Lecture-4.ipynb

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    "# Lecture 4.\n",
    "# Regression problem"
   ]
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    "## Problem statement\n",
    "\n",
    "1. Assume we have $m$ pairs of measurements $(x_i, y_i)$, where $ x_i \\in \\mathbb{R}^n, \\; y_i \\in \\mathbb{R}^p$. We want to find function $f$, such that \n",
    "$$\n",
    "\\frac{1}{2}\\|f(x_i) - y_i \\|^2_2 \\to \\min\n",
    "$$\n",
    "\n",
    "2. Assume we have $m$ pairs of measurements $(x_i, y_i)$, where $ x_i \\in \\mathbb{R}^n, \\; y_i \\in \\mathbb{R}^p$. We want to find *parametric* function $f(x, w)$, such that \n",
    "$$\n",
    "\\frac{1}{2}\\|f(x_i, w) - y_i \\|^2_2 \\to \\min_w\n",
    "$$\n",
    "\n",
    "3. Assume we have $m$ pairs of measurements $(x_i, y_i)$, where $ x_i \\in \\mathbb{R}^n, \\; y_i \\in \\mathbb{R}$. We want to find *parametric* function $f(x, w)$, such that \n",
    "$$\n",
    "\\frac{1}{2} \\sum_{i=1}^m(f(x_i, w) - y_i )^2 \\to \\min_w\n",
    "$$"
   ]
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   "source": [
    "## Linear least squares problem\n",
    "\n",
    "Consider the case of linear dependence between $x_i \\in \\mathbb{R}^n$ and $y_i \\in \\mathbb{R}, \\; i = 1,\\ldots, m$.\n",
    "\n",
    "Then\n",
    "$$\n",
    "f(x, w) = x^{\\top}w\n",
    "$$\n",
    "or\n",
    "$$\n",
    "f(X, W) = XW\n",
    "$$\n",
    "Least squares problem is stated as\n",
    "$$\n",
    "L(w|X, y) = \\frac{1}{2}\\sum\\limits_{i=1}^m (x^{\\top}_i w - y_i)^2 = \\frac{1}{2}\\|Xw - y \\|^2_2 \\to \\min_w\n",
    "$$\n",
    "\n",
    "**Note.** Further $m \\geq n$ and $\\mathrm{rank}(X) = n$ except special cases"
   ]
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    "### Normal equation\n",
    "\n",
    "From the forst order necessary condition and convexity of the norm we get\n",
    "$$\n",
    "L'(w^* | X, y) = 0 \\Rightarrow (X^{\\top}X)w^* = X^{\\top}y\n",
    "$$\n",
    "or\n",
    "$$\n",
    "w^* = (X^{\\top}X)^{-1}X^{\\top}y = X^+y = X^{\\dagger}y,\n",
    "$$\n",
    "where $X^{\\dagger} = X^+ = (X^{\\top}X)^{-1}X^{\\top}$ - *pseudo-inverse matrix*."
   ]
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   "source": [
    "### Direct mehods"
   ]
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   "source": [
    "#### Cholesky factorization\n",
    "\n",
    "**Definition.** Any matrix $A \\in \\mathbb{S}^n_{++}$ has unique Cholesky factorization:\n",
    "$$\n",
    "A = LL^{\\top},\n",
    "$$\n",
    "where $L$ &mdash; lower tringular matrix.\n",
    "\n",
    "Algorithm:\n",
    "1. Compute $X^{\\top}X$ and $X^{\\top}y$\n",
    "2. Compute Cholesky factorization of the matrix $X^{\\top}X$\n",
    "3. Find $w^*$ from the forward and backward substitution"
   ]
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   "source": [
    "### Pro & contra\n",
    "\n",
    "Pro \n",
    "\n",
    "- when $m \\gg n$ storing of $X^{\\top}X$ requires much less memory, than storing $X$\n",
    "- if matrix $X$ is sparse, there exist method, which give sparse Cholesky factor\n",
    "\n",
    "Contra\n",
    "\n",
    "- condition number of $X^{\\top}X$ equals square of the condition number of the matrix $X$. Error is proportional to the condition number.\n",
    "- it's necessary to compute $X^{\\top}X$"
   ]
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   "source": [
    "#### QR factorization\n",
    "\n",
    "**Definition.** Any matrix $A \\in \\mathbb{R}^{m \\times n}$ cam be represented in the form \n",
    "$$\n",
    "A = QR,\n",
    "$$\n",
    "where $Q \\in \\mathbb{R}^{m \\times m}$ is unitary, and $R \\in \\mathbb{R}^{m \\times n}$ is rectangular upper triangular."
