lectures/lecture14/lecture-14.ipynb

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    "# Lecture 14: Structured matrices, FFT, convolutions, Toeplitz matrices"
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    "## Previous lecture\n",
    "\n",
    "- Krylov methods: Arnoldi relation, CG, GMRES\n",
    "- Preconditioners \n",
    "    - Jacobi\n",
    "    - Gauss-Seidel\n",
    "    - SSOR\n",
    "    - ILU and its modifications"
   ]
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    "## Other structured matrices\n",
    "\n",
    "- Up to now, we discussed preconditioning only for **sparse matrices**\n",
    "- But iterative methods work well for any matrices that have fast black-box matrix-by-vector product\n",
    "- Important class of such matrices are **Toeplitz matrices** (and **Hankel matrices**) and their **multilevel** variants\n",
    "\n",
    "They are directly connected to the **convolution** operation and **Fast Fourier Transform**."
   ]
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    "## Convolution \n",
    "\n",
    "- One of the key operation in signal processing/machine learning is the **convolution of two functions**.\n",
    "\n",
    "- Let $x(t)$ and $y(t)$ be two given functions. Their convolution is defined as\n",
    "\n",
    "$$(x * y)(t) = \\int_{-\\infty}^{\\infty} x(\\tau) y(t -  \\tau) d \\tau.$$"
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    "## Convolution theorem and Fourier transform\n",
    "\n",
    "A well-known fact: a *convolution* in the **time domain** is a *product* in the **frequency domain**. \n",
    "\n",
    "- Time-frequency transformation is given by the **Fourier transform**:\n",
    "\n",
    "$$\\widehat{x}(w) = (\\mathcal{F}(x))(w) = \\int_{-\\infty}^{\\infty} e^{i w t} x(t) dt.$$\n",
    "\n",
    "- Then, \n",
    "\n",
    "$$\\mathcal{F}(x * y) = \\mathcal{F}(x) \\mathcal{F}(y).$$\n",
    "\n",
    "- Thus, the \"algorithm\" for the computation of the convolution can be:\n",
    "\n",
    "\n",
    "1. Compute Fourier transform of $x(t)$ and $y(t)$. \n",
    "2. Compute their product\n",
    "3. Compute inverse Fourier transform"
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    "## Discrete convolution operation\n",
    "\n",
    "$$(x * y)(t) = \\int_{-\\infty}^{\\infty} x(\\tau) y(t -  \\tau) d \\tau.$$\n",
    "\n",
    "Let us approximate the integral by a quadrature sum on a uniform grid, and store the signal at equidistant points.\n",
    "\n",
    "Then we are left with the summation\n",
    "\n",
    "$$z_i = \\sum_{j=0}^{n-1} x_j y_{i - j},$$\n",
    "\n",
    "which is called **discrete convolution**. This can be thought as an application of a filter with coefficients $x$ to a signal $y$. \n",
    "\n",
    "There are different possible filters for different purposes, but they all utilize the **shift-invariant structure**."
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   "source": [
    "## Discrete convolution and Toeplitz matrices\n",
    "\n",
    "A discrete convolution can be thought as a matrix-by-vector product:\n",
    "\n",
    "$$z_i = \\sum_{j=0}^{n-1} x_j y_{i - j}, \\Leftrightarrow z = Ax$$\n",
    "\n",
    "where the matrix $A$ elements are given as $a_{ij} = y_{i-j}$, i.e., they depend only on the difference between the row index and the column index.\n"
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    "## Toeplitz matrices: definition\n",
    "\n",
    "A matrix is called **Toeplitz** if its elements are defined as\n",
    "\n",
    "$$a_{ij} = t_{i - j}.$$\n",
    "\n",
    "- A Toeplitz matrix is completely defined by its first column and first row (i.e., $2n-1$ parameters).\n",
    "\n",
    "- It is a **dense matrix**, however it is a **structured matrix** (i.e., defined by $\\mathcal{O}(n)$ parameters).\n",
    "\n",
    "- And the main operation in the discrete convolution is the product of Toeplitz matrix by vector.\n",
    "\n",
    "- Can we compute it faster than $\\mathcal{O}(n^2)$?"
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    "## Toeplitz and circulant matrix\n",
    "\n",
    "- For a special class of Toeplitz matrices, named **circulant matrices** the fast matrix-by-vector product can be done.\n",
    "\n",
    "- A matrix $C$ is called **circulant**, if \n",
    "\n",
    "$$C_{ij} = c_{i - j \\mod n},$$\n",
    "\n",
    "i.e. it periodicaly wraps\n",
    "\n",
    "$$C = \\begin{bmatrix} c_0 & c_1 & c_2 & c_3 \\\\\n",
    " c_3 & c_0 & c_1 & c_2 \\\\\n",
    " c_2 & c_3 & c_0 & c_1 \\\\\n",
    " c_1 & c_2 & c_3 & c_0 \\\\\n",
    " \\end{bmatrix}.\n",
    " $$\n",
    " \n",
    "- These matrices have the same **eigenvectors**, given by the Discrete Fourier Transform (DFT)."
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    "## Spectral theorem for circulant matrices\n",
    "\n",
    "**Theorem:**\n",
    "\n",
    "Any circulant matrix can be represented in the form\n",
    "\n",
    "$$C = \\frac{1}{n} F^* \\Lambda F,$$\n",
    "\n",
    "where $F$ is the **Fourier matrix** with the elements\n",
    "\n",
    "$$F_{kl} = w_n^{kl}, \\quad k, l = 0, \\ldots, n-1, \\quad w_n = e^{-\\frac{2 \\pi i}{n}},$$\n",
    "\n",
    "and matrix $\\Lambda = \\text{diag}(\\lambda)$ is the diagonal matrix and\n",
    "\n",
    "$$\\lambda = F c, $$\n",
    "\n",
    "where $c$ is the first column of the circulant matrix $C$.\n",
    "\n",
    "The proof will be later: now we need to study the FFT matrix."
