{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"<h2>Aufgabe 1: Lineare Regression</h2>\n",
"\n",
"Gegeben sind drei Messpunkte $(x_i, y_i)$ mit den konkreten Messwerten:\n",
"\n",
"<ul>\n",
" <li>(1,1)</li>\n",
" <li>(2,2)</li>\n",
" <li>(3,2)</li>\n",
"</ul>\n",
"\n",
"Gesucht sind zwei Funktionen $f$ und $g$, welche die drei Messpunkte approximieren. Dabei soll $f$ eine Gerade und $g$ ein quadratische Funktion sein."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"Das heißt:\n",
" \n",
"$$\n",
" f(x) = ax + b\\\\\n",
" g(x) = ax^2 + bx + c\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"<h3>Aufgaben: </h3>\n",
"<ol>\n",
" <li>Stell für beide Funktionen das zu lösende LGS auf!</li>\n",
" <li>Löse die LGS und gib die Funktionen $f$ und $g$ konkret an!</li>\n",
"</ol>"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"Für eine Gerade sieht eine mögliche allgemeine Systemmatrix wie folgt aus:\n",
"\n",
"$$\n",
" \\begin{bmatrix}\n",
" 1 & x_1\\\\\n",
" 1 & x_2\\\\\n",
" \\vdots & \\vdots\\\\\n",
" 1 & x_n\n",
" \\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"Für eine quadratische Funktion sieht eine mögliche allgemeine Systemmatrix wie folgt aus:\n",
"\n",
"$$\n",
" \\begin{bmatrix}\n",
" 1 & x_1 & x_1^2\\\\\n",
" 1 & x_2 & x_2^2\\\\\n",
" \\vdots & \\vdots & \\vdots\\\\\n",
" 1 & x_n & x_n^2\n",
" \\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"Für die Gerade sieht ein mögliches LGS so aus:\n",
"\n",
"$$\n",
" \\begin{bmatrix}\n",
" 1 & 1\\\\\n",
" 1 & 2\\\\\n",
" 1 & 3\\\\\n",
" \\end{bmatrix}\n",
" \\begin{bmatrix} b \\\\ a \\end{bmatrix} = \\begin{bmatrix} 1 \\\\ 2 \\\\ 2\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"Lösen mit der Normalengleichung:\n",
" \n",
"$$\n",
"\\begin{bmatrix}\n",
" 1 & 1 & 1\\\\\n",
" 1 & 2 & 3\\\\\n",
" \\end{bmatrix}\n",
" \\begin{bmatrix}\n",
" 1 & 1\\\\\n",
" 1 & 2\\\\\n",
" 1 & 3\\\\\n",
" \\end{bmatrix}\n",
" \\begin{bmatrix} b \\\\ a \\end{bmatrix} = \n",
" \\begin{bmatrix}\n",
" 1 & 1 & 1\\\\\n",
" 1 & 2 & 3\\\\\n",
" \\end{bmatrix}\n",
" \\begin{bmatrix} 1 \\\\ 2 \\\\ 2\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$\n",
" \\begin{bmatrix}\n",
" 3 & 6\\\\\n",
" 6 & 14\\\\\n",
" \\end{bmatrix}\n",
" \\begin{bmatrix} b \\\\ a \\end{bmatrix} = \n",
" \\begin{bmatrix} 5 \\\\ 11\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$\n",
"\\begin{bmatrix} b \\\\ a \\end{bmatrix} = \\begin{bmatrix} \\frac{2}{3} \\\\ \\frac{1}{2} \\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$\n",
