added IPython notebook for Example 1.1.1

exp-111.ipynb

+{
+  "cells": [
+    {
+      "cell_type": "markdown",
+      "metadata": {},
+      "source": [
+        "### IPython notebook for Example 1.1.1 from the lecture"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "import numpy as np\n",
+        "from numpy.linalg import lstsq, matrix_rank, norm\n",
+        "from IPython.display import display, Math\n",
+        "from sympy import Matrix, latex\n",
+        "\n",
+        "def my_latex(expr):\n",
+        "    return latex(expr, mat_delim='(').replace('.0 &', '&').replace('.0\\\\', '\\\\')\n",
+        "\n",
+        "delta_t = 0.15\n",
+        "n = 6\n",
+        "t = np.arange(1, n + 1) * delta_t\n",
+        "xi_hat = np.array([1.2, 0.6, 1.6, 0.9])\n",
+        "xi = xi_hat[:-1]\n",
+        "\n",
+        "A_hat = np.stack((t, np.exp(t), t ** 3, np.sin(t)), axis=1)\n",
+        "A = A_hat[:, :-1]\n",
+        "display(Math(r'\\hat A = h \\begin{pmatrix} t_1 & e^{t_1} & t_1^3 & \\sin(t_1) \\\\ t_2 & e^{t_2} & t_2^3 & \\sin(t_2) \\\\ \\vdots &\\vdots & \\vdots & \\vdots \\end{pmatrix},\\quad ' + \\\n",
+        "             r'A = h \\begin{pmatrix} t_1 & e^{t_1} & t_1^3 \\\\ t_2 & e^{t_2} & t_2^3 \\\\ \\vdots &\\vdots & \\vdots \\end{pmatrix},\\quad ' + \\\n",
+        "             rf't = {my_latex(Matrix(t))}'))"
+      ],
+      "outputs": [],
+      "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "display(Math(r\"\\operatorname{rank}(\\hat A) = \" + f\"{matrix_rank(A_hat)}\"))\n",
+        "display(Math(r\"\\operatorname{rank}(A) =  \" + f\"{matrix_rank(A)}\"))"
+      ],
+      "outputs": [],
+      "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "display(Math(r\"\\hat y=\\hat A\\hat\\xi,\\quad y=A\\xi, \\quad \" + rf\"\\hat\\xi = {my_latex(Matrix(xi_hat))},\\quad \\xi = {my_latex(Matrix(xi))}\"))\n",
+        "y_hat = np.matmul(A_hat, xi_hat)\n",
+        "y = np.matmul(A, xi)\n",
+        "\n",
+        "delta_y1 = 1e-2 * np.array([1, 0, -1, -1, -0.5, 1])\n",
+        "delta_y2 = 1e-2 * np.array([-1, 1, 1, -0.5, -2, 1])\n",
+        "\n",
+        "display(Math(rf\"\\delta y_1 = {my_latex(Matrix(delta_y1))},\\quad \\delta y_2 = {my_latex(Matrix(delta_y2))}\"))\n",
+        "display(Math(f\"\\|\\delta y_1\\|_2 = {norm(delta_y1):.3f}\" + r\",\\quad\" \\\n",
+        "             f\"\\|\\delta y_2\\|_2 = {norm(delta_y2):.3f}\"))"
+      ],
+      "outputs": [],
+      "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "tilde_y1_hat = y_hat + delta_y1\n",
+        "tilde_y2_hat = y_hat + delta_y2\n",
+        "tilde_y1 = y + delta_y1\n",
+        "tilde_y2 = y + delta_y2\n",
+        "\n",
+        "tilde_xi1 = lstsq(A, tilde_y1, rcond=None)[0]\n",
+        "tilde_xi2 = lstsq(A, tilde_y2, rcond=None)[0]\n",
+        "\n",
+        "display(Math(r\"\\tilde \\xi_i=\\operatorname{arg\\,min}\\|A\\xi-(y+\\delta y_i)\\|\"))\n",
+        "\n",
+        "display(Math(rf\"\\|\\tilde \\xi_1-\\xi\\|_2 = {norm(tilde_xi1 - xi):.3f}\" + r\",\\quad\" \\\n",
+        "             rf\"\\|\\tilde \\xi_2-\\xi\\|_2 = {norm(tilde_xi2 - xi):.3f}\"))\n",
+        "\n",
+        "tilde_xi1_hat = lstsq(A_hat, tilde_y1_hat, rcond=None)[0]\n",
+        "tilde_xi2_hat = lstsq(A_hat, tilde_y2_hat, rcond=None)[0]\n",
+        "\n",
+        "display(Math(r\"\\tilde{\\hat{\\xi}}_i=\\operatorname{arg\\,min}\\|\\hat A\\xi-(\\hat y+\\delta y_i)\\|\"))\n",
+        "\n",
+        "display(Math(r\"\\|\\tilde{\\hat{\\xi}}_1-\\hat{\\xi}\\|_2 = \" + f\"{norm(tilde_xi1_hat - xi_hat):.3f}\" + r\",\\quad\" \\\n",
+        "             r\"\\|\\tilde{\\hat{\\xi}}_2-\\hat{\\xi}\\|_2 = \" + f\"{norm(tilde_xi2_hat - xi_hat):.1f}\"))"
+      ],
+      "outputs": [],
+      "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "display(Math(r\"\\frac{\\|\\tilde \\xi_1-\\xi\\|_2}{\\|\\delta y_1\\|_2} = \" + f\"{norm(tilde_xi1 - xi) / norm(delta_y1):.1f}\" + r\",\\quad\" \\\n",
+        "             r\"\\frac{\\|\\tilde{\\hat{\\xi}}_1-\\hat{\\xi}\\|_2}{\\|\\delta y_1\\|_2} = \" + f\"{norm(tilde_xi1_hat - xi_hat) / norm(delta_y1):.1f}\"))\n",
+        "display(Math(r\"\\frac{\\|\\tilde \\xi_2-\\xi\\|_2}{\\|\\delta y_2\\|_2} = \" + f\"{norm(tilde_xi2 - xi) / norm(delta_y2):.1f}\" + r\",\\quad\" \\\n",
+        "             r\"\\frac{\\|\\tilde{\\hat{\\xi}}_2-\\hat{\\xi}\\|_2}{\\|\\delta y_2\\|_2} = \" + f\"{norm(tilde_xi2_hat - xi_hat) / norm(delta_y2):.1e}\"))"
+      ],
+      "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
+}
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