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    "## Fourier matrix\n",
    "\n",
    "The Fourier matrix is defined as:\n",
    "\n",
    "$$\n",
    "F_n =\n",
    "\\begin{pmatrix}\n",
    "1 & 1 & 1 & \\dots & 1 \\\\\n",
    "1 & w^{1\\cdot 1}_n & w^{1\\cdot 2}_n & \\dots & w^{1\\cdot (n-1)}_n\\\\\n",
    "1 & w^{2\\cdot 1}_n & w^{2\\cdot 2}_n & \\dots & w^{2\\cdot (n-1)}_n\\\\\n",
    "\\dots & \\dots & \\dots &\\dots &\\dots \\\\\n",
    "1 & w^{(n-1)\\cdot 1}_n & w^{(n-1)\\cdot 2}_n & \\dots & w^{(n-1)\\cdot (n-1)}_n\\\\\n",
    "\\end{pmatrix},\n",
    "$$\n",
    "\n",
    "or equivalently\n",
    "\n",
    "$$ F_n = \\{ w_n^{kl} \\}_{k,l=0}^{n-1}, $$\n",
    "\n",
    "where \n",
    "\n",
    "$$w_n = e^{-\\frac{2\\pi i}{n}}.$$"
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    "**Properties:**\n",
    "* Symmetric (not Hermitian!)\n",
    "* Unitary up to a scaling factor: $F_n^* F_n = F_n F_n^* = nI$ (check this fact). Therefore $F_n^{-1} = \\frac{1}{n}F^*_n$\n",
    "* Can be multiplied by a vector (called dicrete Fourier transform or DFT) with <font color='red'>$\\mathcal{O}(n \\log n)$</font> complexity (called fast Fourier transform or <font color='red'>FFT</font>)! FFT helps to analyze spectrum of a signal and, as we will see later, helps to do fast mutiplications with certain types of matrices."
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "\n",
    "N = 1000\n",
    "dt = 1.0 / 800.0\n",
    "x = np.linspace(0.0, N*dt, N)\n",
    "y = np.sin(50.0 * 2.0*np.pi*x) + 0.5*np.sin(80.0 * 2.0*np.pi*x) + 0.2*np.sin(300.0 * 2.0*np.pi*x)\n",
    "plt.plot(x, y)\n",
    "plt.xlabel('Time')\n",
    "plt.ylabel('Signal')\n",
    "plt.title('Initial signal')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'Discrete Fourier transform')"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "yf = np.fft.fft(y)\n",
    "xf = np.linspace(0.0, 1.0/(2.0*dt), N//2)\n",
    "plt.plot(xf, 2.0/N * np.abs(yf[0:N//2])) #Note: N/2 to N will give negative frequencies\n",
    "plt.xlabel('Frequency')\n",
    "plt.ylabel('Amplitude')\n",
    "plt.title('Discrete Fourier transform')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Fast Fourier transform (FFT)\n",
    "Here we consider a matrix interpretation of the standard [Cooley-Tukey algorithm (1965)](https://www.ams.org/journals/mcom/1965-19-090/S0025-5718-1965-0178586-1/S0025-5718-1965-0178586-1.pdf), which has underlying **divide and conquer** idea. Note that in packages more advanced versions are used.\n",
    "\n",
    "- Let $n$ be a power of 2. \n",
    "- First of all we <font color='red'> permute the rows </font> of the Fourier matrix such that the first $n/2$ rows of the new matrix had row numbers <font color='red'>$1,3,5,\\dots,n-1$</font> and the last $n/2$ rows had row numbers <font color='red'>$2,4,6\\dots,n$</font>. \n",
    "\n",
    "- This permutation can be expressed in terms of multiplication by permutation matrix $P_n$:\n",
    "\n",
    "$$\n",
    "P_n =\n",
    "\\begin{pmatrix}\n",
    "1 & 0 & 0 & 0 & \\dots & 0 & 0 \\\\\n",
    "0 & 0 & 1 & 0 &\\dots & 0 & 0 \\\\\n",
    "\\vdots & & & & & & \\vdots \\\\\n",
    "0 & 0 & 0 & 0 &\\dots & 1 & 0 \\\\\n",
    "\\hline\n",
    "0 & 1 & 0 & 0 & \\dots & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 1 &\\dots & 0 & 0 \\\\\n",
    "\\vdots & & & & & & \\vdots \\\\\n",
    "0 & 0 & 0 & 0 &\\dots & 0 & 1 \n",
    "\\end{pmatrix},\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Hence,\n",
    "\n",
    "$$\n",
    "P_n F_n =\n",
    "\\begin{pmatrix}\n",
    "1 & 1 & 1 & \\dots & 1 \\\\\n",
    "1 & w^{2\\cdot 1}_n & w^{2\\cdot 2}_n & \\dots & w^{2\\cdot (n-1)}_n\\\\\n",
    "1 & w^{4\\cdot 1}_n & w^{4\\cdot 2}_n & \\dots & w^{4\\cdot (n-1)}_n\\\\\n",
    "\\vdots & & & & \\vdots\\\\\n",
    "1 & w^{(n-2)\\cdot 1}_n & w^{(n-2)\\cdot 2}_n & \\dots & w^{(n-2)\\cdot (n-1)}_n\\\\\n",
    "\\hline\n",
    "1 & w^{1\\cdot 1}_n & w^{1\\cdot 2}_n & \\dots & w^{1\\cdot (n-1)}_n\\\\\n",
    "1 & w^{3\\cdot 1}_n & w^{3\\cdot 2}_n & \\dots & w^{3\\cdot (n-1)}_n\\\\           \n",
    "\\vdots & & & & \\vdots\\\\\n",
    "1 & w^{(n-1)\\cdot 1}_n & w^{(n-1)\\cdot 2}_n & \\dots & w^{(n-1)\\cdot (n-1)}_n\\\\\n",
    "\\end{pmatrix},\n",
    "$$\n",
    "\n",
    "Now let us imagine that we separated its columns and rows by two parts each of size $n/2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "As a result we get <font color='red'>$2\\times 2$ block matrix</font> that has the following form\n",
    "\n",
    "$$\n",
    "P_n F_n =\n",
    "\\begin{pmatrix}\n",
