FixedPointIterationsMethods.ipynb

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   "source": [
    "#math and linear algebra stuff\n",
    "import numpy as np\n",
    "import numpy.linalg as la\n",
    "\n",
    "#plots\n",
    "import matplotlib as mpl\n",
    "mpl.rcParams['figure.figsize'] = (15.0, 15.0)\n",
    "#mpl.rc('text', usetex = True)\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Fixed Point iteration methods\n",
    "\n",
    "Fixed point iteration method stand for a very large class of algorithm that aims at iteratively solving a fixed point problem of the form $AX=B$ with $A$ an operator, and $B$ a given set of data.\n",
    "\n",
    "In this short report, we will only be studying the case where $A$ is a linear operator represented by a square matrix.\n",
    "\n",
    "## Jacobi and Gauss-Seidel methods\n",
    "\n",
    "### Jacobi Method\n",
    "\t \n",
    "Jacobi and Gauss-Seidel method assume that one can decompose the $n \\times n$ matrix $M$ into a sum of matrices $M = G + H$ where $G$ is full rank, and easily invertible, for instance it is set to a diagonal matrix in the Jacobi method.\n",
    "\n",
    "Anyway, this lead us to a fixed point search in the high dimensional space $\\mathbb{R}^n$:\n",
    "\n",
    "\\begin{align*}\n",
    "    M \\vec{x} - \\vec{y} &= 0\\\\\n",
    "    G \\vec{x} + H \\vec{x} &= \\vec{y} \\\\\n",
    "    G \\vec{x} &= -H \\vec{x} + \\vec{y} \\\\\n",
    "    \\vec{x} &= - G^{-1} H \\vec{x} + G^{-1} \\vec{y} \\\\\n",
    "    \\vec{x} &= M' \\vec{x} + \\vec{y'}\n",
    "\\end{align*}\n",
    "\t\n",
    "Where $M' = - G^{-1} H$ and $\\vec{y'} = G^{-1} \\vec{y}$.\n",
    "\n",
    "This fixed point search can be carried out using a simple iterative scheme\n",
    "\t\n",
    "\\begin{equation}\n",
    "  \\vec{x^{k+1}} = M' \\vec{x^k} + \\vec{y'}\n",
    "\\end{equation}\n",
    "\n",
    "or\n",
    "\n",
    "\\begin{equation}\n",
    "  \\vec{x^{k+1}} = G^{-1} \\left( \\vec{y} - H \\vec{x^k} \\right)\n",
    "\\end{equation}\n",
    "\n",
    "According to the Picard fixed point theorem, this class of algorithm converges if the operator $T : \\vec{x} \\rightarrow M' \\vec{x} + \\vec{y'}$ is a strict contraction, i.e if there exists $\\rho \\in [0,1[$ such that\n",
    "\n",
    "\\begin{align*}\n",
    "    \\forall (x,x') \\in \\mathbb{R}^n \\times \\mathbb{R}^n, x \\neq x' :\\quad &\\|Tx - Tx'\\| \\leq \\rho \\|x-x'\\| \\\\\n",
    "    &\\|M'x + \\vec{y'} - M'x' - \\vec{y'}\\| \\leq \\rho \\|x-x'\\| \\\\\n",
    "    &\\|M'x - M'x'\\| \\leq \\rho \\|x-x'\\| \\\\\n",
    "    &\\|M' (x - x')\\| \\leq \\rho \\|x-x'\\| \\\\\n",
    "    &\\|M'\\| \\|(x - x')\\| \\leq \\rho \\|x-x'\\| \\\\\n",
    "    &\\|M'\\| \\leq \\rho\n",
    "\\end{align*}\n",
    "\n",
    "This condition can be interpreted as a $[0,1[$-Lipshitz continuity property of the linear application $M'$, which is equivalent to a constraint over the operator norm of $M'$, that reduces to a constraint over its largest singular value: $0 \\leq \\sigma(M')_{max} < 1 $."
