{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Vector Norms"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Computing norms by hand\n",
        "\n",
        "$p$-norms can be computed in two different ways in numpy:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import numpy.linalg as la"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "x = np.array([1.,2,3])"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "First, let's compute the 2-norm by hand:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "3.7416573867739413"
            ]
          },
          "execution_count": 3,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "np.sum(x**2)**(1/2)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Next, let's use `numpy` machinery to compute it:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "3.7416573867739413"
            ]
          },
          "execution_count": 4,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "la.norm(x, 2)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Both of the values above represent the 2-norm: $\\|x\\|_2$."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "--------------\n",
        "\n",
        "## About the $\\infty$-norm\n",
        "\n",
        "Different values of $p$ work similarly:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "3.0773848853940629"
            ]
          },
          "execution_count": 5,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "np.sum(np.abs(x)**5)**(1/5)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "3.0773848853940629"
            ]
          },
          "execution_count": 6,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "la.norm(x, 5)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "---------------------\n",
        "\n",
        "The $\\infty$ norm represents a special case, because it's actually (in some sense) the *limit* of $p$-norms as $p\\to\\infty$.\n",
        "\n",
        "Recall that: $\\|x\\|_\\infty = \\max(|x_1|, |x_2|, |x_3|)$.\n",
        "\n",
        "Where does that come from? Let's try with $p=100$:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "array([  1.00000000e+00,   1.26765060e+30,   5.15377521e+47])"
            ]
          },
          "execution_count": 7,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "x**100"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "5.1537752073201132e+47"
            ]
          },