    "\\left\\{w^{2kl}_n\\right\\} & \\left\\{w_n^{2k\\left(\\frac{n}{2} + l\\right)}\\right\\} \\\\\n",
    "\\left\\{w_n^{(2k+1)l}\\right\\} & \\left\\{w_n^{(2k+1)\\left(\\frac{n}{2} + l\\right)}\\right\\}\n",
    "\\end{pmatrix},\n",
    "\\quad k,l = 0,\\dots, \\frac{n}{2}-1.\n",
    "$$\n",
    "\n",
    "So far it does not look like something that works faster :) But we will see that in a minute.\n",
    "Lets have a more precise look at the first block $\\left\\{w^{2kl}_n\\right\\}$:\n",
    "\n",
    "$$\n",
    "w^{2kl}_n = e^{-2kl\\frac{2\\pi i}{n}} = e^{-kl\\frac{2\\pi i}{n/2}} = w^{kl}_{n/2}.\n",
    "$$\n",
    "\n",
    "So this block is exactly twice smaller Fourier matrix $F_{n/2}$!\n",
    "\n",
    "<!---\n",
    "Now we can write\n",
    "\n",
    "$$\n",
    "\\begin{pmatrix}\n",
    "F_{n/2} & \\left\\{w_n^{2k\\left(\\frac{n}{2} + l\\right)}\\right\\} \\\\\n",
    "\\left\\{w_n^{(2k+1)l}\\right\\} & \\left\\{w_n^{(2k+1)\\left(\\frac{n}{2} + l\\right)}\\right\\}\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "-->\n",
    "The block $\\left\\{w_n^{(2k+1)l}\\right\\}$ can be written as\n",
    "\n",
    "$$\n",
    "w_n^{(2k+1)l} = w_n^{2kl + l} = w_n^{l} w_n^{2kl} = w_n^{l} w_{n/2}^{kl},\n",
    "$$\n",
    "\n",
    "which can be written as $W_{n/2}F_{n/2}$, where \n",
    "\n",
    "$$W_{n/2} = \\text{diag}(1,w_n,w_n^2,\\dots,w_n^{n/2-1}).$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Doing the same tricks for the other blocks we will finally get\n",
    "\n",
    "$$\n",
    "P_n F_n =\n",
    "\\begin{pmatrix}\n",
    "F_{n/2} & F_{n/2} \\\\\n",
    "F_{n/2}W_{n/2} & -F_{n/2}W_{n/2}\n",
    "\\end{pmatrix} =\n",
    "\\begin{pmatrix}\n",
    "F_{n/2} & 0 \\\\\n",
    "0 & F_{n/2}\n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix}\n",
    "I_{n/2} & I_{n/2} \\\\\n",
    "W_{n/2} & -W_{n/2}\n",
    "\\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "- Thus, we <font color='red'>reduced multiplication by $F_n$ to 2 multiplications by $F_{n/2}$</font> and cheap multiplications by diagonal matrices. \n",
    "- If we apply the obtained expressions recursively to $F_{n/2}$, we will get <font color='red'>$\\mathcal{O}(n\\log n)$ </font> complexity."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Full matvec time = 0.06917304271482862\n",
      "FFT time = 3.3967071485988395e-05\n",
      "Relative error = 1.550139758681484e-12\n"
     ]
    }
   ],
   "source": [
    "#FFT vs full matvec\n",
    "import time\n",
    "import numpy as np\n",
    "import scipy as sp\n",
    "import scipy.linalg\n",
    "\n",
    "n = 10000\n",
    "F = sp.linalg.dft(n)\n",
    "x = np.random.randn(n)\n",
    "\n",
    "y_full = F.dot(x)\n",
    "\n",
    "full_mv_time = %timeit -q -o F.dot(x)\n",
    "print('Full matvec time =', full_mv_time.average)\n",
    "\n",
    "y_fft = np.fft.fft(x)\n",
    "fft_mv_time = %timeit -q -o np.fft.fft(x)\n",
    "print('FFT time =', fft_mv_time.average)\n",
    "\n",
    "print('Relative error =', (np.linalg.norm(y_full - y_fft)) / np.linalg.norm(y_full))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Circulant matrices\n",
    "\n",
    "FFT helps to multiply fast by certain types of matrices. We start from a circulant matrix:\n",
    "\n",
    "$$\n",
    "C =\n",
    "\\begin{pmatrix} \n",
    "c_0 & c_{n-1} & c_{n-2} & \\dots & c_1 \\\\\n",
    "c_{1} & c_{0} & c_{n-1} & \\dots & c_2 \\\\\n",
    "c_{2} & c_{1} & c_0 & \\dots & c_3 \\\\\n",
    "\\dots & \\dots & \\dots & \\dots & \\dots \\\\\n",
    "c_{n-1} & c_{n-2} & c_{n-3} & \\dots & c_0\n",
    "\\end{pmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Theorem.** \n",
    "Let $C$ be a circulant matrix of size $n\\times n$ and let $c$ be it's first column , then\n",
    "\n",
    "$$\n",
    "C = \\frac{1}{n} F_n^* \\text{diag}(F_n c) F_n \n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "*Proof.* \n",
    "- Consider a number \n",
    "\n",
    "$$\\lambda (\\omega) = c_0 + \\omega c_1 + \\dots + \\omega^{n-1} c_{n-1},$$ \n",
    "\n",
    "where $\\omega$ is any number such that $\\omega^n=1$. \n",
    "- Lets multiply $\\lambda$ by $1,\\omega,\\dots, \\omega^{n-1}$:\n",
    "\n",
    "$$\n",
    "\\begin{split}\n",
    "\\lambda & = c_0 &+& \\omega c_1 &+& \\dots &+& \\omega^{n-1} c_{n-1},\\\\\n",
    "\\lambda\\omega & = c_{n-1} &+& \\omega c_0 &+& \\dots &+& \\omega^{n-1} c_{n-2},\\\\\n",
    "\\lambda\\omega^2 & = c_{n-2} &+& \\omega c_{n-1} &+& \\dots &+& \\omega^{n-1} c_{n-3},\\\\\n",
    "&\\dots\\\\\n",
