lecture-9.ipynb

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   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "Lecture 9: Structured matrices"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
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     "source": [
      "## Syllabus\n",
      "**Week 1:** Python intro  \n",
      "**Week 2:** Matrices, vectors, norms, ranks  \n",
      "**Week 3:** Linear systems, eigenvectors, eigenvalues  \n",
      "**Week 4:** Singular value decomposition + test + homework seminar  \n",
      "**Week 5:** Sparse & structured matrices"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "##Recap of the previous lecture\n",
      "- Sparse matrices, compressed sparse row format\n",
      "- Gaussian elimination and graphs"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Today lecture\n",
      "Today we will talk about **structured matrices**.\n",
      "\n",
      "- Shift-invariant structure: Toeplitz matrices\n",
      "- Circulant matrices and FFT\n",
      "- What is an inverse of a Toeplitz matrix?\n",
      "- Displacement rank (and Gohberg-Semencul formula)\n",
      "- Multilevel matrices"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "In many applications, the linear transformation is described by a shift-invariant integral operator,  also called **convolution**:\n",
      "$$\n",
      "   f(x) = \\int_{\\Pi} K(x -y) u(y) dy,\n",
      "$$\n",
      "where $\\Pi$ is a square in $\\mathbb{R}^d$.\n",
      "\n",
      "Example applications:\n",
      "- Image processing (blurring), $K(x) = \\exp(-\\sigma\\Vert x \\Vert^2)$.\n",
      "- Filters in image processing/signal processing\n",
      "- Integral equations, $K(x) = \\frac{1}{\\Vert x \\Vert}$ - electrostatics, $K(x)=\\exp(i k \\Vert x \\Vert)/(\\Vert x \\Vert)$.\n",
      "- Autoregressive processes ($y(t) = \\sum_k c_k y(t - t_k)$)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "After the discretization on a uniform grid, we get **discrete convolution**:  \n",
      "$$\n",
      "     y_i = \\sum_{j} t_{i - j} x_j,\n",
      "$$\n",
      "and the matrix $T_{ij} = t_{i - j}, i, j = 1, \\ldots, n$ is called a **Toeplitz matrix**: on each diagonal we have the same value."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
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       "slide_type": "slide"
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     "source": [
      "For $4 \\times 4$ case it looks as follows:\n",
      "$$\n",
      "     T = \\begin{pmatrix}\n",
      "           t_0 & t_{-1} & t_{-2} & t_{-3} \\\\\n",
      "           t_1 & t_0 & t_{-1} & t_{-2} \\\\\n",
      "           t_2 & t_1 & t_0 & t_{-1} \\\\\n",
      "           t_3 & t_2 & t_1 & t_0 \n",
      "         \\end{pmatrix},\n",
      "$$\n",
      "and the matrix is completely defined by **2n-1** parameters (first column, first row minus the first element).  \n",
      "\n",
      "Now, let us do a short demo on how the Toeplitz matrices can be used."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "Consider a one-dimensional signal"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import matplotlib.pyplot as plt\n",
      "%matplotlib inline\n",
      "n = 128\n",
      "u = np.zeros(n)\n",
      "u[9] = 0.25\n",
      "u[10] = 0.5\n",
      "u[11] = 0.25\n",
      "\n",
      "u[70] = 1\n",
      "plt.plot(u)"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 1,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10e4552d0>]"
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       "text": [
        "<matplotlib.figure.Figure at 0x10c77ad10>"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "Do some blurring of the signal"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "c = 1\n",
      "T = [[np.exp(-0.1 * (i - j) ** 2) for i in xrange(n)] for j in xrange(n)]\n",
      "T = c * np.array(T)\n",
      "y = T.dot(u)\n",
      "p = np.arange(n)\n",
      "plt.plot(p, y, label='blurred')\n",
      "plt.plot(p, u, label='original')\n",