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    "#### Application\n",
    "\n",
    "1. Compute QR factorization of the matrix $X$: $X = QR$.\n",
    "2. $Q = [Q_1, Q_2]$, $Q_1 \\in \\mathbb{R}^{m \\times n}$,\n",
    "$R = \n",
    "\\begin{bmatrix}\n",
    "R_1\\\\\n",
    "0\n",
    "\\end{bmatrix}$,\n",
    "$R_1 \\in \\mathbb{R}^{n \\times n}$ - square upper triangular matrix\n",
    "2. The problem is reformulated as: \n",
    "$$\n",
    "\\|R_1w - Q_1^{\\top}y \\|^2_2 \\to \\min_w\n",
    "$$\n",
    "and normal equation\n",
    "$$\n",
    "R_1w^* = Q_1^{\\top}y\n",
    "$$\n",
    "We get linear system with square upper triangular matrix, which is easily solved by backward substitution."
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   "source": [
    "### Pro & contra\n",
    "\n",
    "Pro \n",
    "\n",
    "- error is proportional to the condition number of the matrix $X$, but not $X^{\\top}X$\n",
    "- more stable than one is based on the Cholesky factorization\n",
    "\n",
    "Contra\n",
    "\n",
    "- we can not control the stability of the solution"
   ]
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   "source": [
    "#### SVD\n",
    "\n",
    "**Definition.** Any matrix $A \\in \\mathbb{R}^{m \\times n}$ can be represented in the form\n",
    "$$\n",
    "A = U\\widehat{\\Sigma} V^* = [U_1, U_2] \\begin{bmatrix} \\Sigma\\\\ 0 \\end{bmatrix} V^*,\n",
    "$$\n",
    "where $U \\in \\mathbb{R}^{m \\times m}$ is unitary, $U_1 \\in \\mathbb{R}^{m \\times n}$, $\\Sigma = \\mathrm{diag}(\\sigma_1, \\ldots, \\sigma_n) \\in \\mathbb{R}^{n \\times n}$ is diagonal with singular values $\\sigma_i$ on the diagonal, and $V \\in \\mathbb{R}^{n \\times n}$ is unitary."
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   "source": [
    "#### Application\n",
    "\n",
    "$$\n",
    "\\| Xw - y\\|^2_2 = \\left\\| \\begin{bmatrix} \\Sigma \\\\ 0 \\end{bmatrix} V^* w - \\begin{bmatrix} U_1^{\\top} \\\\ U_2^{\\top} \\end{bmatrix}y \\right\\|^2_2 \\sim \\| \\Sigma V^* w - U_1^{\\top}y \\|^2_2\n",
    "$$\n",
    "Solution of the linear syste with **square** matrix:\n",
    "$$\n",
    "w^* = V\\Sigma^{-1}U_1^{\\top}y = \\sum\\limits_{i=1}^n \\frac{u_i^{\\top}y}{\\sigma_i} v_i,\n",