    "\\lambda\\omega^{n-1} & = c_{1} &+& \\omega c_{2} &+& \\dots &+& \\omega^{n-1} c_{0}.\n",
    "\\end{split}\n",
    "$$\n",
    "\n",
    "- Therefore,\n",
    "\n",
    "$$\n",
    "\\lambda(\\omega) \\cdot \\begin{pmatrix} 1&\\omega & \\dots& \\omega^{n-1} \\end{pmatrix} =\n",
    "\\begin{pmatrix} 1&\\omega&\\dots& \\omega^{n-1} \\end{pmatrix} \\cdot C.\n",
    "$$\n",
    "\n",
    "- Writing this for $\\omega = 1,w_n, \\dots, w_n^{n-1}$ we get\n",
    "\n",
    "$$\n",
    "\\Lambda F_n = F_n C\n",
    "$$\n",
    "\n",
    "and finally\n",
    "\n",
    "$$\n",
    "C = \\frac{1}{n} F^*_n \\Lambda F_n, \\quad \\text{where}\\quad \\Lambda = \\text{diag}(F_nc) \\qquad\\blacksquare\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Fast matvec with circulant matrix\n",
    "\n",
    "- Representation $C = \\frac{1}{n} F^* \\text{diag}(F_n c) F_n $ gives us an explicit way to multiply a vector $x$ by $C$ in $\\mathcal{O}(n\\log n)$ operations. \n",
    "- Indeed,\n",
    "\n",
    "$$\n",
    "Cx = \\frac{1}{n} F_n^* \\text{diag}(F_n c) F_n x = \\text{ifft}\\left( \\text{fft}(c) \\circ \\text{fft}(x)\\right)\n",
    "$$\n",
    "\n",
    "where $\\circ$ denotes elementwise product (Hadamard product) of two vectors (since $\\text{diag}(a)b = a\\circ b$) and ```ifft``` denotes inverse Fourier transform $F^{-1}_n$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Full matvec time = 0.011294817562946783\n",
      "FFT time = 0.00012443668489951442\n",
      "Relative error = 7.183826224607303e-16\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import scipy as sp\n",
    "import scipy.linalg\n",
    "\n",
    "def circulant_matvec(c, x):\n",
    "    return np.fft.ifft(np.fft.fft(c) * np.fft.fft(x))\n",
    "\n",
    "n = 5000\n",
    "c = np.random.random(n)\n",
    "C = sp.linalg.circulant(c)\n",
    "x = np.random.randn(n)\n",
    "\n",
    "\n",
    "y_full = C.dot(x)\n",
    "full_mv_time = %timeit -q -o C.dot(x)\n",
    "print('Full matvec time =', full_mv_time.average)\n",
    "\n",
    "\n",
    "y_fft = circulant_matvec(c, x)\n",
    "fft_mv_time = %timeit -q -o circulant_matvec(c, x)\n",
    "print('FFT time =', fft_mv_time.average)\n",
    "\n",
    "print('Relative error =', (np.linalg.norm(y_full - y_fft)) / np.linalg.norm(y_full))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Toeplitz matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Now we get back to Toeplitz matrices!\n",
    "\n",
    "$$\n",
    "T =\n",
    "\\begin{pmatrix} \n",
    "t_0 & t_{-1} & t_{-2} & t_{-3}& \\dots & t_{1-n} \\\\\n",
    "t_{1} & t_{0} & t_{-1} & t_{-2}& \\dots & t_{2-n} \\\\\n",
    "t_{2} & t_{1} & t_0 & t_{-1} &\\dots & t_{3-n} \\\\\n",
    "t_{3} & t_{2} & t_1 & t_0 & \\dots & t_{4-n} \\\\\n",
    "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots\\\\\n",
    "t_{n-1} & t_{n-2} & t_{n-3} & t_{n-4} &\\dots &t_0\n",
    "\\end{pmatrix},\n",
    "$$\n",
    "\n",
    "or equivalently $T_{ij} = t_{i-j}$.\n",
    "\n",
    "Matvec operation can be written as\n",
    "\n",
    "$$\n",
    "y_i = \\sum_{j=1}^n t_{i-j} x_j,\n",
    "$$\n",
    "\n",
    "which can be interpreted as a discrete <font color='red'>convolution</font> of filter $t_i$ and signal $x_i$. \n",
    "For simplicity the size of the filter $t$ is such that the sizes of the input and output signals are the same. Generally, filter size can be arbitrary.\n",
    "\n",
    "Fast convolution computation has a variety of applications, for instance, in signal processing or partial differential and integral equations. For instance, here is the smoothing of a signal:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy import signal\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "alpha = 0.01\n",
    "sig = np.repeat([0., 1., 0.], 100)\n",
    "filt = np.exp(-alpha * (np.arange(100)-50)**2)\n",
    "filtered = signal.convolve(sig, filt, mode='same') / sum(filt)\n",
    "\n",
    "fig, (ax_orig, ax_filt, ax_filtered) = plt.subplots(3, 1, sharex=True)\n",
    "ax_orig.plot(sig)\n",
    "ax_orig.margins(0, 0.1)\n",
    "ax_filt.plot(filt)\n",
    "ax_filt.margins(0, 0.1)\n",
    "ax_filtered.plot(filtered)\n",
    "ax_filtered.margins(0, 0.1)\n",
    "\n",
    "ax_orig.set_title('Original signal')\n",
    "ax_filt.set_title('Filter')\n",
    "ax_filtered.set_title('Convolution')\n",
    "\n",
    "fig.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Fast matvec with Toeplitz matrix\n",
    "\n",
    "**Key point**: the multiplication by a Toeplitz matrix can be reduced to the multiplication by a circulant.\n",
    "\n",