      "plt.legend(loc='best')"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "<matplotlib.legend.Legend at 0x10eed7b10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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BlcCXcg+Am4FWYAGwAnimEsWVlcFrkFVg8vJUCpqbvd8vtQ9+4kQJvLCPKBEN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\ny/lHsqAAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x104338a10>"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "Now add some noise"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e = np.random.randn(n)\n",
      "w = y + 1e-3 * e\n",
      "plt.plot(w)\n",
      "plt.plot(y)"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10eddb210>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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SX2O02CsPHSDhPU4+v55LHYhVHOuv/DkAJeYw4lHikzoXQKzBcgHwT48/yhWu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\nERERERERERERERGRcvx/5nGXg9XyuAYAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10e942550>"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "Now let us do the \"deblurring\" (we assume, that we know the matrix T)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "urec = np.linalg.solve(T, w)\n",
      "plt.plot(urec)"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10f46cd10>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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SjUU4vryOr1/5IpxQXkcqlYV2rAit+yJp1s6B0rhzIxyOcPbuBR57zH9dOxFK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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10f377950>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "It fails completely! What to do? Use SVD approximation to the matrix T."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "u1, s1, v1 = np.linalg.svd(T)\n",
      "r = 70\n",
      "u1 = u1[:, :r]\n",
      "s1 = s1[:r]\n",
      "v1 = v1[:r, :]\n",
      "urec_svd = v1.T.dot(u1.T.dot(w) / s1)\n",
      "plt.plot(urec_svd)\n",
      "plt.plot(u)"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 16,
       "text": [
        "[<matplotlib.lines.Line2D at 0x110293b90>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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2xImS7QHYkTitrf1olFIydHbbRMuyPYDyCg0BKcEisAKwcWsLFeGMANSW15K0WvSWvkqv\n6OyyKS/LzgGUV6oAKMEisALQ2BKnysoIQEWkAhNK0hzv7kerlFKhs8umIicEVF6hOQAlWARWADbF\n49SUZQQgFAoRJcbmVp0LoPRMZ3eSaDQ7BFRWrjkAJVgEVgCa2+PURGuz1pUTY4veEE7pgVQKuu38\nEFBUcwBKwAisAMQ748Sisax1leEYTQkVAKUw7e1QFrUpC+dWAWkISAkWgRWA1mR829PAHCojMZr0\njqBKDyQSEK3IPBAGNASkBJPACkAi2cKgymwBqInU0tKpN4RTCpNIQFmFjRVyCUDYIlquISAlWARW\nANpTcQZX5whANEZrl3oASmESCQkBuT2ASDhCmQqAEjACKwCdJs7QmmwBiJXHaO1WAVAK09YmAhBx\n5wBC4gFoDkAJEoEVgC4yj4N0GFQeI2GrACiFSSQgUp7MCwFFopoDUIJFYAWgOxxneG22BzCoMka7\nCoDSA4kElJV5hICiGgJSgkVfCMBxwFvACuByj/fPBJYArwMvAFP7YJ89YluZ5wE7DK6O0WFUAJTC\nJBIQ8QgBlWkISAkYxQqABdyIiMA+wOnAlJw27wKfQTr+a4Bbi9xnj3R1gSlvycsBDK2ppQutAlIK\nIwKQHwKyyjQEpASLYgXgIGAlsBroBu4HZua0eRFoTr9+GRhb5D57JB6HUHmc2px5AMNiMbpC6gEo\nhWlrAysnBGSFLCIaAlICRrECMAZY61pel17nx9eAx4vcZ4/E40A0fybwkJoYqUicZHJnW6CUMokE\nWJHsEFAkHCFSpiEgJVhEem5SELMdbY8CzgMO83pz9uzZ217X1dVRV1e3w0ZtaUpirC6qyqqy1teW\nx7Cq4rS1waBBO7x5JeAkEmkPILcKqEw9AKV/qa+vp76+vs+2V6wArAfGuZbHIV5ALlOB25BcwVav\nDbkFoFg2NsWxkjFCoVDW+lg0RrhCHgqjAqD4kUiAVZ5/KwjNASj9Te7geM6cOUVtr9gQ0EJgT2A8\nEAVOBR7NafMx4GHgLCRfsNNpbI4TScXy1sfKY4TK4/pQGKUgiQSEczyASDiCpTkAJWAU6wEkgYuB\n+UhF0O3Am8CF6fdvAX4IDAZ+l17XjSSPdxqNLS1ETb4A1JbXYqIt+lhIpSBtbWBZOWWgYYtIRHMA\nSrAoVgAA5qX/3Nzien1++m+Xsbk1TjkeHkA0RqpMnwusFCaRgFAkPwQUjmgISAkWgZwJvLUtnvU8\nYIeaaA221UY8vj25a+WjRiIBYcsjBKRJYCVgBFIAmtrjVFm1eeutsIVlKtgc1ySA4k8iASGPEJCl\nISAlYARSAJrb41SX5XsAAFETY1NcJ4Mp/mzzAPJCQOoBKMEikALg9ThIhygxtuiD4ZUCtLUB4fxb\nQYQszQEowSKYAtDdQqzcWwAqQ7VsadP7ASn+OCGg3LuBqgegBI1ACkBbMp73OEiHSitGsz4XWClA\nIgGhcP7dQMOW5gCUYBFIAWi34wz2EYAqK0ZLpwqA4k8iASaUEwIKWRCWEJDRIjIlIARSADo8ngfs\nUF0WI64CoBQgkQDC+SEgQjaWBd3d/WebovQlgRSATuIMi+WXgYLcDiKuzwVWfDAGifOH8stAbWNT\nUYGGgZTAEEgB6A7FGVbr7QEMKq8loQKg+NDeDuXlYBs7LwRkp2zKy9FEsBIYgikA4RZGDPIRgMoY\nCVurgBRvEgmoqoJkKudWEGGLZCpJRYUKgBIcAicAxkAqkv88YIfdqmK0p9QDULxxBMBO5T8QRkNA\nStAInAC0tQEVcQZXeQvAkOoYnagAKN5sEwANASkfAQInAPE4hDweB+kwtCZGlwqA4oOGgJSPEoET\ngJYWgylr9Z0JPDQWIxlWAVC8aWuD6moJAeXeDdQJAakAKEEhcALQ2JQglCrPit+6GTGolqQV18k8\niifuEFDuTGA7pTkAJVgETgA2NrV4Pg7SYXBVjFBFi17Eiic9hYA0B6AEicAJQGNLnLICAuA8F1if