   ]
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   "source": [
    "### Preconditionning interpretation\n",
    "\n",
    "The generic matrix decomposition scheme $M=G+H$ presented in the previous section can also be viewed as a preconditionning problem:\n",
    "\n",
    "\\begin{align*}\n",
    "    M \\vec{x} - \\vec{y} &= 0\\\\\n",
    "    G \\vec{x} + H \\vec{x} &= \\vec{y} \\\\\n",
    "    G \\vec{x} &= -H \\vec{x} + \\vec{y} \\\\\n",
    "    \\vec{x} &= - G^{-1} H \\vec{x} + G^{-1} \\vec{y} \\\\\n",
    "    (G^{-1} H + Id) \\vec{x} &= G^{-1} \\vec{y} \\\\\n",
    "    G^{-1} (H + G) \\vec{x} &= G^{-1} \\vec{y} \\\\\n",
    "    G^{-1} M \\vec{x} &= G^{-1} \\vec{y} \\\\\n",
    "\\end{align*}\n",
    "\n",
    "The action of casting the equivalent problem $G^{-1} M \\vec{x} = G^{-1} \\vec{y}$ instead of $M \\vec{x} = \\vec{y}$ is called preconditionning.\n",
    "\n",
    "This method is actually useful when the condition number of $G^{-1} M$ is lower than the condition number of $M$, and of course when $G$ is easily invertible."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "determinant of M is 54.0\n",
      "Condition number of M is 14.5981649059\n",
      "spectral radius of M1 is 8.35889894354\n"
     ]
    }
   ],
   "source": [
    "\"\"\"\n",
    "Let's try to solve a simple linear system using Jacobi iterations\n",
    "\"\"\"\n",
    "\n",
    "M = np.array([[1,1,1,1],\n",
    "            [1,1,4,4],\n",
    "            [1,4,1,4],\n",
    "            [1,4,4,1]])\n",
    "Y = np.ones(M.shape[0])\n",
    "\n",
    "print \"determinant of M is \"+str(np.linalg.det(M))\n",
    "print \"Condition number of M is \"+str(np.linalg.cond(M))\n",
    "\n",
    "#Now perform the decomposition\n",
    "G = np.diag(np.diagonal(M))\n",
    "H = M-G\n",
    "#New variables\n",
    "M1=np.dot(-np.diag(1./np.diagonal(G)),H)\n",
    "Y1=np.dot(np.diag(1./np.diagonal(G)),Y)\n",
    "u,lamb,vh=np.linalg.svd(M1)\n",
    "spectRadM1 = lamb[0]\n",
    "print \"spectral radius of M1 is \"+str(spectRadM1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Weighted Jacobi Method\n",
    "\n",
    "The problem of the Jacobi Method is the strong assumption that $\\rho(M')<1$. Because this assumption is very strong, it rarely holds, and in those case, the method behave very poorly (exponential divergence).\n",
    "\n",
    "One of the evolution that has been proposed in various iterative methods is the over/under relaxation. The idea is simply to consider that, from the point of view of the current solution, the new iterate may be in the right direction toward the solution, but it goes too far. The simple solution is then, to pick-up the next solution between the current one, and the nex Jacobi estimate, using a convex combination of the two.\n",
    "\n",
    "This method can be formalized this way:\n",
    "\n",
    "\\begin{align*}\n",
    "      \\vec{x^{k+1}} &= \\omega( \\underbrace{M' \\vec{x^k} + \\vec{y'}}_{\\text{next Jacobi iterate}} ) + (1-\\omega)\\vec{x^{k}} \\\\\n",
    "        &= \\vec{x^{k}} + \\omega \\left( \\underbrace{M' \\vec{x^k} + \\vec{y'}}_{\\text{next Jacobi iterate}} - \\vec{x^{k}} \\right)\n",
    "\\end{align*}\n",
    "\n",
    "or equivalently\n",
    "\n",
    "\\begin{align*}\n",
    "      \\vec{x^{k+1}} &= \\omega( M' \\vec{x^k} + \\vec{y'} ) + (1-\\omega)\\vec{x^{k}} \\\\\n",
    "        &= \\left( \\omega M' + (1-\\omega) Id \\right) \\vec{x^{k}} - \\omega \\vec{y'} \\\\\n",
    "        &= M'' \\vec{x^k} + \\vec{y''}\n",
    "\\end{align*}\n",
    "\n",
    "Where we can now choose $\\omega$ so that the spectral radius condition for $M''$ is less violated. For instance $\\omega = \\frac{1}{\\sigma(M')}$ get us closer to this condition, for large $\\sigma(M'')$ but the condition won't be respected anyway."