    "- Indeed, every Toeplitz matrix of size $n\\times n$ can be embedded into a Circulant matrix $C$ of size $2n \\times 2n$:\n",
    "\n",
    "$$\n",
    "C = \n",
    "\\begin{pmatrix}\n",
    "T & \\dots \\\\\n",
    "\\dots & \\dots\n",
    "\\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "- The $3\\times 3$ matrix $T = \\begin{pmatrix} \n",
    "t_0 & t_{-1} & t_{-2} \\\\\n",
    "t_{1} & t_{0} & t_{-1} \\\\\n",
    "t_{2} & t_{1} & t_0 \\\\\n",
    "\\end{pmatrix}$ \n",
    "can be embedded as follows\n",
    "\n",
    "$$\n",
    "C = \n",
    "\\begin{pmatrix} \n",
    "t_0 & t_{-1} & t_{-2} & t_{2} & t_{1}\\\\\n",
    "t_{1} & t_{0} & t_{-1} & t_{-2} & t_{2} \\\\\n",
    "t_{2} & t_{1} & t_0 & t_{-1} & t_{-2} \\\\\n",
    "t_{-2}& t_{2} & t_{1} & t_0 & t_{-1}  \\\\\n",
    "t_{-1} & t_{-2} & t_{2} & t_{1} & t_0  \n",
    "\\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "- For matvec \n",
    "$ \\begin{pmatrix} y_1 \\\\ y_2 \\\\ y_3 \\end{pmatrix}=\n",
    "\\begin{pmatrix} \n",
    "t_0 & t_{-1} & t_{-2} \\\\\n",
    "t_{1} & t_{0} & t_{-1} \\\\\n",
    "t_{2} & t_{1} & t_0 \\\\\n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix} x_1 \\\\ x_2 \\\\ x_3 \\end{pmatrix}$\n",
    "we pad vector $x$ with zeros:\n",
    "\n",
    "$$\n",
    "\\begin{pmatrix} y_1 \\\\ y_2 \\\\ y_3 \\\\ \\star \\\\ \\star \\end{pmatrix} =\n",
    "\\begin{pmatrix} \n",
    "t_0 & t_{-1} & t_{-2} & t_{2} & t_{1}\\\\\n",
    "t_{1} & t_{0} & t_{-1} & t_{-2} & t_{2} \\\\\n",
    "t_{2} & t_{1} & t_0 & t_{-1} & t_{-2} \\\\\n",
    "t_{-2}& t_{2} & t_{1} & t_0 & t_{-1}  \\\\\n",
    "t_{-1} & t_{-2} & t_{2} & t_{1} & t_0  \n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix} x_1 \\\\ x_2 \\\\ x_3 \\\\ 0 \\\\ 0 \\end{pmatrix}=\n",
    "\\text{ifft}(\\text{fft}(\\begin{pmatrix} t_0 \\\\ t_{1} \\\\ t_{2} \\\\ t_{-2} \\\\ t_{-1} \\end{pmatrix})\\circ \\text{fft}(\\begin{pmatrix} x_1 \\\\ x_2 \\\\ x_3 \\\\ 0 \\\\ 0 \\end{pmatrix})).\n",
    "$$\n",
    "\n",
    "- Note that you **do not need to form and store** the whole matrix $T$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Multilevel Toeplitz matrices\n",
    "\n",
    "The 2-dimensional convolution is defined as \n",
    "\n",
    "$$\n",
    "y_{i_1i_2} = \\sum_{j_1,j_2=1}^n t_{i_1-j_1, i_2-j_2} x_{j_1 j_2}.\n",
    "$$\n",
    "\n",
    "Note that $x$ and $y$ are 2-dimensional arrays and $T$ is 4-dimensional. \n",
    "To reduce this expression to matrix-by-vector product we have to reshape $x$ and $y$ into long vectors:\n",
    "\n",
    "$$\n",
    "\\text{vec}(x) = \n",
    "\\begin{pmatrix}\n",
    "x_{11} \\\\ \\vdots \\\\ x_{1n} \\\\ \\hline \\\\ \\vdots \\\\ \\hline \\\\ x_{n1} \\\\ \\vdots \\\\ x_{nn}\n",
    "\\end{pmatrix},\n",
    "\\quad\n",
    "\\text{vec}(y) = \n",
    "\\begin{pmatrix}\n",
    "y_{11} \\\\ \\vdots \\\\ y_{1n} \\\\ \\hline \\\\ \\vdots \\\\ \\hline \\\\ y_{n1} \\\\ \\vdots \\\\ y_{nn}\n",
    "\\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "In this case matrix $T$ is **block Toeplitz with Toeplitz blocks:** (BTTB)\n",
    "\n",
    "$$\n",
    "T = \n",
    "\\begin{pmatrix} \n",
    "T_0 & T_{-1} & T_{-2} &  \\dots & T_{1-n} \\\\\n",
    "T_{1} & T_{0} & T_{-1} & \\dots & T_{2-n} \\\\\n",
    "T_{2} & T_{1} & T_0 & \\dots & T_{3-n} \\\\\n",
    "\\dots & \\dots & \\dots &  \\dots & \\dots\\\\\n",
    "T_{n-1} & T_{n-2} & T_{n-3}  &\\dots &T_0\n",
    "\\end{pmatrix},\n",
    "\\quad \\text{where} \\quad\n",
    "T_k = t_{k, i_2 - j_2}\\quad  \\text{are Toeplitz matrices}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Fast matvec with multilevel Toeplitz matrix\n",
    "\n",
    "To get fast matvec we need to embed block Toeplitz matrix with Toeplitz blocks into the block circulant matrix with circulant blocks. The analog of\n",
    "$$\\begin{pmatrix} y_1 \\\\ y_2 \\\\ y_3 \\\\ \\star \\\\ \\star \\end{pmatrix} =\n",
    "\\text{ifft}(\\text{fft}(\\begin{pmatrix} t_0 \\\\ t_{1} \\\\ t_{2} \\\\ t_{-2} \\\\ t_{-1} \\end{pmatrix})\\circ\\text{fft}(\\begin{pmatrix} x_1 \\\\ x_2 \\\\ x_3 \\\\ 0 \\\\ 0 \\end{pmatrix})).$$\n",
    "will look like\n",
    "$$ \\begin{pmatrix} y_{11} & y_{12} & y_{13} & \\star & \\star \\\\ \n",
    "y_{21} & y_{22} & y_{23} & \\star & \\star \\\\  \n",
    "y_{31} & y_{32} & y_{33} & \\star & \\star \\\\ \n",
    "\\star & \\star & \\star & \\star & \\star \\\\  \n",
    "\\star & \\star & \\star & \\star & \\star \\\\  \n",
    "\\end{pmatrix} = \\text{ifft2d}(\\text{fft2d}(\\begin{pmatrix} t_{0,0} & t_{1,0} & t_{2,0} & t_{-2,0} & t_{-1,0} \\\\ \n",