\nCqZ44a4CyksC9yIEtH49JJO7wFClX3ntNQLxbPHACcDmeJyox+MgHWLRGKgA7HJaW+H8XjwYtKkJ\nWvpxmoZfCCgSjvQYAnr9dfjEJ+CRR3aRsUq/ccEF8NRT/W1F8QRPAFrjVIQKewBGnwvcp9x1FyxZ\nUrjNW2/B7bfDhx8WbvfDH8LPf953tm0vWUngnBCQbfzLQFevhhNOgNGj4f33d529yq7HtuHNN+GD\nD/rbkuIJnAA0JeJUhr3vAwRQbpVDKMXWlq5daFWwueMO+SvEu+/K/4ULC7dbsgSWL+8buwrR2Ai/\n/33++qwcQE4IyK8MtKsLjjsOLr8cTj8dNm70368WH5Q+q1fLOaACMABpao9THfH3AEKhEJZdS2Oz\nloL2FQ0N8I9/FO7cGhogHC4sAMbA0qXiLRSisxOOOELu27Oj1NfD9dfnr+9NCChXANavF1tmzYJR\no2DDBu99XnMN/PjHO26zMjBwBigqAAOQ5s4W3+cBO5SlYjT2Z6C5hEilCr/f0SGj6fZ2WLHCv11D\nAxxzTGEBcEIn775b+JbL8+fDggWyzR1l2TJYsyZ/P1lJYJ8QUG4OoKkJhgyR1yNH+gvAsmXwxz8W\nFkpj4KKLtNS0v1izBh54oHCbZctgzz2DEeoLnAC0dsZ9J4E5RImxpU09gJ547TXYb7/CndGqVfDx\nj0v8+/HH/du9+y6ceiq8+qp/B7h0KUybBmPGZEJGXtx3H1iWvwB8+9s9J2KXLRNxW7s2e/2OhIC2\nboXddpPXo0b5h4Dee0+OV6F8yYoVEpp6/fXC9is7h7/8Bb73vcIivXw5HHusegADktbuOLU9CEBF\nSAWgN7z8MrzxBvzud/5t3n0XJk6EE0+UMJAfDQ1w1FFSIuk3clq6FPbdF/be2z8M1NYmQnPKKf4C\n8Oc/wznnyMXsx7JlMHSodMi52992LyAPD8BLAJqaYPBgeV3IA1i9Wux+6CF/uxYskP+vvebfBuCl\nlwp/v74kCB0dwHPPSb6mEC+/LIOCQmHIZctUAAYsiWSc3XxuBOdQEY7RlFAB6ImlS+HrX4frrpNO\nzouGBpg0ScI7L72EZ3VVZ6d0ih/7GHz60/5hoN4IwGOPwSGHwPTp3gKQSMiI/Omn4VvfgkcfzW/T\n1SUd/3/9V76nkUhkqoB6Wwa6dWu2AGzcmD+C7OiALVvEO3nwQf8R5oIFUkpaSAC6u0XgbrjBv01v\n+O1v4d//Ltxm0SKYMMH7d3WzeLF0jAOVRAKOP16OfSFeflnyS08+6f1+KiXnZl0dbNokFUGlTOAE\noD3lfydQhyorRnPHjgmA3+guiCxdCqedBjNnigh44XgAsRgcfDA880x+m9WrYdw4iETgwAMlDOS3\nv333hSlT/AXgvvuk0mbSJO8w0apVMH48fOpT8Mtfwp135rdZsULEaMqUfA+g4N1AfcpA3SGgigoR\nkK1bs9usWQNjx8JBB4kALV3q/f2ef15EopAA3HSTCM7ixd45mpYW+H//r+fR7q9/DddeW7jNgw+K\n4D37rH8bY6Qufs6cwtsyBu6+u+8nyq1e3XMZ8vPPS9jwttv822zcKAOdiy/2F4D33pN8j/PXU1nz\nQGdACkAqJYnFnrj77vwYboeJM7TGvwwUoDpSS0uOAPzmNzKq/NrX4G9/8/7cmjUSn+7v+OwLL3iP\nbHN5/PGek7jNzd6je2Mk/PPJT0rlyu23e8e2HQ8AJA/gFQZ6991MmwMP9PYAkkl4+20Z/e69t9RZ\n57J1q1TvnHyyiI6XB7BqlbwHsM8+3m2WLZP9TJjgLwDbczdQdwgIvMNA770nwhQKwZe+5D0S3bBB\nRpVnnCEC6JV7aWyUTvuOO6C2Nt9+gCeeEBGZNk06Pi9Wr5ZR/UsviW1eGCPhqtNPLxzee+EFOTee\neqqw6DzzDHz1qz2PwreXb35TbCx0rj/5JPzP/8h59c473m1eflkGMcceK8fN6/g75w7A7ruXfhio\nLwTgOOAtYAVwuU+bG9LvLwGm9bTBBx6AvfaSDsGP11+H886D++/PXt8VamForLAHUBONEe/KrgJ6\n4gnpWKZNg3PP9T6ZHnoIBg3yHw33BWvXyijljDP8R4nXXgtf+YpcTM3N3m2WLCkcl3/2Wfm+I0fC\nN76R//66dTKaHT5cTvRPfQr+85/8do4HAHD44d4jV7dIOAKQGwJ55x0R1+rqTAgot838+XDkkdLx\nTZgggpzrgrvtmTRJ9p27HbcAeIWAfB8IUyAE5HgA4J0IXr1akuUgAvDXv+YfpwUL4NBD5RjssYf3\n7z97Npx5pngv06ZJiCaXhQvhyitFuE8+2Xs7zz0nYbszzxRvwYs33pAO/eqrZTDhF7b6zW/gssvk\nd/vnP73bgJTBnnmmlN/2VAm1caNcC5s2+bcDOSeXLJFztVDi/8kn4aST5Jrx+74vvSQCMGSIDB68\nwmPLl8t7oAIAYAE3IiKwD3A6MCWnzQnAHsCewNeBAilF4ZFHxFU+8URvT8AYUf2jj84kzZz1XaE4\nI32eBuZQWx4j3pXtASxfDl/+srh/w4Z5xzMfeghuvVU6z0JJos5O6Ui8+PBDf3f6nXckBFJfL5+/\n++78Nu3tMjpZtkyeXVtX530x3X477L8/zJ2b/55tS2jnhBNkW14nuhOOcZg8OX/klEplj7gnTvQO\nyzQ0ZNqMHi12r17tv79hwyRclNuJvvmmVCVBRpxyPUC3ANTWSmeeOxp3BGDiRO8kcEVlCoMhHMpc\nHoVCQLkegNdcAMcDAOm4V67M386CBRJ/BhFcLzF95BG45JLMdrwE4NVXJdfyxS/Cf/+3nE+5PPus\nJOUvvFDOFa+y24c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       "text": [
        "<matplotlib.figure.Figure at 0x11019bcd0>"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Questions\n",
      "1. What is the complexity of taking Toeplitz matrix-by-vector product?\n",
      "2. What is the complexity of solving linear systems with Toeplitz matrices?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Matrix-by-vector product\n",
      "A Toeplitz matrix is defined by $2n-1$ parameters, so we migh expect a $\\mathcal{O}(n \\log n)$ algorithm.  \n",
      "How we can do that?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Convolutions on the continious level\n",
      "There is a well-known result that convolution is  a product in the **Fourier domain**:  \n",
      "$$f = g * h, \\quad \\mbox{or } f(x) = \\int g(x) h(x - y) dy,$$  \n",
      "$$\\widehat{f} = \\widehat{g} \\widehat{h},$$\n",
      "where\n",
      "$$\n",
      "   \\widehat{f}(k) = \\int e^{i k x} f(x) dx.\n",
      "$$\n",
      "is the **Fourier transform of a function**."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "Thus, the algorithm can look as follows:  \n",
      "1. Compute Fourier transforms $\\widehat{g}, \\widehat{h}$.\n",
      "2. Multiply them $\\widehat{f} = \\widehat{g} \\widehat{h}$.\n",