" f(x) = \\frac{1}{2}x + \\frac{2}{3}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"<img src=\"imgs/aufg1.png\" style=\"margin:auto;\" width=\"40%\" />"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"Für die quadratische Funktion sieht ein möglich LGS so aus:\n",
" \n",
"$$\n",
" \\begin{bmatrix}\n",
" 1 & 1 & 1\\\\\n",
" 1 & 2 & 4\\\\\n",
" 1 & 3 & 9\\\\\n",
" \\end{bmatrix}\n",
" \\begin{bmatrix} c \\\\ b \\\\ a \\end{bmatrix} = \\begin{bmatrix} 1 \\\\ 2 \\\\ 2\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"Dieses LGS hat eine Lösung!\n",
"\n",
"$$\n",
" \\begin{bmatrix} c \\\\ b \\\\ a \\end{bmatrix} = \\begin{bmatrix} -1 \\\\ 2.5 \\\\ -0.5 \\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"Und mit der Normalengleichung?\n",
" \n",
"$$\n",
"\\begin{bmatrix}\n",
" 1 & 1 & 1\\\\\n",
" 1 & 2 & 3\\\\\n",
" 1 & 4 & 9\\\\\n",
" \\end{bmatrix}\n",
" \\begin{bmatrix}\n",
" 1 & 1 & 1\\\\\n",
" 1 & 2 & 4\\\\\n",
" 1 & 3 & 9\\\\\n",
" \\end{bmatrix}\n",
" \\begin{bmatrix} c \\\\ b \\\\ a \\end{bmatrix} = \\begin{bmatrix}\n",
" 1 & 1 & 1\\\\\n",
" 1 & 2 & 3\\\\\n",
" 1 & 4 & 9\\\\\n",
" \\end{bmatrix}\n",
" \\begin{bmatrix} 1 \\\\ 2 \\\\ 2\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$\n",
" \\begin{bmatrix}\n",
" 3 & 6 & 14\\\\\n",
" 6 & 14 & 36\\\\\n",
" 14 & 36 & 98\\\\\n",
" \\end{bmatrix}\n",
" \\begin{bmatrix} c \\\\ b \\\\ a \\end{bmatrix} = \n",
" \\begin{bmatrix} 5 \\\\ 11 \\\\ 27 \\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"Gleiche Lösung!\n",
"\n",
"$$\n",
" \\begin{bmatrix} c \\\\ b \\\\ a \\end{bmatrix} = \\begin{bmatrix} -1 \\\\ 2.5 \\\\ -0.5 \\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$\n",
" g(x) = -0.5x^2 + 2.5x - 1\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"<div class=\"text-center\">\n",
"<b>Das ist eine Interpolation!</b>\n",
" \n",
"<img src=\"imgs/aufg2.png\" style=\"margin:auto;\" width=\"40%\" />\n",
"</div>"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"<h2>Aufgabe 2: Cholesky-Zerlegung</h2>"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"Berechne die Cholesky-Zerlegung folgender Matrizen:\n",
" \n",
"$$ A =\n",
" \\begin{bmatrix}\n",
" 4 & 0 & 0 & 0\\\\\n",
" 0 & 9 & 0 & 0\\\\\n",
" 0 & 0 & 36 & 0\\\\\n",
" 0 & 0 & 0 & 16\\\\\n",
" \\end{bmatrix} \\quad\n",
" B = \\begin{bmatrix}\n",
" 4 & 2 & 2 & 8 \\\\\n",
" 2 & 10 & 10 & 7 \\\\\n",
" 2 & 10 & 11 & 9 \\\\\n",
" 8 & 7 & 9 & 22\n",
"\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"Für A:\n",
" \n",
"$$\n",
" L = \n",
" \\begin{bmatrix}\n",
" 2 & 0 & 0 & 0\\\\\n",
" 0 & 3 & 0 & 0\\\\\n",
" 0 & 0 & 6 & 0\\\\\n",
" 0 & 0 & 0 & 4\\\\\n",
" \\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"source": [