   ]
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  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Found a solution in 119/2000 iteration within absolute precision 0.001\n"
     ]
    },
    {
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     "execution_count": 20,
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x4b760d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\"\"\"\n",
    "    Implementing the Jacobi and weighted Jacobi Method\n",
    "\"\"\"\n",
    "Xk=np.zeros(M.shape[1])\n",
    "omega = 1./spectRadM1\n",
    "nbIter = 2000\n",
    "i = 0\n",
    "eps=1e-3\n",
    "residual = np.ones_like(Y)*(1+eps)\n",
    "err=[]\n",
    "\n",
    "while (i<nbIter) and not np.allclose(residual,0,atol=eps):\n",
    "    nextIterate = np.dot(M1,Xk)+Y1\n",
    "    #Classic jacobi method\n",
    "    if( spectRadM1 < 1 ):\n",
    "        Xk = nextIterate\n",
    "    #Weighted jacobi method\n",
    "    else:\n",
    "        Xk += omega*(nextIterate - Xk)\n",
    "    residual = np.dot(M,Xk)-Y\n",
    "    err.append(np.linalg.norm(residual))\n",
    "    i += 1\n",
    "\n",
    "if i<nbIter:\n",
    "    print \"Found a solution in \"+str(i)+\"/\"+str(nbIter)+\" iteration within absolute precision \"+str(eps)\n",
    "else:\n",
    "    print \"Jacobi iteration failed, residual norm is \"+str(np.linalg.norm(residual))\n",
    "    \n",
    "plt.figure()\n",
    "plt.suptitle(\"Jacobi method convergence\")\n",
    "plt.plot(np.arange(i),err)\n",
    "plt.xlabel(\"Nb of iterations\")\n",
    "plt.ylabel(\"Residual norm\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###  Successive Over-Relaxation Method (SOR)\n",
    "\n",
    "In practice, the weighted Jacobi method is not really efficient either, Young and Frankel, derived an interesting algorithm, called Successive Over-Relaxation, or SOR, based on a simple trick we will explain later:\n",
    "\n",
    "Let's get back to our first matrix decomposition, and slightly modify it:\n",
    "\n",
    "\\begin{align*}\n",
    "    (G + Id + H) \\vec{x} &= \\vec{y} \\\\\n",
    "    \\omega (G + Id + H) \\vec{x} &= \\omega \\vec{y} \\\\\n",
    "    (G + (\\omega-1)G + \\omega Id + \\omega H) \\vec{x} &= \\omega \\vec{y} \\\\\n",
    "    (G + \\omega Id ) \\vec{x} &= \\omega \\vec{y} - ((\\omega-1)G + \\omega H)\\vec{x} \\\\\n",
    "\\end{align*}\n",
    "\n",
    "With $\\omega$ a relaxation parameter that can be used for underrelaxation ($\\omega<1$) or overrelaxation ($\\omega>1$).\n",
    "The update equation become:\n",
    "\\begin{align*}\n",
    "    \\vec{x^{k+1}} &= - (G+\\omega Id)^{-1}((\\omega-1)G + \\omega H)\\vec{x^{k}} + \\omega (G+\\omega Id)^{-1}\\vec{y} \\\\\n",
    "    \\vec{x^{k+1}} &= M'\\vec{x} + \\vec{y'}\n",
    "\\end{align*}\n",
    "\n",
    "Where $M' = - (G+\\omega Id)^{-1}((\\omega-1)G + \\omega H)$ and $\\vec{y'} = \\omega (G+\\omega Id)^{-1}\\vec{y}$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Found a solution in 8/2000 iteration within absolute precision 0.001\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x69b1cd0>"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x4d86590>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\"\"\"\n",
    "Implementing the SOR variation of Jacobi method\n",
    "\"\"\"\n",
    "\n",
    "#Now perform the new decomposition\n",
    "G = np.diag(np.diagonal(M))\n",
    "H = M-G-np.identity(H.shape[0])\n",
    "omega = 1.5\n",
    "gwId = np.diag(1./(np.diagonal(G)+omega))\n",
    "M1=-np.dot(gwId,(omega-1.)*G+omega*H)\n",
    "Y1=omega*np.dot(gwId,Y)\n",
    "\n",
    "#Now solve\n",
    "Xk=np.zeros(M.shape[1])\n",
    "nbIter = 2000\n",
    "i = 0\n",
    "eps=1e-3\n",
    "residual = np.ones_like(Y)*(1+eps)\n",
    "err=[]\n",
    "\n",