      "3. Compute inverse Fourier transform\n",
      "   $$\n",
      "      f(x) = \\int e^{-ik x} \\widehat{f}(k) dk.\n",
      "   $$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "But we have to work on the discrete level, so we might replace the continious Fourier transform by the  **discrete Fourier transform** (DFT). "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Discrete Fourier transform\n",
      "\n",
      "The DFT matrix is defined as\n",
      "$$\n",
      "   F_{kl} = w^{kl}, \\quad k, l = 0, \\quad n-1,\n",
      "$$\n",
      "and $w = \\exp(\\frac{2 \\pi i}{n})$ is the root of unity.  \n",
      "It is a discretization of the integral Fourier transform.  "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "But a Toeplitz matrix is defined by $(2n-1)$ parameters!  \n",
      "To apply Fast Fourier Transform (FFT) we have to consider a subclass of Toeplitz matrices:  \n",
      "**circulant matrices**."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Circulant matrices\n",
      "A circulant matrix is a Toeplitz matrix with  \n",
      "$$\n",
      "   t_{n - i} = t_i.\n",
      "$$\n",
      "Example:\n",
      "$$\n",
      "     C = \\begin{pmatrix}\n",
      "           c_0 & c_3 & c_2 & c_1 \\\\\n",
      "           c_1 & c_0 & c_3 & c_2 \\\\\n",
      "           c_2 & c_1 & c_0 & c_3 \\\\\n",
      "           c_3 & c_2 & c_1 & c_0 \n",
      "         \\end{pmatrix}.\n",
      "$$\n",
      "A circulant matrix is uniquely defined by its first column and has $n$ independent parameters."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Spectral theorem \n",
      "Any circulant matrix $C$ is diagonalized by the **Fourier matrix** F.  \n",
      "$$\n",
      "    C = F^* \\Lambda F, \n",
      "$$\n",
      "and $\\Lambda_{ii} = \\lambda_i$ is the diagonal matrix and  \n",
      "$$\n",
      "   \\lambda = Fc, \n",
      "$$\n",
      "where $c$ is the first column of the circulant matrix."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Proof\n",
      "Fourier matrix is unitary, $FF^* = I$. Thus, it is sufficient to show that the rows of the Fourier matrix are eigenvectors of the circulant matrix. And this is obvious from the definition. The eigenvalues also follow immediately."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Fast circulant matrix-by-vector product\n",
      "Given the spectral theorem, it is easy to provide a $\\mathcal{O}(n \\log n)$ algorithm for circulant matrix-by-vector product via FFT.  "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "Indeed, \n",
      "$$\n",
      "   y = Cx = F^* \\Lambda F x = F^* \\Big( \\mathrm{diag}(Fc) \\Big) Fx,\n",
      "$$\n",
      "so you see a direct analogue with the continious case: \n",
      "1. Compute two direct FFTs\n",
      "2. Compute their elementwise product (diagonal matrix by vector)\n",
      "3. Compute one inverse FFT"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "A quick demo on the spectral theorem"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import scipy.linalg\n",
      "n = 50\n",
      "b = np.random.randn(n)\n",
      "mm = scipy.linalg.circulant(b)\n",
      "mm1 = np.fft.ifft(mm, axis = 0)\n",
      "mm1 = np.fft.fft(mm1, axis = 1) \n",
      "#mm1 = np.fft.fft2(mm)\n",
      "print np.linalg.norm(mm1 - np.diag(np.diag(mm1)))\n",
      "w1 = np.linalg.eigvals(mm) \n",
      "w2 = np.fft.ifft(b) * n\n",
      "plt.plot(w1.real, w1.imag, ls='', marker='x')\n",
      "plt.plot(w2.real, w2.imag, ls='', marker='.')"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "1.04643809636e-14\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 34,
       "text": [
        "[<matplotlib.lines.Line2D at 0x111bd3dd0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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s2a+6s62BtAZ79is1BnklxZ79So1BXsmwZ79SZE5eybBnv2JgP3lJipgLr5KkZQZ5SYqY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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x11051edd0>"
       ]
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Toeplitz matrix-by-vector product\n",
      "Fine, but how it helps is with the Toeplitz matrices.  \n",