    "t_{0,1} & t_{1,1} & t_{2,1} & t_{-2,1} & t_{-1,1} \\\\  \n",
    "t_{0,2} & t_{1,2} & t_{2,2} & t_{-2,2} & t_{-1,2} \\\\ \n",
    "t_{0,-2} & t_{1,-2} & t_{2,-2} & t_{-2,-2} & t_{-1,-2} \\\\ \n",
    "t_{0,-1} & t_{1,-1} & t_{2,-1} & t_{-2,-1} & t_{-1,-1}\n",
    "\\end{pmatrix}) \\circ \\text{fft2d}(\\begin{pmatrix}x_{11} & x_{12} & x_{13} & 0 & 0 \\\\ \n",
    "x_{21} & x_{22} & x_{23} & 0 & 0 \\\\  \n",
    "x_{31} & x_{32} & x_{33} & 0 & 0 \\\\ \n",
    "0 & 0 & 0 & 0 & 0 \\\\  \n",
    "0 & 0 & 0 & 0 & 0 \\\\  \n",
    "\\end{pmatrix})),$$\n",
    "where fft2d is 2-dimensional fft that consists of one-dimensional transforms, applied first to rows and and then to columns (or vice versa)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 576x576 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Blurring and Sharpening Lena by convolution\n",
    "\n",
    "from scipy import signal\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib  inline\n",
    "from scipy import misc\n",
    "import imageio\n",
    "\n",
    "filter_size = 3\n",
    "filter_blur = np.ones((filter_size, filter_size)) / filter_size**2\n",
    "lena = imageio.imread('./lena512.jpg')\n",
    "#lena = misc.face()\n",
    "#lena = lena[:, :, 0]\n",
    "blurred = signal.convolve2d(lena, filter_blur, boundary='symm', mode='same')\n",
    "\n",
    "fig, ax = plt.subplots(2, 2, figsize=(8, 8))\n",
    "ax[0, 0].imshow(lena[200:300, 200:300], cmap='gray')\n",
    "ax[0, 0].set_title('Original Lena')\n",
    "ax[0, 1].imshow(blurred[200:300, 200:300], cmap='gray')\n",
    "ax[0, 1].set_title('Blurred Lena')\n",
    "ax[1, 0].imshow((lena - blurred)[200:300, 200:300], cmap='gray')\n",
    "ax[1, 0].set_title('Lena $-$ Blurred Lena')\n",
    "ax[1, 1].imshow(((lena - blurred)*3 + blurred)[200:300, 200:300], cmap='gray')\n",
    "ax[1, 1].set_title('$3\\cdot$(Lena $-$ Blurred Lena) + Blurred Lena')\n",
    "fig.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Solving linear systems with Toeplitz matrix\n",
    "\n",
    "- Convolution is ok; but what about **deconvolution**, or solving linear systems with Toeplitz matrices?\n",
    "\n",
    "$$T x = f.$$\n",
    "\n",
    "- For the periodic case, where $T = C$ is circulant, \n",
    "\n",
    "we have the spectral theorem\n",
    "\n",
    "$$C = \\frac{1}{n}F^* \\Lambda F, \\quad C^{-1} = \\frac{1}{n}F^* \\Lambda^{-1} F,$$\n",
    "\n",
    "but for a general Toeplitz matrices, it is not a trivial question."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Iterative methods\n",
    "\n",
    "- Not-a-bad recipe for Toeplitz linear system is to use **iterative method** (fast matvec is available).\n",
    "\n",
    "- A good choice for a preconditioner is a **circulant matrix**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Circulant preconditioner\n",
    "\n",
    "- A natural idea is to use **circulants** as preconditioners, since they are easy to invert.\n",
    "\n",
    "- The first preconditioner was the preconditioner by **Raymond Chan and Gilbert Strang**, who proposed to take the first column of the matrix and use it to generate the circulant.\n",
    "\n",
    "- The second preconditioner is the **Tony Chan** preconditioner, which is also very natural: \n",
    "\n",
    "$$C = \\arg \\min_P \\Vert P - T \\Vert_F.$$\n",
    "\n",
    "- A simple formula for the entries of $C$ can be derived."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import scipy.linalg\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "n = 100\n",
    "c = np.zeros(n)\n",
    "c[0] = -2\n",
    "c[1] = 1\n",
    "Tm = sp.linalg.toeplitz(c, c)\n",
    "\n",
    "\n",
    "c1 = sp.linalg.circulant(c) #Strang preconditioner\n",
    "Fmat = 1.0/np.sqrt(n) * np.fft.fft(np.eye(n)) #Poor man's Fourier matrix\n",
    "\n",
    "d2 = np.diag(Fmat.conj().dot(Tm).dot(Fmat))\n",
    "c2 = Fmat.dot(np.diag(d2)).dot(Fmat.conj().T)\n",
    "\n",
    "\n",
    "mat = np.linalg.inv(c1).dot(Tm)\n",
    "ev = np.linalg.eigvals(mat).real\n",
    "plt.plot(np.sort(ev), np.ones(n), 'o')\n",
    "plt.xlabel('Eigenvalues for Strang preconditioner')\n",
    "plt.gca().get_yaxis().set_visible(False)\n",
    "\n",
    "mat = np.linalg.inv(c2).dot(Tm)\n",
    "ev = np.linalg.eigvals(mat).real\n",
    "plt.figure()\n",
    "plt.plot(np.sort(ev), np.ones(n), 'o')\n",
    "plt.xlabel('Eigenvalues for T. Chan Preconditioner')\n",
    "plt.gca().get_yaxis().set_visible(False)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Direct methods for Toeplitz matrices\n",
    "\n",
    "- The idea with preconditioners works for 2D/3D convolutions, but much worse.\n",
