{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Relative cost of matrix factorizations"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import numpy.linalg as npla\n",
        "import scipy.linalg as spla\n",
        "\n",
        "import matplotlib.pyplot as pt\n",
        "\n",
        "from time import time"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "array([  10,   14,   20,   30,   43,   63,   92,  133,  193,  279,  404,\n",
              "        585,  848, 1228, 1778], dtype=int32)"
            ]
          },
          "execution_count": 3,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "n_values = (10**np.linspace(1, 3.25, 15)).astype(np.int32)\n",
        "n_values"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "-----> lu\n",
            "10\n",
            "14\n",
            "20\n",
            "30\n",
            "43\n",
            "63\n",
            "92\n",
            "133\n",
            "193\n",
            "279\n",
            "404\n",
            "585\n",
            "848\n",
            "1228\n",
            "1778\n",
            "-----> matmat\n",
            "10\n",
            "14\n",
            "20\n",
            "30\n",
            "43\n",
            "63\n",
            "92\n",
            "133\n",
            "193\n",
            "279\n",
            "404\n",
            "585\n",
            "848\n",
            "1228\n",
            "1778\n",
            "-----> svd\n",
            "10\n",
            "14\n",
            "20\n",
            "30\n",
            "43\n",
            "63\n",
            "92\n",
            "133\n",
            "193\n",
            "279\n",
            "404\n",
            "585\n",
            "848\n",
            "1228\n",
            "1778\n"
          ]
        },
        {
          "data": {
            "text/plain": [
              "<matplotlib.text.Text at 0x7f676d879278>"
            ]
          },
          "execution_count": 5,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
            "image/png": 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lmnOu0b56VW+cvEtnEaB/k5UCcvJzmLhkIiuOrGBw28E8NOAhMppkuB1LRRn9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\n4CulVJT4/067i56nKbwoAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f676f599780>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "def matmat(A):\n",
        "    A @ A\n",
        "\n",
        "for name, f in [\n",
        "        (\"lu\", spla.lu_factor),\n",
        "        (\"matmat\", matmat),\n",
        "        (\"svd\", npla.svd)\n",
        "        ]:\n",
        "\n",
        "    times = []\n",
        "    print(\"----->\", name)\n",
        "    \n",
        "    for n in n_values:\n",
        "        print(n)\n",
        "\n",
        "        A = np.random.randn(n, n)\n",
        "        \n",
        "        start_time = time()\n",
        "        f(A)\n",
        "        times.append(time() - start_time)\n",
        "        \n",
        "    pt.plot(n_values, times, label=name)\n",
        "\n",
        "pt.grid()\n",
        "pt.legend(loc=\"best\")\n",
        "pt.xlabel(\"Matrix size $n$\")\n",
        "pt.ylabel(\"Wall time [s]\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "* The faster algorithms make the slower ones look bad. But... it's all relative.\n",
        "* Is there a better way of plotting this?\n",
        "* Can we see the asymptotic cost ($O(n^3)$) of these algorithms from the plot?"
      ]
    }
  ],
  "metadata": {
    "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": "3.5.2+"
    },
    "widgets": {
      "state": {},
      "version": "1.1.2"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}