    "$$\n",
    "where $v_i$ и $u_i$ are columns of the matrices $V$ and $U_1$ "
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   "source": [
    "### Pro & contra\n",
    "\n",
    "Pro \n",
    "\n",
    "- effect of the perturbation in $y$\n",
    "- stability control: small singular values can be discarded\n",
    "- if the matrix is close to singular, then only SVD shows this\n",
    "\n",
    "Contra\n",
    "\n",
    "- SVD computation is more computationally expensive than compuations of Cholesky and QR factorization"
   ]
  },
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   "source": [
    "### Experiments"
   ]
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     "text": [
      "1.00975619131e-05\n"
     ]
    }
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   "source": [
    "import numpy as np\n",
    "\n",
    "n = 1000\n",
    "m = 2 * n\n",
    "X = np.random.randn(m, n)\n",
    "w = np.random.randn(n)\n",
    "y = X.dot(w) + 1e-5 * np.random.randn(m)\n",
    "\n",
    "w_est = np.linalg.solve(X.T.dot(X), X.T.dot(y))\n",
    "print np.linalg.norm(w - w_est)"
   ]
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   "source": [
    "import scipy.linalg as sclin\n",
    "import scipy.sparse.linalg as scsplin\n",
    "\n",
    "def CholSolve(X, y):\n",
    "    res = sclin.cho_factor(X.T.dot(X), lower=True)\n",
    "    return sclin.cho_solve(res, X.T.dot(y))\n",
    "\n",
    "def QRSolve(X, y):\n",
    "    Q, R = sclin.qr(X)\n",
    "    return sclin.solve_triangular(R[:R.shape[1], :], Q[:, :R.shape[1]].T.dot(y))\n",
    "\n",
    "def SVDSolve(X, y):\n",
    "    U, s, V = sclin.svd(X, full_matrices=False)    \n",
    "    return V.T.dot(np.diagflat(1.0 / s).dot(U.T.dot(y)))\n",
    "\n",
    "def CGSolve(X, y):\n",
    "    def mv(x):\n",
    "        return X.T.dot(X.dot(x))\n",
    "    LA = scsplin.LinearOperator((X.shape[1], X.shape[1]), matvec=mv)\n",
    "    w, _ = scsplin.cg(LA, X.T.dot(y), tol=1e-10)\n",
    "    return w\n",
    "\n",
    "def NPSolve(X, y):\n",
    "    return np.linalg.solve(X.T.dot(X), X.T.dot(y))\n",
    "\n",
    "def LSQRSolve(X, y):\n",
    "    res = scsplin.lsqr(X, y)\n",
    "    return res[0]"
   ]
