From 4e9d6f32239a9593bfc73acc37aaf37c07c815d5 Mon Sep 17 00:00:00 2001
From: Benjamin Berkels <berkels@aices.rwth-aachen.de>
Date: Wed, 28 Apr 2021 10:56:14 +0200
Subject: [PATCH] added IPython notebook for Remark 1.5.1
---
rem-151.ipynb | 82 +++++++++++++++++++++++++++++++++++++++++++++++++++
1 file changed, 82 insertions(+)
create mode 100644 rem-151.ipynb
diff --git a/rem-151.ipynb b/rem-151.ipynb
new file mode 100644
index 0000000..3f16778
--- /dev/null
+++ b/rem-151.ipynb
@@ -0,0 +1,82 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### IPython notebook for Remark 1.5.1 from the lecture"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "metadata": {},
+ "source": [
+ "import numpy as np\n",
+ "from IPython.display import display, Math\n",
+ "from sympy import Float, latex\n",
+ "\n",
+ "\n",
+ "def k(s, t, d):\n",
+ " return d / np.power(d ** 2 + (s - t) ** 2, 3 / 2)\n",
+ "\n",
+ "\n",
+ "d = 0.1\n",
+ "n = 80\n",
+ "h = 1 / n\n",
+ "display(Math(rf\"n = {n},\\; d = {d},\\; h ={h}\"))\n",
+ "\n",
+ "s = (np.arange(n) + 0.5) * h\n",
+ "t = (np.arange(n) + 0.5) * h\n",
+ "display(Math(rf't_j=s_j=(j-\\tfrac12)h,\\; j=1,\\ldots,{n}'))\n",
+ "A = h * k(s[:, np.newaxis], t[np.newaxis, :], d)\n",
+ "display(Math(r'A_{i,j}:=h \\, k(s_i,t_j),\\; k(s,t) := \\frac{d}{(d^2 + (s-t)^2)^{3/2}}'))"
+ ],
+ "outputs": [],
+ "execution_count": null
+ },
+ {
+ "cell_type": "code",
+ "metadata": {},
+ "source": [
+ "eigvals = np.linalg.eig(A)[0]\n",
+ "lambda_max = np.amax(eigvals)\n",
+ "lambda_min = np.amin(eigvals)\n",
+ "cond_A = lambda_max / lambda_min\n",
+ "display(Math(r\"\\lambda_\\text{max} = \" + latex(Float(lambda_max, 4), max=1)))\n",
+ "display(Math(r\"\\lambda_\\text{min} = \" + latex(Float(lambda_min, 4))))\n",
+ "display(Math(r\"\\operatorname{cond(A)} = \" + latex(Float(cond_A, 2))))"
+ ],
+ "outputs": [],
+ "execution_count": null
+ },
+ {
+ "cell_type": "code",
+ "metadata": {},
+ "source": [],
+ "outputs": [],
+ "execution_count": null
+ }
+ ],
+ "metadata": {
+ "anaconda-cloud": {},
+ "kernelspec": {
+ "display_name": "Python 3",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.6.1"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
\ No newline at end of file
--
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