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exp-321.ipynb

+{
+  "cells": [
+    {
+      "cell_type": "markdown",
+      "metadata": {},
+      "source": [
+        "### IPython notebook for Example 3.2.1 from the lecture"
+      ]
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "import numpy as np\n",
+        "from IPython.display import display, Math\n",
+        "\n",
+        "\n",
+        "def kappa(z):\n",
+        "    return np.power(z, -1.5)*np.exp(-1/(4*z)) / (2*np.sqrt(np.pi))\n",
+        "\n",
+        "\n",
+        "n = 20\n",
+        "h = 1/n\n",
+        "\n",
+        "display(Math(r'\\text{Create }A,b_1,b_2\\text{ from Example 1.7.1}'))\n",
+        "A = np.zeros((n, n))\n",
+        "for i in range(n):\n",
+        "    for j in range(i+1):\n",
+        "        A[i, j] = h * kappa((i+1)*h - (j+0.5)*h)\n",
+        "\n",
+        "b_1 = A[:, 0]\n",
+        "b_2 = A[:, 1]"
+      ],
+      "outputs": [],
+      "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "display(Math(r'U^TAV=\\Sigma'))\n",
+        "U, sigma, VT = np.linalg.svd(A)"
+      ],
+      "outputs": [],
+      "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "print('sigma_1 = {:.4f}'.format(sigma[0]))\n",
+        "print('sigma_19 = {:.4f}'.format(sigma[18]))\n",
+        "print('sigma_20 = {:.2e}'.format(sigma[19]))"
+      ],
+      "outputs": [],
+      "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "display(Math(r'\\Sigma_{\\rm reg}^\\dagger:= {\\rm diag}(\\sigma_1^{-1}, \\ldots, \\sigma_{19}^{-1},0)'))\n",
+        "sigma_dagger = 1 / sigma\n",
+        "sigma_dagger[-1] = 0\n",
+        "display(Math(r'A_{\\rm reg}^\\dagger:=V \\Sigma_{\\rm reg}^\\dagger U^T'))\n",
+        "A_dagger_reg = np.matmul(np.matmul(VT.T, np.diag(sigma_dagger)), U.T)\n",
+        "display(Math(r'\\tilde z_{\\rm reg}= A_{\\rm reg}^\\dagger(b_1+b_2)'))\n",
+        "z_reg_tilde = np.matmul(A_dagger_reg, b_1+b_2)\n",
+        "z = np.zeros(n)"
+      ],
+      "outputs": [],
+      "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [
+        "z[0:2] = 1\n",
+        "print('||z_reg_tilde-z||_2 = {:.1e}'.format(np.linalg.norm(z_reg_tilde-z)))\n",
+        "display(Math(r'\\text{In Example 1.7.1, we had }\\|\\tilde  z\\|_2=9.5\\cdot10^{24}\\text{ for }\\tilde z=A\\setminus (b_1+b_2)'))\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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