      "The eigenvectors of the Toeplitz matrices are not the Fourier vectors (although they are **asymptotically similar**).  The idea is to embed a Toeplitz matrix into larger circulant matrix."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "$$\n",
      "     T = \\begin{pmatrix}\n",
      "           t_0 & t_{-1} & t_{-2} & t_{-3} \\\\\n",
      "           t_1 & t_0 & t_{-1} & t_{-2} \\\\\n",
      "           t_2 & t_1 & t_0 & t_{-1} \\\\\n",
      "           t_3 & t_2 & t_1 & t_0 \n",
      "         \\end{pmatrix} \\rightarrow \\begin{pmatrix}\n",
      "           t_0 & t_{-1} & t_{-2} & t_{-3} &  t_3& t_2 & t_1  \\\\\n",
      "           t_1 & t_0 & t_{-1} & t_{-2} & t_{-3} & t_3 & t_2\\\\\n",
      "           t_2 & t_1 & t_0 & t_{-1} & t_{-2} & t_{-3} & t_3\\\\\n",
      "           t_3 & t_2 & t_1 & t_0  & t_{-1} & t_{-2} & t_{-3} \\\\\n",
      "           t_{-3}    & t_3    & t_2    & t_1     & t_0 & t_{-1} & t_{-2}\\\\\n",
      "           t_{-2}    & t_{-3}       & t_3    & t_2    & t_1     & t_0 & t_{-1} \\\\\n",
      "           t_{-1}    &  t_{-2}       &  t_{-3}      & t_3    & t_2    & t_1     & t_0 \\\\\n",
      "         \\end{pmatrix}\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "or \n",
      "$$\n",
      "    C = \\begin{bmatrix}\n",
      "         T & * \\\\\n",
      "         * & * \n",
      "        \\end{bmatrix},\n",
      "$$\n",
      "i.e. $T$ is a submatrix of a circulant matrix, and \n",
      "$$\n",
      "     C \\begin{bmatrix}\n",
      "         q \\\\\n",
      "         0\n",
      "       \\end{bmatrix} = \\begin{bmatrix}\n",
      "       T q \\\\\n",
      "       *\n",
      "        \\end{bmatrix}\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n = 512\n",
      "a = np.random.randn(2*n)\n",
      "b = np.random.randn(2*n-1)\n",
      "%timeit np.fft.fft(a)\n",
      "%timeit np.fft.fft(b)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "100000 loops, best of 3: 12.5 \u00b5s per loop\n",
        "10000 loops, best of 3: 26.4 \u00b5s per loop"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n"
       ]
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Matrix-by-vector product by a Toeplitz matrix\n",
      "Final algorithm:\n",
      "\n",
      "1. $c = [t_i; t_{n-i}]$, $x = [x; 0]$\n",
      "2. Compute circulant matrix-by-vector product\n",
      "3. Take the part of the solution\n",
      "\n",
      "This is an $\\mathcal{O}(n \\log n)$ algorithm."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Multidimensional convolution\n",
      "In two dimensions (one of the most important cases) the discrete convolution is defined as\n",
      "$$\n",
      "   y_{ii'} = \\sum_{jj'} T_{(i-i')(j-j')} x_{jj'},\n",
      "$$\n",
      "which leads to **block Toeplitz matrices**: block Toeplitz with Toeplitz blocks (BTTB).\n",
      "(draw a picture on the board)."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Block case\n",
      "The situation is the same: embed into **two-level circulant matrix** (of size $[(2n-1) \\times (2n-1)]^2$),\n",
      "do **two-dimensional FFT**, multiply and do inverse **two-dimensional FFT**.\n",
      "\n",
      "A two-dimensional FFT is simple:  it it just Kronecker product of two one-dimensional FFT. \n",
      "$$\n",
      "   F_2 = F \\otimes F.\n",
      "$$\n",
      "In practice, it means \"apply FFT to columns, and apply FFT to rows\"."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Inversion of Toeplitz matrices\n",
      "\n",
      "Inversion of the circulant matrix is trivial:  $$C = F D F^* \\rightarrow C^{-1} = F D^{-1} F^*,$$\n",
      "and the inverse of a circulant matrix is a circulant matrix.  \n",
      "Inversion of Toeplitz matrices is notoriously difficult."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "Inverse of Toeplitz  matrix is not a Toeplitz matrix!"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import scipy\n",
      "from scipy.linalg import toeplitz\n",
      "n = 6\n",
      "c = np.arange(n)\n",
      "t = scipy.linalg.toeplitz(c**3)\n",
      "print t\n",
      "ii = np.linalg.inv(t)\n",
      "print 'Inverse:'\n",
