--(0.3,0.51)); // druhé svislítko + label("$p_{i-\frac{1}{2}}$",(0.3,0.53),N); + draw((0.5,0.49)--(0.5,0.51)); // třetí svislítko + label("$i$",(0.5,0.49),S); + label("$p_i$",(0.5,0.55),N); + label("$u_i$",(0.5,0.51),N); + draw((0.7,0.49)--(0.7,0.51)); // čtvrté svislítko + label("$p_{i+\frac{1}{2}}$",(0.7,0.53),N); + draw((0.9,0.49)--(0.9,0.51)); // páté svislítko + label("$i+1$",(0.9,0.49),S); + label("$p_{i+1}$",(0.9,0.55),N); + label("$u_{i+1}$",(0.9,0.51),N); + \end{asy} $$ (pu')' \approx \frac{pu'|_{i + \frac{1}{2}} - pu'|_{i - \frac{1}{2}}}{h} = \frac{1}{h^2}(p_{i + \frac{1}{2}}(y_{i+1} - y_i) - p_{i - \frac{1}{2}}(y_i - y_{i-1}))$$ \begin{align*} @@ -253,4 +289,14 @@ \end{align*} \end{enumerate} + Máme úlohu $ -(py')' + qy = F(y) $ na $(a,b),\ y(a)= \gamma_1,\ y(b)= \gamma_2,\ F(0) = 0,\ + g \geq 0,\ F$ omezená, lipschitzovská. + + Použijeme 1. schéma: + $$ -\frac{1}{h} \left( p_{i+1} \frac{u_{i+1} - u_i}{h} - p_{i} \frac{u_i - u_{i-1}}{h} \right) + + q_i u_i = F(u_i), \quad i \in \widehat{m-1} $$ + Maticový zápis: + $$ \bbA_h u= \bbF(u)+\varphi_h \qquad \bbA_h \in \R^{m-1,m-1};\ \varphi_h \in \R^{m-1};\ + \bbF(n) \in \R ^{m-1} $$ + \end{document} diff --git "a/cvic\314\214eni\314\201 NUM - kindle.pdf" b/cviceni_NUM_A4.pdf similarity index 63% rename from "cvic\314\214eni\314\201 NUM - kindle.pdf" rename to cviceni_NUM_A4.pdf index f451451..fc035d8 100644 Binary files "a/cvic\314\214eni\314\201 NUM - kindle.pdf" and b/cviceni_NUM_A4.pdf differ diff --git "a/cvic\314\214eni\314\201 NUM - A4.tex" b/cviceni_NUM_A4.tex similarity index 100% rename from "cvic\314\214eni\314\201 NUM - A4.tex" rename to cviceni_NUM_A4.tex diff --git "a/cvic\314\214eni\314\201 NUM - A4.pdf" b/cviceni_NUM_kindle.pdf similarity index 63% rename from "cvic\314\214eni\314\201 NUM - A4.pdf" rename to cviceni_NUM_kindle.pdf index ab9647b..1c12859 100644 Binary files "a/cvic\314\214eni\314\201 NUM - A4.pdf" and b/cviceni_NUM_kindle.pdf differ diff --git "a/cvic\314\214eni\314\201 NUM - kindle.tex" b/cviceni_NUM_kindle.tex similarity index 100% rename from "cvic\314\214eni\314\201 NUM - kindle.tex" rename to cviceni_NUM_kindle.tex diff --git a/mexson1.pdf b/mexson1.pdf deleted file mode 100644 index 06112c4..0000000 Binary files a/mexson1.pdf and /dev/null differ diff --git a/mexson1.vdesigner/QuickLook/Preview.pdf b/mexson1.vdesigner/QuickLook/Preview.pdf deleted file mode 100644 index a0092d8..0000000 Binary files a/mexson1.vdesigner/QuickLook/Preview.pdf and /dev/null differ diff --git a/mexson1.vdesigner/QuickLook/Thumbnail.jpg b/mexson1.vdesigner/QuickLook/Thumbnail.jpg deleted file mode 100644 index c69f492..0000000 Binary files a/mexson1.vdesigner/QuickLook/Thumbnail.jpg and /dev/null differ diff --git a/mexson1.vdesigner/VectorDesigner b/mexson1.vdesigner/VectorDesigner deleted file mode 100644 index 6f75f9e..0000000 Binary files a/mexson1.vdesigner/VectorDesigner and /dev/null differ diff --git a/mexson2.pdf b/mexson2.pdf deleted file mode 100644 index b1d8a7f..0000000 Binary files a/mexson2.pdf and /dev/null differ diff --git a/mexson2.vdesigner/QuickLook/Preview.pdf b/mexson2.vdesigner/QuickLook/Preview.pdf deleted file mode 100644 index 30fc6e4..0000000 Binary files a/mexson2.vdesigner/QuickLook/Preview.pdf and /dev/null differ diff --git a/mexson2.vdesigner/QuickLook/Thumbnail.jpg b/mexson2.vdesigner/QuickLook/Thumbnail.jpg deleted file mode 100644 index b9446b3..0000000 Binary files a/mexson2.vdesigner/QuickLook/Thumbnail.jpg and /dev/null differ diff --git a/mexson2.vdesigner/VectorDesigner b/mexson2.vdesigner/VectorDesigner deleted file mode 100644 index 56aaada..0000000 Binary files a/mexson2.vdesigner/VectorDesigner and /dev/null differ