          "execution_count": 8,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "np.sum(x**100)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Compare to last value in vector: the addition has essentially taken the maximum:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 9,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "3.0"
            ]
          },
          "execution_count": 9,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "np.sum(x**100)**(1/100)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Numpy can compute that, too:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 10,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "3.0"
            ]
          },
          "execution_count": 10,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "la.norm(x, np.inf)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "-------------\n",
        "\n",
        "## Unit Balls\n",
        "\n",
        "Once you know the set of vectors for which $\\|x\\|=1$, you know everything about the norm, because of semilinearity. The graphical version of this is called the 'unit ball'.\n",
        "\n",
        "We'll make a bunch of vectors in 2D (for visualization) and then scale them so that $\\|x\\|=1$."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 26,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "(-1.5, 1.5)"
            ]
          },
          "execution_count": 26,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
            "image/png": 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RTsAGVT0ky/1ZT8SUtf79Yflydx4Snz2RecCRIjJcRLoClwIz9ptnJpD5SeyLgFkJLLdo\nAqppxgQndhFJ9ziuA54BFgPTVfUtEfm5iJyfnu0+YICIrAAmAzfEXW6xiFhPJCrLFN2cOSnfERKT\nSE9EVZ8CPrffdbe0uLwTsB+iNIbyW7O1fWdyOO88+N733LkxSejXD95+252HxPadMaZEBPQZmQgr\nIhFYTyQayxTdyy+nfEdIjBURY4qs3NZErCeSg/VETNIqK2H1anceEuuJGFMiAvqMTIQVkQisJxKN\nZYqunLYTsSKSQ7kd+8H4V25rItYTyeH88+Gaa9y5MUk4+GBYt86dh8R6IsaUiIA+IxNhRSSChQtT\nviO0EuJY3zJFZz0RY0zeym1NxHoiOVhPxCStVy94/313HhLriRhTIgL6jEyEFZEIrCcSjWWK7qWX\nUr4jJMaKSATl9slh/Cq315P1RHKoqYFvfcudG5OEgw6CDz5w5yGxnkiBdO4Mu3f7TmHKSUCfkYmw\nIpJD587w5psp3zFaCXGsb5mi2b3bjidyQOncGZqafKcw5ULVFZFOnXwnSY71RHKorYXqanduTFy7\nd0O3bmF+MFlPpECsJ2KStHs3dOniO0WyrIjk0LkzLFmS8h2jlRDH+pYpt8ZG95oKLVccVkRysJ6I\nSVJjY/mtiVhPJIfJk2H4cPjRj3wnMeVg0yY49ljYvNl3ktasJ1IgXbpYT8QkpxzXRKyI5NC5Myxb\nlvIdo5UQx9SWKbddu6Br1/ByxWFFJIeePWHHDt8pTLnYujW8QwDEFauIiEhfEXlGRJaJyNMi0ifL\nfE0iMl9E3hCRx+Iss9h69YL+/at9x2ilurrad4RWLFNu27a511RoueKIuyZyA/Ccqn4OmAXcmGW+\nbap6kqqOVtULYy6zqHr3dp8exiRh61a3dltO4haRGuCB9OUHgGwFomR/eKFXL1i5MuU7Rishjqkt\nU26Z4UxoueKIW0QOUdWNAKr6PnBIlvm6ichrIjJXREpqp/revaGhwXcKUy7KcU2kc64ZRORZYFDL\nqwAFbm5j9mwbeAxX1Q0icgQwS0QWqOqqbMusra1lxIgRAFRWVlJVVbVnDJmp4MWaXrEixfbte7MV\ne/mlNF1dXR1UnoxUKhVMntdfT/Hpp2E8P6lUimnTpgHseb/lI9bGZiLyFlCtqhtF5FBgtqoek+Nv\n7gdmquqfs9we1MZmixbBJZfA4sW+k5hy8POfu+2ObrvNd5LWfG1sNgOoTV++Enh8/xlEpFJEuqYv\nDwDGAUtiLrdoBg2CdetSvmO0EuKY2jLltnkzDBwYXq444haR24GzRGQZcCbwKwAROVlE7knPcwzw\nuoi8ATwP/FJVl8ZcbtH07+/GsY2NvpOYcpApIuXE9p2J4PDDYd48GDzYdxJT6k4/HW6+Gc4803eS\n1mzfmQIaNAg2bvSdwpSD996Dww7znSJZVkQi6N49xdq1vlPsK8QxtWVqX3MzrFkDI0aElSsuKyIR\nDB4MK1b4TmFK3YYN0LdveD8VEZf1RCK46y5YsAD++799JzGl7KWXYMoUmDvXd5K2WU+kgI46ytZE\nTHzLl8ORR/pOkTwrIhHU16dYvDisHx0KcUxtmdpXVwdVVe5ySLnisiISwcCBroCsW+c7iSllLYtI\nObGeSETnnQff/jZMnOg7iSlFTU3Qrx+8847bgDFE1hMpsH/4B3jtNd8pTKl680230WKoBSQOKyIR\npFIpTj8dnn/ed5K9QhxTW6bsZs2CM87YOx1KriRYEYnoi19039Bs2uQ7iSlFs2e7Td7LkfVEOuDC\nC2HSJLj8ct9JTCn55BMYOtRtrVpZ6TtNdtYTKYKJE2H6dN8pTKmZMQPGjw+7gMRhRSSCzPh14kSY\nMyeMnfFCHFNbprb93//BxRfve10IuZJiRaQDevWCmhr4wx98JzGlYuVK961eOW8aYD2RDpo/3xWS\nlSvdL5kZ054f/9j9bObtt/tOklu+PRErInk45xy3evrtb/tOYkK2cSOMGgVvvAHDhvlOk5s1Vgto\n//Hrz34Gt97qfs3MlxDH1JZpX7fdBldc0XYBCfG5ypcVkTyccgqcdhr88pe+k5hQzZ8Pf/wj3HST\n7ySFZ8OZPK1fDyedBDNnwpgxvtOYkOza5V4TP/kJ/OM/+k4TnQ1nimzwYHewossug48/9p3GhORH\nP3KHQDxQNkq0IhJBtvHrpEkwYQJ87Wvu0yeETD5ZJpg61e1j9cADIO18pof4XOXLikhMv/61+23V\nK66w36Y50D30kGu4z5gBffr4TlM81hNJQEMDXHQRVFTAww9Djx6+E5limzrVfWv33HNw7LG+0+TH\neiIe9egBjz7q9o0YN85tiGYODLt2wQ9/CP/+7/Dii6VbQOKwIhJBlPFrly5uHPyd77jDBtx7r/ud\nEZ+Ziu1Ay7RggfsWZtUqeOUVd0DvEHIVmxWRBInA978Pzz7rVm/Hj3cvLlNeNm1y/+czz4TJk+Hx\nx93vyRyorCdSIE1N8Pvfu60Wjz0WbrwRTj21/Y69CduaNXDnnXD//e7r23/+5/I63KGXnoiITBKR\nRSLSJCIntTPfuSKyVESWi8iUOMssFZ06wVVXuaOh1dTANde4/Sj+4z/ci9GUhq1b3a78NTVu40JV\neP11uOOO8iogccQdziwEvgq8kG0GEakA7gTOAY4Fvi4iI2Mut6jijF+7dXMFZPFiuOced/75z7sX\nZKabv3VrcTMVSjlkUoWFC12RuOACt1Hhgw+6XflXr3YN1BEjip8rZJ3j/LGqLgMQaXclfQywQlXX\npOedDtQAS+Msu9SIuOHMqafC7t3upxSffNJtV1BXByNHwgknwHHHudPIke7o4J1j/YdMNqqut7Fq\nFSxb5v4HmVO/fq7fcdllbkg6YIDvtGFLpCciIrOBn6jq/DZu+xpwjqpenZ6+HBijqtdnua+y6Il0\nxI4dboetRYv2npYuhQ8+gEMPdXuBDh3qXsz9+7sXeebUs6f7ivmgg9x55tS9uytAnTq57VfKhar7\n1qupyZ03NrrtdLZv33vKTH/6KXz4oXseM6fNm2HtWrdW0aMHHHGE+1alqgpGj4YTT4RDDvH9KP3I\ntyeS83NORJ4FBrW8ClDgn1R1ZkcXaFrr3t1tXzJu3L7X79rldvRbu9adPvwQPvoI3n577+Vt29yb\nJnPKvIkaGtwbranJ3VdFhSso+58qKjpWZDrSGO7IvM3Ne0+ZAtHW5cxjafl4evZ0RXT/U8+erugO\nGOCK8OjRbnroUDckOfjg6PlMdjmLiKqeFXMZ64GWR1QYkr4uq9raWkakB56VlZVUVVVRXV0N7B1L\nFnO6rq6OyZMnF335XbvCmjVu+rLL9r09M0+U+1OFU0+tpqkJZs9O0dwM48ZV09wML77obj/lFDf/\nyy+7v29rWrX92zOXwd0/wNy5qZzTmXwVFW66ogLGj3fTc+a46dNPd9Mvvtix5/OOO+5o9fr56CM4\n4YRof1+o6cx1vpafWfa0adMA9rzf8qKqsU/AbODkLLd1At4GhgNdgTrgmHbuS0Mze/Zs3xFasUzR\nhJhJNcxc6fdeh9//sXoiInIh8DtgAFAP1KnqBBE5DJiqquen5zsX+A3u26D7VPVX7dynxslkjMmP\nHWPVGBOL7YBXQCF+p2+ZogkxE4SbKx9WRIwxsdhwxhgD2HDGGOOJFZEIQhy/WqZoQswE4ebKhxUR\