added IPython notebook for Remark 1.5.1

4e9d6f32 · Benjamin Berkels · 70d41905 · 4e9d6f32
Commit 4e9d6f32 authored 4 years ago by Benjamin Berkels
--- a/rem-151.ipynb
+++ b/rem-151.ipynb
+{
+  "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
+%% Cell type:markdown id: tags:
+
+### IPython notebook for Remark 1.5.1 from the lecture
+
+%% Cell type:code id: tags:
+
+``` python
+import numpy as np
+from IPython.display import display, Math
+from sympy import Float, latex
+
+
+def k(s, t, d):
+    return d / np.power(d ** 2 + (s - t) ** 2, 3 / 2)
+
+
+d = 0.1
+n = 80
+h = 1 / n
+display(Math(rf"n = {n},\; d = {d},\; h ={h}"))
+
+s = (np.arange(n) + 0.5) * h
+t = (np.arange(n) + 0.5) * h
+display(Math(rf't_j=s_j=(j-\tfrac12)h,\; j=1,\ldots,{n}'))
+A = h * k(s[:, np.newaxis], t[np.newaxis, :], d)
+display(Math(r'A_{i,j}:=h \, k(s_i,t_j),\; k(s,t)  := \frac{d}{(d^2 + (s-t)^2)^{3/2}}'))
+```
+
+%% Cell type:code id: tags:
+
+``` python
+eigvals = np.linalg.eig(A)[0]
+lambda_max = np.amax(eigvals)
+lambda_min = np.amin(eigvals)
+cond_A = lambda_max / lambda_min
+display(Math(r"\lambda_\text{max} = " + latex(Float(lambda_max, 4), max=1)))
+display(Math(r"\lambda_\text{min} = " + latex(Float(lambda_min, 4))))
+display(Math(r"\operatorname{cond(A)} = " + latex(Float(cond_A, 2))))
+```
+
+%% Cell type:code id: tags:
+
+``` python
+```