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     "text": [
      "1.00975619204e-05\n",
      "1.00975619118e-05\n",
      "1.00975619024e-05\n",
      "1.00970182607e-05\n",
      "1.00975619131e-05\n",
      "1.00975617103e-05\n"
     ]
    }
   ],
   "source": [
    "w_chol = CholSolve(X, y)\n",
    "print np.linalg.norm(w - w_chol)\n",
    "w_qr = QRSolve(X, y)\n",
    "print np.linalg.norm(w - w_qr)\n",
    "w_svd = SVDSolve(X, y)\n",
    "print np.linalg.norm(w - w_svd)\n",
    "w_cg = CGSolve(X, y)\n",
    "print np.linalg.norm(w - w_cg)\n",
    "w_np = NPSolve(X, y)\n",
    "print np.linalg.norm(w - w_np)\n",
    "w_lsqr = LSQRSolve(X, y)\n",
    "print np.linalg.norm(w - w_lsqr)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
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     "name": "stdout",
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     "text": [
      "10 loops, best of 3: 52.7 ms per loop\n",
      "1 loop, best of 3: 351 ms per loop\n",
      "1 loop, best of 3: 778 ms per loop\n",
      "10 loops, best of 3: 110 ms per loop\n",
      "10 loops, best of 3: 58.6 ms per loop\n",
      "10 loops, best of 3: 153 ms per loop\n"
     ]
    }
   ],
   "source": [
    "%timeit w_chol = CholSolve(X, y)\n",
    "%timeit w_qr = QRSolve(X, y)\n",
    "%timeit w_svd = SVDSolve(X, y)\n",
    "%timeit w_cg = CGSolve(X, y)\n",
    "%timeit w_np = NPSolve(X, y)\n",
    "%timeit w_lsqr = LSQRSolve(X, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
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   "source": [
    "%matplotlib inline\n",
    "import time\n",
    "import matplotlib.pyplot as plt\n",
    "n = [10, 100, 1000, 2000, 5000]\n",
    "chol_time = []\n",
    "qr_time = []\n",
    "svd_time = []\n",
    "cg_time = []\n",
    "np_time = []\n",
    "lsqr_time = []\n",
    "for dim in n:\n",
    "    m = int(1.5 * dim)\n",
    "    X = np.random.randn(m, dim)\n",
    "    w = np.random.randn(dim)\n",
    "    y = X.dot(w) + 1e-5 * np.random.randn(m)\n",
    "    st = time.time()\n",
    "    w_chol = CholSolve(X, y)\n",
    "    chol_time.append(time.time() - st)\n",
    "    st = time.time()\n",
    "    w_qr = QRSolve(X, y)\n",
    "    qr_time.append(time.time() - st)\n",
