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exp-233.ipynb

+{
+  "cells": [
+    {
+      "cell_type": "markdown",
+      "metadata": {},
+      "source": [
+        "### IPython notebook for Example 2.3.3 from the lecture"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "import numpy as np\n",
+        "from IPython.display import display, Math\n",
+        "\n",
+        "display(Math(r'A \\in \\mathbb{R}^{6 \\times 3}, A_1 \\in \\mathbb{R}^{6 \\times 4}\\text{ from Example 1.1.1}'))\n",
+        "delta_t = 0.15\n",
+        "n = 6\n",
+        "t = np.arange(1, n+1) * delta_t\n",
+        "A_1 = np.stack((t, np.exp(t), t**3, np.sin(t)), axis=1)\n",
+        "A = A_1[:, :-1]\n",
+        "print(\"A_1 = \")\n",
+        "print(A_1)\n",
+        "print(\"A = \")\n",
+        "print(A)"
+      ],
+      "outputs": [],
+      "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "display(Math(r'U^TAV=\\Sigma'))\n",
+        "U, sigma, VT = np.linalg.svd(A)\n",
+        "np.set_printoptions(precision=4, suppress=True)\n",
+        "print(\"sigma(A) =\", sigma)\n",
+        "print(\"kappa(A) = {:.1f}\".format(sigma[0]/sigma[-1]))"
+      ],
+      "outputs": [],
+      "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "display(Math(r'U_1^TA_1V_1=\\Sigma_1'))\n",
+        "U_1, sigma_1, VT_1 = np.linalg.svd(A_1)\n",
+        "np.set_printoptions(precision=5)\n",
+        "print(\"sigma(A_1) =\", sigma_1)\n",
+        "print(\"kappa(A_1) = {:.1e}\".format(sigma_1[0]/sigma_1[-1]))"
+      ],
+      "outputs": [],
+      "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "display(Math(r'\\delta y^1=10^{-2} \\begin{pmatrix} 1 & 0 & -1 & -1 & -0.5 & 1 \\end{pmatrix}^T'))\n",
+        "delta_y1 = 1e-2*np.array([1, 0, -1, -1, -0.5, 1])\n",
+        "display(Math(r'\\delta y^2=10^{-2}\\begin{pmatrix} -1 & 1 & 1 & -0.5 & -2 & 1 \\end{pmatrix}^T'))\n",
+        "delta_y2 = 1e-2*np.array([-1, 1, 1, -0.5, -2, 1])"
+      ],
+      "outputs": [],
+      "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "np.set_printoptions(precision=4)\n",
+        "print(\"U_1^T\\delta y^1 =\", np.matmul(U_1.T, delta_y1))\n",
+        "print(\"U_1^T\\delta y^2 =\", np.matmul(U_1.T, delta_y2))\n",
+        "display(Math(r'\\Rightarrow\\ \\delta y^1\\text{ is dominated by }u_3, \\delta y^2\\text{ dominated by }u_4'))\n"
+      ],
+      "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": 1
+}
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