    "while (i<nbIter) and not np.allclose(residual,0,atol=eps):\n",
    "    #SOR Jacobi method\n",
    "    Xk = np.dot(M1,Xk)+Y1\n",
    "    residual = np.dot(M,Xk)-Y\n",
    "    err.append(np.linalg.norm(residual))\n",
    "    i += 1\n",
    "\n",
    "if i<nbIter:\n",
    "    print \"Found a solution in \"+str(i)+\"/\"+str(nbIter)+\" iteration within absolute precision \"+str(eps)\n",
    "else:\n",
    "    print \"Jacobi iteration failed, residual norm is \"+str(np.linalg.norm(residual))\n",
    "\n",
    "plt.figure()\n",
    "plt.suptitle(\"SOR-Jacobi method convergence\")\n",
    "plt.plot(np.arange(i),err)\n",
    "plt.xlabel(\"Nb of iterations\")\n",
    "plt.ylabel(\"Residual norm\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Analysing the SOR trick\n",
    "\n",
    "Where does this $\\omega$ trick comes from ? Although it resemble the weighted Jacobi method, the SOR flavour is far more satisfactory from the theoretical point of view.\n",
    "\n",
    "Let's get back to our spectral condition:\n",
    "\n",
    "\\begin{equation}\n",
    " \\lambda_{max} \\left( M' \\right) < 1\n",
    "\\end{equation}\n",
    "\n",
    "We would like to find another decomposition that would allow us to generate a $M'$ with a small spectral radius ($<1$).\n",
    "What about $M' = - (G+\\omega Id)^{-1}((\\omega-1)G + \\omega H)$ ?\n",
    "\n",
    "We will decompose this expression as the sum of two matrices : $M_1 = (G+\\omega Id)^{-1}(\\omega-1)G$ and $M_2=(G+\\omega Id)^{-1}\\omega H$\n",
    "\n",
    "As both $G$ and $Id$ are diagonal, we can consider that they are simultaneously diagonalizable in the same eigen space : the one spanned by the identity matrix, so that many expression simplifies when considering their spectral radius (absolute value of maximum eigenvalue):\n",
    "\\begin{align*}\n",
    "    \\sigma(M_1) = & \\sigma((G+\\omega Id)^{-1}(\\omega-1)G)\\\\\n",
    "    = &\\sigma(G+\\omega Id)^{-1} \\sigma((\\omega-1)G)\\\\\n",
    "    = &\\frac{|\\omega-1|\\sigma(G)}{\\sigma(G)+\\omega}\\\\\n",
    "\\end{align*}\n",
    "\n",
    "The spectral radius of this matrix can be easily controled using the value of $\\omega$, see the following numerical experiment:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fca883a7310>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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kE0YzjJtplvFIVoNQXuUherGMpDRDoyHeML8iDDsFf5gDMl9fmDUBdu6B90dC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SRtCYsSJg+izX84qleOLpOmiLXcf23wqUoeUdrVYrL4Q2OmyXq7WhnlPWbQAfm2JUnVxy8+M9d7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\nYBNCCCGECHP/D4JNf2xHip0kAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fca8af3b650>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "w = np.linspace(0,2,100)\n",
    "\n",
    "NB_SAMPLE = 8\n",
    "sigGl = np.logspace(-1,1,NB_SAMPLE)\n",
    "\n",
    "cm = plt.get_cmap('gist_rainbow')\n",
    "colors = [cm(1.*i/NB_SAMPLE) for i in range(NB_SAMPLE)]\n",
    "\n",
    "plt.figure(figsize=(10,5))\n",
    "plt.suptitle(\"Spectral radius of the matrix $M_1$ for various values of $\\sigma(G)$\")\n",
    "plt.ylabel(\"$\\sigma(G_1)$\")\n",
    "plt.xlabel(\"$\\omega$\")\n",
    "\n",
    "for sigG,color in zip(sigGl,colors):\n",
    "    s = np.abs(((w-1)*sigG)/(sigG+w))\n",
    "    plt.plot(w,s,color=color,label=\"{0:.1f}\".format(sigG))#r'\\frac{\\sigma(H)}{\\sigma(G)} = '+\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And now, what about $M_2$ :\n",
    "\n",
    "\\begin{equation}\n",
    "    M_2 = (G+\\omega Id)^{-1}\\omega H\n",
    "\\end{equation}\n",
    "\n",
    "Unfortunately, the spectral radius of $M_2$ cannot be computed easily, because there is no proof that $H$ and $G$ can be simultaneously diagonalizable, so that the equality $\\sigma(G)\\sigma(H)=\\sigma(GH)$ does not necessarily hold. \n",