      "print ii"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[  0   1   8  27  64 125]\n",
        " [  1   0   1   8  27  64]\n",
        " [  8   1   0   1   8  27]\n",
        " [ 27   8   1   0   1   8]\n",
        " [ 64  27   8   1   0   1]\n",
        " [125  64  27   8   1   0]]\n",
        "Inverse:\n",
        "[[ 0.04638091 -0.16018559  0.18135219 -0.06522316  0.07954043 -0.02896156]\n",
        " [-0.16018559  0.58152894 -0.73685373  0.36588599 -0.22669023  0.07954043]\n",
        " [ 0.18135219 -0.73685373  1.15422006 -0.88002651  0.36588599 -0.06522316]\n",
        " [-0.06522316  0.36588599 -0.88002651  1.15422006 -0.73685373  0.18135219]\n",
        " [ 0.07954043 -0.22669023  0.36588599 -0.73685373  0.58152894 -0.16018559]\n",
        " [-0.02896156  0.07954043 -0.06522316  0.18135219 -0.16018559  0.04638091]]\n"
       ]
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Displacement rank\n",
      "It is important to have a structure that defines the inverse of a Toeplitz matrices in $\\mathcal{O}(n \\log n)$ parameters (otherwise we will have to store $\\mathcal{O}(n^2)$ parameters).  \n",
      "The key idea here is the **displacement rank** concept."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "Introduce a shift matrix \n",
      "$$\n",
      "  Z_f = \\begin{bmatrix}\n",
      "  0 & 0 & 0 & f \\\\\n",
      "  1 & 0 & 0 & 0 \\\\\n",
      "  0 & 1 & 0 & 0 \\\\\n",
      "  0 & 0 & 1 & 0 \\\\\n",
      "  \\end{bmatrix}.\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "For any Toeplitz matrix $T$ it holds that  the matrix\n",
      "$$\n",
      "  M = L(T) = Z_f T - T Z_e\n",
      "$$\n",
      "has **rank 2**.\n",
      "This is the definition of **displacement rank**: for a fixed pair $Z_f$, $Z_e$, $L(T)$ has rank at most $r$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Displacement rank is preserved by inversion!\n",
      "**Theorem** If a matrix has displacement rank $r$ for a pair of generators $Z_f, Z_e$, then the inverse also has displacement rank for anot higher then $r$ for a pair of generators $Z^{\\top}_e, Z^{\\top}_f$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Proof\n",
      "$$Z_f T - T Z_e = UV^{\\top},$$\n",
      "therefore\n",
      "$$\n",
      "    T^{-1} Z_f - Z_e T^{-1} = T^{-1} U V^{\\top} T^{-1},\n",
      "$$\n",
      "and the right-hand side has rank at most $r$. Moreover, if the right-hand side is given,\n",
      "it is  easy to recover the elements of the matrix, or even compute matrix-by-vector products.  \n",
      "Further reading is a [paper by Victor Pan](http://comet.lehman.cuny.edu/vpan/pdf/188b.pdf)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "This is not an inversion algorithm, it is only the structure of the inverse.  \n",
      "**Gohberg-Semencul** algorithm was the first work on the subject and presented an $\\mathcal{O}(n^2)$ algorithm.  \n",
      "Superfast algorithms ($\\mathcal{O}(n \\log n)$) are typically quite complicated, not stable and require the recursion."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Cauchy matrices\n",
      "A matrix is called a Cauchy-like matrix (another example of a structured matrix), if  \n",
      "$$\n",
      "   a_{ij} = \\frac{b_{ij}}{(x_i - y_j)},\n",
      "$$\n",
      "where matrix $B$ is of low rank.  A Cauchy matrix corresponds to the case when $b_{ij} = 1$. "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "##Displacement structure of the Cauchy matrix\n",
      "Displacement structure of the Cauchy-like matrix is simple:  \n",
      "$$\n",
      "     x_i a_{ij} - a_{ij} y_j = b_{ij} (x_i - y_j),\n",
      "$$\n",
      "i.e. $L(A) = D_1 A - A D_2$.  \n",
      "We can transform generators from one case to another. If we diagonalize $Z_e, Z_f$, we get the diagonal matrices,  \n",
      "i.e. by FFT we can reduce the Toeplitz matrix to a Cauchy-like matrix.  "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "Good news about the Cauchy-like matrix is that there is an explicit inverse formula,\n",
      "$$\n",
      "   a_{ij} = \\frac{b_{ij}}{(x_i - y_j)},\n",
      "$$\n",
      "which also allows for **pivoting**  \n",