    "\n",
    "- In 1D case it is also possible to find direct solvers for the Toeplitz matrix, based on the structure of the inverse.\n",
    "\n",
    "- But the inverse of a Toeplitz matrix is not Toeplitz!\n",
    "\n",
    "- What to do?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Low displacement rank structure\n",
    "\n",
    "- The Toeplitz matrices belong to a more general class of **low displacement rank** matrices.\n",
    "\n",
    "- Define the **scaled periodic shift matrix** $Z_e$ that takes the vector $x$ and transforms it to a vector\n",
    "\n",
    "$$Z_e x = \\begin{bmatrix}\n",
    "e x_{n-1} \\\\\n",
    "x_1 \\\\\n",
    "x_2 \\\\\n",
    "\\vdots \\\\\n",
    "x_{n-2}\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "- What is the matrix form of this linear operator?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Shift matrices, displacement matrices and Toeplitz matrices\n",
    "\n",
    "Given a Toeplitz matrix $T$, we select any $e, f$ such that $ef \\ne 1$ and define the displacement operator as\n",
    "\n",
    "$$L(T) = Z_e T - T Z_f.$$\n",
    "\n",
    "For a Toeplitz matrix, $L(T)$ has **rank 2** (it has only first row and last column non-zero)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-1.   0.   0.   0.5  0. ]\n",
      " [ 0.   0.   0.   0.  -0.5]\n",
      " [ 0.   0.   0.   0.   0. ]\n",
      " [ 0.   0.   0.   0.   0. ]\n",
      " [ 0.   0.   0.   0.   1. ]]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import scipy.linalg\n",
    "n = 5 \n",
    "c = np.zeros(n)\n",
    "c[0] = -2\n",
    "c[1] = 1\n",
    "T = sp.linalg.toeplitz(c, c)\n",
    "e = 0.5\n",
    "f = 0.5\n",
    "def Z_shift(e):\n",
    "    return  np.diag(np.ones(n-1), -1)  + e * np.diag(np.ones(1), n-1)\n",
    "\n",
    "Z1 = Z_shift(e)\n",
    "Z2 = Z_shift(f)\n",
    "\n",
    "print(Z1.dot(T) - T.dot(Z2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "What about the inverse?\n",
    "\n",
    "It is also of rank $2$ !"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.02062073e+00, 1.02062073e+00, 2.60439840e-16, 6.95949729e-17,\n",
       "       2.21601201e-17])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import scipy.linalg\n",
    "n = 5 \n",
    "c = np.zeros(n)\n",
    "c[0] = -2\n",
    "c[1] = 1\n",
    "T = sp.linalg.toeplitz(c, c)\n",
    "e = 0.5\n",
    "f = 0.5\n",
    "def Z_shift(e):\n",
    "    return  np.diag(np.ones(n-1), -1)  + e * np.diag(np.ones(1), n-1)\n",
    "\n",
    "Z1 = Z_shift(e)\n",
    "Z2 = Z_shift(f)\n",
    "\n",
    "Tinv = np.linalg.inv(T)\n",
    "\n",
    "p1 = Z1.dot(Tinv) - Tinv.dot(Z2)\n",
    "np.linalg.svd(p1)[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Small displacement rank: definition\n",
    "\n",
    "- The matrix is said to be of **displacement rank $r$** with respect to the pair of generators $Z_e, Z_f$, if\n",
    "\n",
    "$$L(T) = Z_e T - T Z_f = GH^{\\top},$$\n",
    "\n",
    "where $G$ is $n \\times r$ and $H$ is $n \\times r$.\n",
    "\n",
    "- It is similar to \"discrete derivative\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Theorem on the inverse structure\n",
    "\n",
    "Let $T$ satisfy\n",
    "\n",
    "$$Z_e T - T Z_f = GH ^{\\top},$$\n",
    "\n",
    "and let it be invertible.\n",
    "\n",
    "Then we have\n",
    "\n",
    "$$T^{-1} (Z_e T - T Z_f) T^{-1} = T^{-1} Z_e - Z_f T^{-1} = T^{-1} G H^{\\top} T^{-1},$$\n",
    "\n",
    "i.e. the inverse has **small displacement rank** with the reversed pair of generators $Z_e, Z_f$ (why?)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Recovering a matrix from the displacement representation\n",
    "\n",
    "- Does the low-rank representation after the displacement operator describe the structure?\n",
    "\n",
    "- I.e. we need to solve the equation of the form\n",
    "\n",
    "$$Z_e T - T Z_f = GH^{\\top} = B$$\n",
    "\n",
    "for a given right-hand side.\n",
    "\n",
    "- This is a linear system of equations in disguise! (Do you see that this is a linear system? What is the size of this linear system?)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Sylvester equation\n",
    "\n",
    "- The equation above is a special case of the **Sylvester matrix equation** which has the general form\n",
    "\n",
    "$$AX - X B = C,$$\n",
    "\n",
    "with $A$, $B$ and $C$ given.\n",
    "\n",
    "- In general, this is a linear system with $\\mathcal{O}(n^2)$ unknowns, and the computational cost is expected to be $n^6$.\n",
    "\n",