    "\n",
    "We should however be able to prove that $\\sigma(M') < 1  \\forall (\\omega,M) \\in [0,2[\\times S_{++}^n$ but we don't know how.\n",
    "This assertion has been proved in the more general framework of the LU based decomposition used in Gauss-Seidel SOR in the following work: \"W. Kahan, Gauss–Seidel methods of solving large systems of linear equations\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Notes\n",
    "\n",
    "In 2015, a nice refinement to the basic Jacobi method has been derived, see [https://arxiv.org/abs/1511.04292](Scheduled Relaxation Jacobi method: improvements and applications).\n",
    "\n",
    "We can notice that the author exploited the famous Chebyshev polynomial acceleration scheme we present in the notebook called ChebyshevPolynomialAcceleration."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gauss Seidel method\n",
    "\n",
    "In the gauss seidel method, it is assumed that the matrix $M$ can be decomposed as follows:\n",
    "\n",
    "\\begin{equation}\n",
    "  M = D + (L + U)\n",
    "\\end{equation}\n",
    "\n",
    "Where, $L$ is a strictly lower triangular matrix, $D$ is a non singular diagonal matrix, and $U$ is a stricly upper triangular matrix. The method exposed in the previous part can be easily generalized:\n",
    "\n",
    "\\begin{align*}\n",
    "    M \\vec{x} - \\vec{y} &= 0\\\\\n",
    "    (D+L+U) \\vec{x} &= \\vec{y} \\\\\n",
    "    (D+L) \\vec{x} + U \\vec{x} &= \\vec{y} \\\\\n",
    "    (D+L) \\vec{x} &= -U \\vec{x} + \\vec{y} \\\\\n",
    "    \\vec{x} &= - (D+L)^{-1} U \\vec{x} + (D+L)^{-1} \\vec{y} \\\\\n",
    "    \\vec{x} &= M' \\vec{x} + \\vec{y'}\n",
    "\\end{align*}\n",
    "\t\n",
    "Where $M' = - (D+L)^{-1} U$ and $\\vec{y'} = (D+L)^{-1} \\vec{y}$.\n",
    "\n",
    "This fixed point search can be carried out using a simple iterative scheme\n",
    "\t\n",
    "\\begin{equation}\n",
    "  \\vec{x^{k+1}} = M' \\vec{x^k} + \\vec{y'}\n",
    "\\end{equation}\n",
    "\n",
    "Although the $D+L$ matrix may not appear directly to be easily invertible, as it is lower triangular, we can perform a simple forward substitution process, from the first row to the last, in order to solve the inverse at a given point:\n",
    "\n",
    "If we denote $X_{ij}$ as the element at the $i^{th}$ row and $j^{th}$ column of the matrix $X$, and $x_i$ the $i^{th}$ component of the vector $x$, we can express $o$ the inverse of the matrix $DL \\in S^n_{++}$ at point $x$ as follows:\n",
    "\n",
    "\\begin{align*}\n",
    "    \\forall i \\in 0,1,\\dots n-1 \\, \\text{do: }& \\\\\n",
    "    &o_i = \\frac{x_i}{DL_{ii}} - \\sum_{j=0}^{i-1} DL_{ij} o_j\n",
    "\\end{align*}\n",
    "\n",
    "### Notes\n",
    "\n",
    "If $M\\vec{x} - \\vec{y} = 0$ has a solution, the Gauss Seidel method converges if $M$ is symmetric, positive definite, or, as formalized in chapter 1 of Convex Optimization from Boyd & Vandenberghe, it lies in the positive definite cone $S^n_{++}$.\n",
    "\n",
    "### SOR for Gauss-Seidel algorithm\n",
    "\n",
    "As the same strict contraction condition should always hold, we directly get back to our first matrix decomposition to derive the SOR version of the Gauss-Seidel method:\n",
    "\n",
    "\\begin{align*}\n",
    "    (D + L + U) \\vec{x} &= \\vec{y} \\\\\n",
    "    \\omega (D + L + U) \\vec{x} &= \\omega \\vec{y} \\\\\n",
    "    (D + (\\omega-1)D + \\omega L + \\omega U) \\vec{x} &= \\omega \\vec{y} \\\\\n",
    "    (D + \\omega L ) \\vec{x} &= \\omega \\vec{y} - ((\\omega-1)D + \\omega U)\\vec{x} \\\\\n",
    "\\end{align*}\n",
    "\n",