      "(permuting rows and columns of a Cauchy-like matrix leaves it a Cauchy-like matrix)."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## FFT of a Toeplitz matrices and low-rank\n",
      "The transformation between the Cauchy-like matrices and Toeplitz matrices is just a Fourier transform:  \n",
      "$$\n",
      "     P = F T F^*.\n",
      "$$\n",
      "For a circulant matrix we will get the diagonal matrix.  \n",
      "But what we will get for a general Toeplitz matrix?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import scipy.sparse\n",
      "n = 100\n",
      "a = [[1.0 / (i - j + 0.5) for i in xrange(n)] for j in xrange(n)]\n",
      "a = np.array(a)\n",
      "#ex = np.ones(n);\n",
      "#a = scipy.sparse.spdiags(np.vstack((ex,  -2*ex, ex)), [-1, 0, 1], n, n, 'csr'); \n",
      "#a = a.todense()\n",
      "mm1 = np.fft.ifft(a, axis=0)\n",
      "mm1 = np.fft.fft(mm1, axis=1) "
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "outputs": [],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import matplotlib.pyplot as plt\n",
      "from mpl_toolkits.mplot3d import axes3d\n",
      "\n",
      "fig = plt.figure()\n",
      "ax = fig.add_subplot(111, projection='3d')\n",
      "#fig, ax = ppl.subplots(111, projection='3d')\n",
      "t = np.arange(n)\n",
      "x, y = np.meshgrid(t, t)\n",
      "ax.plot_surface(x, y, mm1.real, rstride=10, cstride=10)\n",
      "ax.autoscale_view(tight=True)\n",
      "ax.view_init(elev=10., azim=40)\n",
      "fig.tight_layout()\n",
      "#X, Y, Z = axes3d.get_test_data(0.05)\n",
      "#ax.plot_wireframe(X, Y, Z, rstride=10, cstride=10)\n"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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2b2kOmSrLpa20SpvHEoQ2Bule0+XGDCMnkDEoy7LYunUrK1eupLm5mQcffJCl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\nkUgkoc7/B2UvB5gz2CHZAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x111e5dc90>"
       ]
      }
     ],
     "prompt_number": 43
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "source": [
      "The singularity is along the diagonal!  \n",
      "Off the diagonal we have nice and **low-rank blocks!**"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "b11 = mm1[n/2:, :][:, :n/2]\n",
      "s = np.linalg.svd(b11)[1]\n",
      "plt.semilogy(s)"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 44,
       "text": [
        "[<matplotlib.lines.Line2D at 0x112eb9e50>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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rxH/3qBh4BZgIHIj7Nz/dj2NYk1gzYDn26zeWX+7FNcBurM3bq+E9frkXYLMp\nfwacjrX7x//aP+m9yKYKIJ3PFeSqcqyMAzgD+5/fL+phX/4LsSYg8Pf9AJtDaylwHv68FxcDP8Sa\nPkqBK7H/P/x4L8C+/AH+BvwBm54noXuRTQlgNdCZ6HMFw4l28vjV68BN4b9vIvpFmO8KgGewUS+z\nY4778X60IjqSoxFwFfYL2I/34gHsh2En4AbgP7HJJ/14LxoDp4b/bgJcjfUf5vS98PNzBZFnLQ4T\nfdaiBdbGl3NDupJ0KdbssQb7svsLNkTYj/ejJ/ABdi/WYu3f4M97Eev7RH8g+vFedML+n1iDDZWO\nfF/68V6IiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiOSv/wedFoPNSSj4cwAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x111e4af10>"
       ]
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Block-low rank matrices\n",
      "In fact there is more: every block outside the main diagonal has low rank. This class is called **quasiseparable matrices**. There exist exact $\\mathcal{O}(n)$ algorithms for their inversion (and the inverse of a quasiseparable matrix).  Thus, the algorithm:\n",
      "\n",
      "1. Compute quasiseparable representation: here you can do it in a fast way!\n",
      "2. Compute the inverse"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## A note on the block case\n",
      "For multilevel Toeplitz matrices there is not a lot known about the inversion.  \n",
      "The best known algorithms have complexity $\\mathcal{O}(N^{3/2})$. There is no explicit representation of the inverse.  \n",
      "However, there are a lot of possibilities for the **iterative algorithms**, since matrix-by-vector product is fast."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "## Take home message\n",
      "- Toeplitz matrices\n",
      "- Circulant matrices and FFT\n",
      "- Displacement structure\n",