"Für Diagonalmatrizen einfach die Wurzeln der Diagonalterme bilden"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"Für B:"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$\n",
"l_{1,1} = \\sqrt{\\color{red}4} = \\color{blue}2\n",
"$$\n",
"\n",
"$$\n",
"B = \\begin{bmatrix}\n",
" \\color{red}4 & 2 & 2 & 8 \\\\\n",
" 2 & 10 & 10 & 7 \\\\\n",
" 2 & 10 & 11 & 9 \\\\\n",
" 8 & 7 & 9 & 22\n",
"\\end{bmatrix}\n",
"\\quad\n",
"L = \\begin{bmatrix}\n",
"\\color{blue}2 & 0 & 0 & 0\\\\\n",
"? & ? & 0 & 0\\\\\n",
"? & ? & ? & 0\\\\\n",
"? & ? & ? & ?\\\\\n",
"\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$\n",
"l_{2,1} = \\frac{1}{\\color{green}2} \\color{red}2 = \\color{blue}1\n",
"$$\n",
"\n",
"$$\n",
"B = \\begin{bmatrix}\n",
" 4 & 2 & 2 & 8 \\\\\n",
" \\color{red}2 & 10 & 10 & 7 \\\\\n",
" 2 & 10 & 11 & 9 \\\\\n",
" 8 & 7 & 9 & 22\n",
"\\end{bmatrix}\n",
"\\quad\n",
"L = \\begin{bmatrix}\n",
"\\color{green}2 & 0 & 0 & 0\\\\\n",
"\\color{blue}1 & ? & 0 & 0\\\\\n",
"? & ? & ? & 0\\\\\n",
"? & ? & ? & ?\\\\\n",
"\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$\n",
"l_{2,2} = \\sqrt{\\color{red}{10} - \\color{green}1^2} = \\color{blue}3\n",
"$$\n",
"\n",
"$$\n",
"B = \\begin{bmatrix}\n",
" 4 & 2 & 2 & 8 \\\\\n",
" 2 & \\color{red}{10} & 10 & 7 \\\\\n",
" 2 & 10 & 11 & 9 \\\\\n",
" 8 & 7 & 9 & 22\n",
"\\end{bmatrix}\n",
"\\quad\n",
"L = \\begin{bmatrix}\n",
"2 & 0 & 0 & 0\\\\\n",
"\\color{green}1 & \\color{blue}3 & 0 & 0\\\\\n",
"? & ? & ? & 0\\\\\n",
"? & ? & ? & ?\\\\\n",
"\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$\n",
"l_{3,1} = \\frac{1}{\\color{green}2} \\color{red}2 = \\color{blue} 1\n",
"$$\n",
"\n",
"$$\n",
"B = \\begin{bmatrix}\n",
" 4 & 2 & 2 & 8 \\\\\n",
" 2 & 10 & 10 & 7 \\\\\n",
" \\color{red}2 & 10 & 11 & 9 \\\\\n",
" 8 & 7 & 9 & 22\n",
"\\end{bmatrix}\n",
"\\quad\n",
"L = \\begin{bmatrix}\n",
"\\color{green}2 & 0 & 0 & 0\\\\\n",
"1 & 3 & 0 & 0\\\\\n",
"\\color{blue}1 & ? & ? & 0\\\\\n",
"? & ? & ? & ?\\\\\n",
"\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$\n",
"l_{3,2} = \\frac{1}{\\color{green}3} (\\color{red}{10} - \\color{orange}1 \\cdot \\color{orange}1) = \\color{blue} 3\n",
"$$\n",
"\n",
"$$\n",
"B = \\begin{bmatrix}\n",
" 4 & 2 & 2 & 8 \\\\\n",
" 2 & 10 & 10 & 7 \\\\\n",
" 2 & \\color{red}{10} & 11 & 9 \\\\\n",
" 8 & 7 & 9 & 22\n",
"\\end{bmatrix}\n",
"\\quad\n",
"L = \\begin{bmatrix}\n",
"2 & 0 & 0 & 0\\\\\n",
"\\color{orange}1 & \\color{green}3 & 0 & 0\\\\\n",
"\\color{orange}1 & \\color{blue}3 & ? & 0\\\\\n",
"? & ? & ? & ?\\\\\n",
"\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$\n",
"l_{3,3} = \\sqrt{\\color{red} {11} - \\color{green} 3^2 - \\color{green} 1^2} = \\color{blue}1\n",
"$$\n",
"\n",