    "- For Sylvester equation we can solve it in $\\mathcal{O}(n^3)$ (will see in the next part on matrix equations)\n",
    "\n",
    "- But we can do better for specific $A, B$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Back to the particular case\n",
    "\n",
    "For the particular case, we have\n",
    "\n",
    "$$Z_e T - T Z_f = GH^{\\top} = B,$$\n",
    "\n",
    "and the solution is given by\n",
    "\n",
    "$$ (e - f) T = \\sum_{j = 1}^r Z_e(g_j) Z_f( J h_j), $$\n",
    "\n",
    "where $Z_e$ is the **e-scaled circulant** generated by the vector, and $g_j$, and $h_j$ are the columns of the matrices $G$ and $H$, and $J$ is the **per-identity matrix** (which has ones on the anti-diagonal)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Gohberg-Semencul formula\n",
    "\n",
    "Based on this idea and for the special case when $e = 0, f \\rightarrow \\inf$, we get the following famous formula for the inverse of the Toeplitz matrices as a sum \n",
    "of two products of **triangular** Toeplitz matrices:\n",
    "\n",
    "Let $A$ be a Toeplitz matrix, and\n",
    "\n",
    "$$A \\begin{bmatrix} x_0 \\\\ x_1 \\\\ \\vdots \\\\ x_n \\end{bmatrix}=\\begin{bmatrix} 1 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{bmatrix},\n",
    "\\quad \n",
    "A \\begin{bmatrix} y_0 \\\\ y_1 \\\\ \\vdots \\\\ y_n \\end{bmatrix}=\\begin{bmatrix} 0 \\\\ 0 \\\\ \\vdots \\\\ 1 \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "then \n",
    "\n",
    "$$A^{-1} = \\frac{1}{x_0} \\begin{bmatrix} x_0 & 0 & \\ldots & 0 \\\\ x_1 & x_0 & 0 & \\ldots \\\\ \\ldots & \\ldots \\\\ x_n & \\ldots & \\ldots & x_0 \\end{bmatrix}\\begin{bmatrix} u_0 & u_1 & \\ldots & 0 \\\\ 0 & u_0 & u_1 & \\ldots \\\\ \\ldots & \\ldots \\\\ 0 & \\ldots & \\ldots & u_0 \\end{bmatrix}-\\frac{1}{x_0} \\begin{bmatrix} 0 & 0 & \\ldots & 0 \\\\ y_0 & 0 & 0 & \\ldots \\\\ y_1 & y_0 & \\ldots  \\\\ \\ldots & \\ldots & \\\\ y_{n-1} & \\ldots & y_0 & 0 \\end{bmatrix}\\begin{bmatrix} 0 & v_0 & \\ldots & 0 \\\\ 0 & 0 & v_0 & v_1 \\\\ \\ldots & \\ldots   \\\\ \\ldots & \\ldots & \\ldots & v_0 \\\\ 0 & \\ldots & \\ldots & 0\\end{bmatrix},$$\n",
    "\n",
    "where $u_y = y_{n-i}, \\quad v_i = x_{n-i}$.\n",
    "\n",
    "The main meaning: the inverse can be recovered from **the first column** and the **last column** of the matrix.\n",
    "\n",
    "For details, I refer to the book by Tyrtyshnikov \"Brief introduction to numerical analysis\". "
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    "## Fast and superfast direct methods\n",
    "\n",
    "- For many years, these formulas gave rise to the fast $\\mathcal{O}(n^2)$ and superfast $\\mathcal{O}(n \\log n)$ methods for Toeplitz matrices.\n",
    "\n",
    "- The basic idea is that you use **augmentation method**. \n",
    "\n",
    "- Consider that you have computed the **inverse** of a $(n-1) \\times (n-1)$ block of the Toeplitz matrix.\n",
    "\n",
    "- You only need to store two vectors to represent the inverse.\n",
    "\n",
    "- Then, the bigger matrix can be written in the block form.\n",
    "\n",
    "$$T_n = \\begin{bmatrix} T_{n-1} & a \\\\ b^{\\top} & c \\end{bmatrix}.$$\n",
    "\n",
    "- We only need to recompute the last and the first column!\n"
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    "## Recomputing the last and the first columns\n",
    "\n",
    "- Let us split the vector as \n",
    "\n",
    "$$x = \\begin{bmatrix} x_1 & x_2 \\end{bmatrix}.$$\n",
    "\n",
    "- Then,\n",
    "\n",
    "$$T_{n-1} x_1 + a x_2 = e_1, \\quad b^{\\top} x_1 + c x_2 = 0.$$\n",
    "\n",
    "- Or, \n",
    "\n",
    "$$  x_1 = T^{-1}_{n-1} e_1 - T^{-1}_n a x_2.$$\n",
    "\n",
    "- Application of $T^{-1}_{n-1}$ costs $\\mathcal{O}(n \\log n)$ operations, thus $x_2$ will be recovered in the same number of operations as well. The total complexity is then $\\mathcal{O}(n^2 \\log n)$ operations.\n",
    "\n",
    "- **Superfast algorithms** are obtained in terms of reducing the problem to the computations with polynomials and using **block** elimination (in the Fourier style)."
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    "## Other types of low-displacement rank matrices\n",
    "\n",
    "- Hankel matrices\n",
    "- Cauchy matrices\n",
    "- Vandermonde matrices"
   ]
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   "source": [
    "## Take home message\n",
    "\n",
    "- Toeplitz matrices, Circulant matrices, Spectral theorem, FFT\n",
    "- Multilevel Toeplitz matrices"
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