nY0ws1hMxxgDWEzHGeGJFJIIQx6+WKZoQM0G4ufJhRcQYE4v1RIwxgPVEjDGeWBGJIMTxq2WKJsRM\nEG6ufFgRMcbEYj0RYwxgPRFjjCdWRCIIcfxqmaIJMROEmysfVkSMMbFYT8QYA1hPxBjjiRWRCEIc\nv1qmaELMBOHmyocVEWNMLNYTMcYA1hMxxnhiRSSCEMevlimaEDNBuLnyYUXEGBOL9USMMYD1RIwx\nnsQqIiIySUQWiUiTiJzUznyrReRNEXlDRF6Ls0wfQhy/WqZoQswE4ebKR9w1kYXAV4EXcszXDFSr\n6mhVHRNzmUVXV1fnO0IrlimaEDNBuLny0TnOH6vqMgARyTWOEkp46FRfX+87QiuWKZoQM0G4ufJR\nrDe2Ak+LyDwRuapIyzTGFEHONREReRYY1PIqXFH4J1WdGXE5p6jqBhEZCDwrIm+p6pyOx/Vj9erV\nviO0YpmiCTEThJsrH4l8xSsis4GfqOr8CPPeAnyqqv8vy+32/a4xnuTzFW+snsh+2ly4iBwEVKjq\nVhHpCZwN/DzbneTzIIwx/sT9ivdCEVkLjAWeEJEn09cfJiJPpGcbBMwRkTeAV4CZqvpMnOUaY8IR\n3BarxpjS4vVr11A3VutArnNFZKmILBeRKQXO1FdEnhGRZSLytIj0yTJfk4jMTz9XjxUoS7uPW0S6\nish0EVkhIn8TkWGFyNHBTFeKyKb0czNfRL5VhEz3ichGEVnQzjy/TT9PdSJS5TuTiIwXkfoWz9PN\nOe9UVb2dgM8BRwGzgJPame8doG9IuXAF+G1gONAFqANGFjDT7cBP05enAL/KMt8nBX5ucj5u4Frg\n7vTlS4DpAWS6EvhtsV5D6WV+CagCFmS5fQLwl/TlLwCvBJBpPDCjI/fpdU1EVZep6gqyNGVbKOrG\nahFzjQFWqOoaVW0EpgM1BYxVAzyQvvwAcGGW+QrdmI7yuFtm/RNwZgCZoPDPzT7UbcawpZ1ZaoAH\n0/O+CvQRkUHtzF+MTNDB56lUtiINcWO1wcDaFtPr0tcVyiGquhFAVd8HDskyXzcReU1E5opIIYpa\nlMe9Zx5VbQLqRaRfAbJ0JBPAxPSw4RERGVLAPFHtn3s9hX0NRTU2PRz+i4iMyjVzkl/xtinUjdUS\nypWodjK1NS7N1hEfnn6ujgBmicgCVV2VcNSOCuFr+xnAQ6raKCJX49aUCr2GVIr+jnsNbReRCcBj\nwNHt/UHBi4iqnpXAfWxIn28WkUdxq6+xikgCudYDLRuGQ9LX5a29TOlm2CBV3SgihwKbstxH5rla\nJSIpYDSQZBGJ8rjXAUOB90SkE3Cwqn6UYIYOZ1LVlqvw9wL/WsA8Ua3HPU8ZsV9Dcanq1haXnxSR\nu0WkX3v/v5CGM1k3VhORXunLmY3VFvnOBcwDjhSR4SLSFbgU92lXKDOA2vTlK4HH959BRCrTWRCR\nAcA4YEnCOaI87pnpjAAX4RrUhZQzU7rwZtSQ/POSjZD9NTQDuAJARMYC9Zkhq69MLXsyIjIGtxlI\n+x8AxexWt9EJvhA3JmwANgBPpq8/DHgiffkIXLf9DdyhB24IIVd6+lxgGbCi0LmAfsBz6eU9A1Sm\nrz8ZuCd9+YvAgvRz9SZQW6AsrR43bivk89OXuwGPpG9/BRhRhP9Zrky/wH34vAE8DxxdhEwPAe8B\nO4F3gW8C3wWubjHPnbhvlt6knW8oi5UJ+H6L52ku8IVc92kbmxljYglpOGOMKUFWRIwxsVgRMcbE\nYkXEGBOLFRFjTCxWRIwxsVgRMcbEYkXEGBPL/wfrar+/+XGlfgAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f82450f8278>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "alpha = np.linspace(0, 2*np.pi, 2000, endpoint=True)\n",
        "x = np.cos(alpha)\n",
        "y = np.sin(alpha)\n",
        "\n",
        "vecs = np.array([x,y])\n",
        "\n",
        "p = 5\n",
        "norms = np.sum(np.abs(vecs)**p, axis=0)**(1/p)\n",
        "norm_vecs = vecs/norms\n",
        "\n",
        "import matplotlib.pyplot as pt\n",
        "pt.grid()\n",
        "pt.gca().set_aspect(\"equal\")\n",
        "pt.plot(norm_vecs[0], norm_vecs[1])\n",
        "pt.xlim([-1.5, 1.5])\n",
        "pt.ylim([-1.5, 1.5])"
      ]
    },
    {
      "cell_type": "code",
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