"$$\n",
"B = \\begin{bmatrix}\n",
" 4 & 2 & 2 & 8 \\\\\n",
" 2 & 10 & 10 & 7 \\\\\n",
" 2 & 10 & \\color{red}{11} & 9 \\\\\n",
" 8 & 7 & 9 & 22\n",
"\\end{bmatrix}\n",
"\\quad\n",
"L = \\begin{bmatrix}\n",
"2 & 0 & 0 & 0\\\\\n",
"1 & 3 & 0 & 0\\\\\n",
"\\color{green}1 & \\color{green}3 & \\color{blue}1 & 0\\\\\n",
"? & ? & ? & ?\\\\\n",
"\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$\n",
"l_{4,1} = \\frac{1}{\\color{green}2} \\color{red} 8 = \\color{blue}4\n",
"$$\n",
"\n",
"$$\n",
"B = \\begin{bmatrix}\n",
" 4 & 2 & 2 & 8 \\\\\n",
" 2 & 10 & 10 & 7 \\\\\n",
" 2 & 10 & 11 & 9 \\\\\n",
" \\color{red}8 & 7 & 9 & 22\n",
"\\end{bmatrix}\n",
"\\quad\n",
"L = \\begin{bmatrix}\n",
"\\color{green}2 & 0 & 0 & 0\\\\\n",
"1 & 3 & 0 & 0\\\\\n",
"1 & 3 & 1 & 0\\\\\n",
"\\color{blue}4 & ? & ? & ?\\\\\n",
"\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$\n",
"l_{4,2} = \\frac{1}{\\color{green}3} \\color{red} (\\color{red}7 - \\color{orange}4 \\cdot \\color{orange}1) = \\color{blue}1\n",
"$$\n",
"\n",
"$$\n",
"B = \\begin{bmatrix}\n",
" 4 & 2 & 2 & 8 \\\\\n",
" 2 & 10 & 10 & 7 \\\\\n",
" 2 & 10 & 11 & 9 \\\\\n",
" 8 & \\color{red}7 & 9 & 22\n",
"\\end{bmatrix}\n",
"\\quad\n",
"L = \\begin{bmatrix}\n",
"2 & 0 & 0 & 0\\\\\n",
"\\color{orange}1 & \\color{green}3 & 0 & 0\\\\\n",
"1 & 3 & 1 & 0\\\\\n",
"\\color{orange}4 & \\color{blue}1 & ? & ?\\\\\n",
"\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$\n",
"l_{4,3} = \\frac{1}{\\color{green}1} \\color{red} (\\color{red}9 - \\color{orange}4 \\cdot \\color{orange}1 - \\color{purple} 1 \\cdot \\color{purple} 3) = \\color{blue}2\n",
"$$\n",
"\n",
"$$\n",
"B = \\begin{bmatrix}\n",
" 4 & 2 & 2 & 8 \\\\\n",
" 2 & 10 & 10 & 7 \\\\\n",
" 2 & 10 & 11 & 9 \\\\\n",
" 8 & 7 & \\color{red}9 & 22\n",
"\\end{bmatrix}\n",
"\\quad\n",
"L = \\begin{bmatrix}\n",
"2 & 0 & 0 & 0\\\\\n",
"1 & 3 & 0 & 0\\\\\n",
"\\color{orange}1 & \\color{purple}3 & \\color{green}1 & 0\\\\\n",
"\\color{orange}4 & \\color{purple}1 & \\color{blue}2 & ?\\\\\n",
"\\end{bmatrix}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$\n",
"l_{4,4} = \\sqrt{\\color{red}{22} - \\color{green}4^2 - \\color{green}1^2 - \\color{green}2^2} = \\color{blue}1\n",
"$$\n",
"\n",
"$$\n",
"B = \\begin{bmatrix}\n",
" 4 & 2 & 2 & 8 \\\\\n",
" 2 & 10 & 10 & 7 \\\\\n",
" 2 & 10 & 11 & 9 \\\\\n",
" 8 & 7 & 9 & \\color{red}{22}\n",
"\\end{bmatrix}\n",
"\\quad\n",
"L = \\begin{bmatrix}\n",
"2 & 0 & 0 & 0\\\\\n",
"1 & 3 & 0 & 0\\\\\n",
"1 & 3 & 1 & 0\\\\\n",
"\\color{green}4 & \\color{green}1 & \\color{green}2 & \\color{blue}1\\\\\n",
"\\end{bmatrix}\n",
"$$"
]
}
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