      "- Low-rank blocks in the Fourier transform (and quasiseparable matrices)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "source": [
      "##### Questions?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from IPython.core.display import HTML\n",
      "def css_styling():\n",
      "    styles = open(\"./styles/custom.css\", \"r\").read()\n",
      "    return HTML(styles)\n",
      "css_styling()"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "skip"
      }
     },
     "outputs": [
      {
       "html": [
        "<link href='http://fonts.googleapis.com/css?family=Fenix' rel='stylesheet' type='text/css'>\n",
        "<link href='http://fonts.googleapis.com/css?family=Alegreya+Sans:100,300,400,500,700,800,900,100italic,300italic,400italic,500italic,700italic,800italic,900italic' rel='stylesheet' type='text/css'>\n",
        "<link href='http://fonts.googleapis.com/css?family=Source+Code+Pro:300,400' rel='stylesheet' type='text/css'>\n",
        "<style>\n",
        "    @font-face {\n",
        "        font-family: \"Computer Modern\";\n",
        "        src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunss.otf');\n",
        "    }\n",
        "    div.cell{\n",
        "        /*width:80%;*/\n",
        "        /*margin-left:auto !important;\n",
        "        margin-right:auto;*/\n",
        "    }\n",
        "    h1 {\n",
        "        font-family: 'Alegreya Sans', sans-serif;\n",
        "    }\n",
        "    h2 {\n",
        "        font-family: 'Fenix', serif;\n",
        "    }\n",
        "    h3{\n",
        "\t\tfont-family: 'Fenix', serif;\n",
        "        margin-top:12px;\n",
        "        margin-bottom: 3px;\n",
        "       }\n",
        "\th4{\n",
        "\t\tfont-family: 'Fenix', serif;\n",
        "       }\n",
        "    h5 {\n",
        "        font-family: 'Alegreya Sans', sans-serif;\n",
        "    }\t   \n",
        "    div.text_cell_render{\n",
        "        font-family: 'Alegreya Sans',Computer Modern, \"Helvetica Neue\", Arial, Helvetica, Geneva, sans-serif;\n",
        "        line-height: 1.2;\n",
        "        font-size: 160%;\n",
        "        /*width:70%;*/\n",
        "        /*margin-left:auto;*/\n",
        "        margin-right:auto;\n",
        "    }\n",
        "    .CodeMirror{\n",
        "            font-family: \"Source Code Pro\";\n",
        "\t\t\tfont-size: 90%;\n",
        "    }\n",
        "/*    .prompt{\n",
        "        display: None;\n",
        "    }*/\n",
        "    .text_cell_render h1 {\n",
        "        font-weight: 200;\n",
        "        font-size: 50pt;\n",
        "\t\tline-height: 110%;\n",
        "        color:#CD2305;\n",
        "        margin-bottom: 0.5em;\n",
        "        margin-top: 0.5em;\n",
        "        display: block;\n",
        "    }\t\n",
        "    .text_cell_render h5 {\n",
        "        font-weight: 300;\n",
        "        font-size: 16pt;\n",
        "        color: #CD2305;\n",
        "        font-style: italic;\n",
        "        margin-bottom: .5em;\n",
        "        margin-top: 0.5em;\n",
        "        display: block;\n",
        "    }\n",
        "    \n",
        "    .warning{\n",
        "        color: rgb( 240, 20, 20 )\n",
        "        }  \n",
        "</style>\n",
        "<script>\n",
        "    MathJax.Hub.Config({\n",
        "                        TeX: {\n",
        "                           extensions: [\"AMSmath.js\"]\n",
        "                           },\n",
        "                tex2jax: {\n",
        "                    inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
        "                    displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
        "                },\n",
        "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
        "                \"HTML-CSS\": {\n",
        "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
        "                }\n",
        "        });\n",
        "</script>\n"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 236,
       "text": [
        "<IPython.core.display.HTML at 0x114027510>"
       ]
      }
     ]
    }
   ],
   "metadata": {}
  }
 ]
}