    "With $\\omega$ a relaxation parameter that can be used for underrelaxation ($\\omega<1$) or overrelaxation ($\\omega>1$).\n",
    "The update equation become:\n",
    "\\begin{align*}\n",
    "    \\vec{x^{k+1}} &= - (D+\\omega L)^{-1}((\\omega-1)D + \\omega U)\\vec{x^{k}} + \\omega (D+\\omega L)^{-1}\\vec{y} \\\\\n",
    "    \\vec{x^{k+1}} &= M'\\vec{x} + \\vec{y'}\n",
    "\\end{align*}\n",
    "\n",
    "Where $M' = - (D+\\omega L)^{-1}((\\omega-1)D + \\omega U)$ and $\\vec{y'} = \\omega (D+\\omega L)^{-1}\\vec{y}$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Found a solution in 4/2000 iteration within absolute precision 0.001\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x6fc4710>"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x5a03450>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\"\"\"\n",
    "Let's try to solve a Positive Definite (PD) linear system\n",
    "using Gauss-Seidel iterations\n",
    "\"\"\"\n",
    "\n",
    "#need a PD matrix\n",
    "M=np.identity(4)\n",
    "M = np.dot(M,M.T)\n",
    "\n",
    "#Now perform the decomposition\n",
    "D = np.diag(np.diagonal(M))\n",
    "ltMask = np.tri(D.shape[0])\n",
    "L = ltMask*(M-D)\n",
    "U = (1-ltMask)*M\n",
    "\n",
    "#define forward substitution for a matrix DL\n",
    "def forSDL( DL, x ):\n",
    "    out = np.zeros_like(x)\n",
    "    out[0] = x[0]/DL[0,0]\n",
    "    for i in range(1,DL.shape[0]):\n",
    "        out[i] = x[i]/DL[i,i]-np.dot(out[:i],DL[i,:i])\n",
    "    return out\n",
    "\n",
    "#SOR variable change\n",
    "omega=0.9\n",
    "DL=-(D+omega*L)\n",
    "U1=(omega-1)*D+omega*U\n",
    "Y1=omega*forSDL(-DL,Y)\n",
    "\n",
    "#Now solve\n",
    "Xk=np.zeros(M.shape[1])\n",
    "nbIter = 2000\n",
    "i = 0\n",
    "eps=1e-3\n",
    "err=[]\n",
    "residual = np.ones_like(Y)*(1+eps)\n",
    "\n",
    "while (i<nbIter) and not np.allclose(residual,0,atol=eps):\n",
    "    Xk = forSDL(DL, np.dot(U1,Xk))+Y1\n",
    "    residual = np.dot(M,Xk)-Y\n",
    "    err.append(np.linalg.norm(residual))\n",
    "    i += 1\n",
    "\n",
    "if i<nbIter:\n",
    "    print \"Found a solution in \"+str(i)+\"/\"+str(nbIter)+\" iteration within absolute precision \"+str(eps)\n",
    "else:\n",
    "    print \"Gauss-Seidel iteration failed, residual norm is \"+str(np.linalg.norm(residual))\n",
    "\n",
    "plt.figure()\n",
    "plt.suptitle(\"SOR Gauss-Seidel method convergence\")\n",
    "plt.plot(np.arange(i),err)\n",
    "plt.xlabel(\"Nb of iterations\")\n",
    "plt.ylabel(\"Residual norm\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### SSOR for Gauss-Seidel algorithm\n",
    "\n",
    "The Symmetric successive overrelaxation method is based on a preconditionning method dedicated to symmetric matrices, which reads:\n",
    "\\begin{equation}\n",
    "    M=(D+L) D^{-1} (D+L)^T\n",
    "\\end{equation}\n",
    "\n",
    "With $D$ a diagonal and non singular matrix, and $L$ a lower tridiagonal matrix, $L^T$ its transpose being upper tridiagonal.\n",
    "\n",
    "As seen previously, this decomposition can also be parametrized by a parameter $\\omega$::\n",
    "\n",
    "\\begin{equation}\n",
    "    M(\\omega)=\\frac{\\omega}{2-\\omega} \\left( \\frac{1}{\\omega} D + L \\right) D^{-1} \\left( \\frac{1}{\\omega} D + L \\right)^T\n",
    "\\end{equation}\n",
    "\n",
    "We can set up a new fixed point search using this method, which reads\n",
    "\n",
    "\\begin{align*}\n",
    "    M(\\omega) \\vec{x} &= \\vec{y} \\\\\n",
    "\\end{align*}\n",
    "\n",
    "The update equation becomes:\n",
    "\\begin{align*}\n",
    "    o\n",
    "\\end{align*}\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### SSOR for Gauss-Seidel algorithm"
   ]
  }
 ],
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