    "    st = time.time()\n",
    "    w_svd = SVDSolve(X, y)\n",
    "    svd_time.append(time.time() - st)\n",
    "    st = time.time()\n",
    "    w_cg = CGSolve(X, y)\n",
    "    cg_time.append(time.time() - st)\n",
    "    st = time.time()\n",
    "    w_np = NPSolve(X, y)\n",
    "    np_time.append(time.time() - st)\n",
    "    st = time.time()\n",
    "    w_lsqr = LSQRSolve(X, y)\n",
    "    lsqr_time.append(time.time() - st)"
   ]
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Qi/WgmkF0KNPBxl2l49R6+KGt5cDonAe6/vpUBEZ4NkLjqaTPZa3UyyR9zthQ\nl8hUVapUYfHixTz33HMsXryY1157DV9fX/Lly8err77K6tWrU+3v6OhI7969UUoxb968VLWVK1dy\n69Yt2rRpQ8GCBZ+onyJFigBw69atJ3tDQgghnojBZODDnR+yPmy9xfonNT/htbKv2birdBz80fqk\n3W4+iZN2l25s+76yyLMQGpMndWqmaVqq96tpmjtQG4gB/rF1YyJRp06duHTpEiEhIXz88ce0bt0a\nk8nEihUraNu2LT169ODh2ZX69OmDg4MDCxYswGg0pmyfM2cOkHgJ+0kln+dxL20LIYR4cgaTgQ93\nfci6sHUW6x+//DGvl33dxl2lQSnY8QWsGgLKaF73Kpk4B2PhKrbvLQs99aFRKXUO2AD4AoMfKQcD\nuYEflVLRNm5NPESn09GsWTPGjh3L6tWriYiI4JdffiF37twsWrSIlStXpuxbokQJmjZtytWrV1m7\ndi0AYWFhbNq0iRIlStC8efMn7uPq1asAyMNMQghhG0aTkY92fcS6C9YDY6dy2ejyrskIa0emP2l3\nPj/b9mUDOfJBGE3T2gPtk75Mvg5ZU9O0hUl/jlBKjXjoJYNIXEbwW03TGgOhQA0S53A8DXyUSX21\nAdqULl063X3T8qQPlTxNHB0d6dSpE0ePHuWzzz5jy5YttG/fPqXer18/QkJCmDNnDm3atGHevHko\npQgMDMThCZ9MM5lM7NixA4AaNWpkyvsQQghhndFk5KO/PmLthbUW62NqjMlegVEfB7/3hdBVluul\nGkDnxeDibsuubCZHhkbAH+jxyLZSSR8AF4GU0KiUOqdpWgCJU+a0AFoC14CpQLBSysLdq49PKbUa\nWB0QEPDk10dFKu7uif/jPbr4T9u2bSlYsCBr164lPDycBQsWpNzv+KQWLlzIpUuXKFSoUMo0P0II\nIbKG0WTk478+Zs35NRbrH9b4kM7lzWfJsJvYe/BzN7i4y3I9B0/anVE58vK0UipIKaWl8eFr4TXh\nSqleSqlCSilnpVQJpdTwzAqM4sksXbqUjRs3YjKZzGrXr19PuU+xXr16qWpOTk707NkTo9FIt27d\nuHLlCi1btkx5kOVxGAwG5syZw+DBg9E0jSlTpuDq6vpkb0gIIUS6jCYjn+z+hNXnV1usf1jjQ94o\nn40WWbh/FRa0tB4Yaw6BV2c/1YERcu5Io3hK7Nmzh6lTp1KwYEHq1KlDyZIlAbhw4QJr1qwhNjaW\ndu3a0bE7vLDAAAAgAElEQVRjR7PX9u3bl0mTJrFz507A8gowj1qxYgVhYWEAREdHc+nSJXbu3Mm1\na9fw9PRk9uzZFud/FEIIkTmSA+Oqc5Yv8X5Q/YPsFRhvnYbFHSAy3HK92WdQa6hte7ITCY3Crt57\n7z3KlCnDpk2bOHLkCCEhIcTFxZEvXz4aNGhA165d6dq1q8WnmUuVKkXjxo3ZtGkTRYsW5ZVXXkn3\nfCtXrmTlypU4ODiQO3du8ufPT/Xq1WnSpAldu3bF29s7K96mEEIIwKRMBP0dlGZg7Fahm427SkP4\nPljy+lM/aXdGaY/eKyb+u4CAALV///509wsNDaVChQo26EgI6+TnUAhhCyZl4tPdn7Li7AqL9fdf\nep83n3/Txl2l4dR6WNbT8hyMutzQ+cenZg5GTdMOKKUC0tsvR97TKIQQQoicw2Ay8PFfH1sNjCMD\nRmavwJgNJu2+HxFL+Ik7WXqOxyWXp4UQQgiRZRKMCYzcPpIt4Vss1kcEjOCtim/ZuCsrlIKdX1qf\ng9HLF978PcvnYIy5n8CqqYeJuh1Hk17PU+al57L0fBkloTETZdY8jUIIIcTTIEYfw7Ctw/jnmuVF\n10YEjKBHxUdn0LMTkxHWvQ/75liuF6oM3X6DPAWytI34WAOrpx0m8lbiKOeG+ceJj9FTqX7RLD1v\nRsjl6UyklFqtlOrn6elp71aEEEIIu4qMj6Tvxr5WA+O71d7NPoFRHwe/9bIeGEs1gJ5rsjww6hOM\nrPnuXyLCH/x/o4LtS0/z72YrT2/bkIw0CiGEECJTRcRG0G9jP87cPWOx/lGNj+hSvouNu7IivUm7\nK3VMfEo6i+dgNBpNbJhzjGtnI81qLm5OFC3vlaXnzwgJjUIIIYTINFceXKHfhn5cirpkVnPUHBlX\nexxt/NrYoTML7l+Dxa/BzeOW6zWHQNNx8ITL02aUMim2LAol7Ohts5qTiyOth1QmX5E8WdpDRkho\nFEIIIUSmOH/vPH039uVmzE2zmrODM1/W/5KGxbPJMq3ZZNJupRS7lp3h9J4bZjUHR41X+leiYKns\ncdubhEYhhBBC/GfHbx9n4MaB3I03nwjbzcmNaY2mUb1QdTt0ZkH4PljSCWItTGlj40m7968N48jW\ny+YFDZr0ep7iz+ezSR8ZIaFRCCGEEP/J/uv7GbJlCNH6aLOap4sn3zf+nhfyv2CHziw4HQK/9sgW\nk3Yf3XaZvasvWKw16FqOMgHZY6qdZBIahRBCCPHEdlzewbvb3iXeGG9Wy58rP7OazqKMVxk7dGbB\nwR9h9TBQRvOamw90WwZFqtqkldP7rrPjl9MWay+3L0XFukVs0sfjkCl3hBBCCPFE1l9Yz7AtwywG\nxiJ5ivDDKz9kj8CoFOz4AlYNsRwYvXwhcIPNAmPY0Qg2LwgFCys5+zcpRtXmJYg3GDkcfs8m/WSU\nhEYhhBBCPLZlp5cxascoDMpgVvPz9GPRK4so5l7MDp09wmSEtSOtr/JSqDIEbszyVV6SXT17j5DZ\nxzCZzBNj+ZoFqfVaaeINJgb8eIBOs/5m55lbNukrIyQ0CiGEEOKxzD82n7F/j0VZGCqrlK8SC1ss\npIBb1k6EnSHZZNLuZBGXH7DmuyMY9CazWsnKPjR8szzxBhP9fjzA1lO3SDCY6PPDfnafjbBJf+mR\n0JiJNE1ro2na7MhI84k5hRBCiJxOKcXUg1OZcmCKxfpLBV9ibvO55HXNa+POLIiLTJyD8cRKy/VK\nHaHrMnBxt0k7kbdiWPXtYRJizUdmi5TLS7M+FYk3Kvr8sJ8dpxNHF/1vnYG4WHr/sI9/zpvP4Whr\nEhozkSwj+N+cPHmSoUOHUqlSJTw9PXF2dqZw4cK0atWKefPmER9vfs/M6dOneffdd6latSre3t7o\ndDq8vb2pUaMGI0aM4MCBA3Z4J0II8fQxKRPj94xn7tG5FusNijZgRuMZ5NbltnFnFty/BgtaWl/l\n5eXB0GFOlq/ykiz6Xjyrph4m9n6CWS1/cXdaDniRBKXovXAfu85G4GKIZ/C/y5nw1ywCj68hwWDi\nxv04m/SaFk0pC3dhiv8kICBA7d+/P939QkNDqVChgg06yv7Gjh1LcHAwJpOJmjVrEhAQgLu7Ozdu\n3GDHjh2cOXOGatWqkfx9VUoxduxYxo4di8lkomrVqlSvXh1vb2+ioqI4cuQIf//9NwkJCUyfPp3B\ngwfb+R1mX/JzKIRIj96kZ8yuMay9sNZivVWpVoyrPQ6dg87GnVmQTSbtThYXreePrw5y56r5dER5\nn3Ojw4iqGJ0d6L1wH3sv3KHcnYuMOLCUotH/vyR9ZfQEmvRon2U9app2QCkVkN5+MuWOsLvPP/+c\nTz/9lGLFirFs2TJq1Khhts/69euZPHlyytdjx44lKCiIYsWKsXTpUmrXrm32mps3b/LNN98gtwsI\nIcSTizfGM2LbCLZd3max3rlcZz6s8SEOWja4eJnepN3tZkDlzjZrRx9vZM13/1oMjHm8XGg7zB+D\nTqPX/L0cuhDBm6c20eX0ZhxV6nseS8z9GkPb+jh52Xf9aRlpzAIy0phxYWFhlC1bFoCDBw9SqVIl\nq/vGx8fj4uLC+fPnKVeuHA4ODhw8eJCKFSumeQ6DwYCTk/x+ZI38HAohrInWRzN0y1D2Xd9nsd73\nhb4MrTIUTdNs3JkF6U7avQhKN7FZO0aDibUzjnDphHmAdc2jo8OIqjjmdabH/L3cOn6akQeWUPae\nhZVhkuQfPhyfAf2zpFcZaczBQsvnvH/AK5wMfaLXLViwAL1eT5cuXdIMjAAuLi4przEYDHTt2jXd\nwAhIYBRCiCdwL+4eAzcN5NjtYxbr71R7h96Vetu4KysOLYZVb2eLSbsBTCbFpoUnLAZGnYsjbYZW\nxsHTme5z/qH4zrV8euxPXEzmD8gAaC4uFBg5Eq+ub2R12+mSf02FXe3alXiTcuPGGV+y6a+//gKg\nUaNGWdKTEEI8625E36D/xv6cizxnVtPQ+Ljmx7xe9nU7dPYIpWDnV7BlnOW6ly+8+bvN5mBMbEmx\n4+fTnN1/06zm4KTRcuALOOd3ZeA363h93Tyq3bS8KgyAa6VKFJ48CZdSpbKy5QyT0Cjs6tq1awAU\nLVo0w6+5fv06AEWKmC+xFBYWxsKFC1Nty5s3L8OHD3/yJoUQ4hkSHhVO3w19ufLgilnNSXNiQt0J\ntCjZwg6dPcJkhPUfwN7ZlusFX4Ruv4G7bddv3rv6Asd3mH/vNA2a96mEW7E8TH5/OsO3LcZdb+FS\nOoCjIz79++MzcACaLhs8XJREQqN4qoSFhREcHJxqW4kSJSQ0CiFEBpy5e4b+G/tzK9Z8FRIXRxe+\nbvA19YrWs0Nnj9DHwR/9rM/BWKoBdF5sszkYk/27OZz9a8Ms1hp2L497AVjbuS/dz+61egznEiUo\nPHkSuSpXzqIun1w2eNRJPMsKFSoEwJUr5r+VWVOwYEEArl69alZr0KABSimUUuj1+sxpUgghngFH\nbx2lV0gvi4Exty43M5vMzB6BMS4SfuqYbSbtTnbyn2vsWnbGYq3Wa6XxiDnD2TZtqZZGYMz7RhdK\n/vF7SmCMuR9JfIz5k9f2IiON2dCTPlSSE9WpU4ctW7awefNmAgMDM/Sa2rVrs3XrVjZv3kzv3tnk\nJmwhhMjB9l7by9AtQ4kxxJjVvFy8+L7p91TMl/6Dh1nu/rXEwHjD8sM5vDw4cR5GB9uOiV349xZb\nFp20WKvSuAh5//qB+z8vwdo6OSavfJSY9Dl56iWGcmUycXTrRnb+tIDydRrQuPeALOr88chIo7Cr\nXr16odPpWL58OSdOnEhz3+QVYXr27ImTkxO//fYboaHPTsAWQoissPXSVgZuGmgxMBZwK8DCFguz\nR2CMOAPzmlkPjE3HQYvPbR4Yr5y+S8ic4yiT+RSG5SrmwnveO8T9vMTq6w11GlBu7eqUwHjr4gWW\nfjqKjbOnERf9gMMb1nD9nOURTFuT0CjsytfXl6CgIBISEmjVqhXW5rdct24dLVok3njt5+fHmDFj\nSEhI4JVXXmH37t0WX3Pv3r0s61sIIZ4Gq8+t5p1t75BgMl/errh7cRa9sohSebPBk7uX9ycGxshL\n5jUHJ3h1NtR+2+Zt3boUxZoZRzAaTGa1YnmjKDyzPyrsgsXXxuhciR/5MZXmzMDJy4uEuFi2L57P\njx8M49rph0YtlWLjnOmYTBamE7IxuTwt7O7DDz/EYDAQHBzMSy+9RK1atQgICCBPnjyplhEMCPj/\nvKOffPIJSinGjRtH7dq1qVatWsoygvfu3SMsLIxNmzYBUK9eNrgHRwghspmfT/7M+D3jLdbKeJVh\ndtPZ+OTysXFXFmSzSbuT3bsRw+pph9HHmYc5H/1l/FZ+gaYsz714rEBpykz5Av9q5QE4u+8ftiyY\nRdRt8/tJAW5eOMe/G9dRpXnrzHsDT0BWhMkCsiLMkwkNDWXGjBls3bqVS5cuERcXR758+fD396dj\nx468+eabKRN8Jzt16hQzZ85k69athIWFER0djbu7O35+ftSpU4fu3btTtartJnTNieTnUIhni1KK\nuUfn8u2hby3WX8z/IjMaz8DTxdPGnVmQ7qTdv0KRajZvK+pOHL9/eYAHd+LNah4PLuJ/aCpORvOa\n3sGRpZXb0HXSKF4o7sX9WzfZvGAm5w9YfzgG4Pm6DanfPRA3T2t3Rf43siKMyHEqVKjAtGnTHus1\n5cqVY8qUKVnUkRBCPF2UUkw5MIUFxxdYrNcoVINvG36Lm87Nxp09Ir1Ju/OWgO5/2HTS7mSxDxJY\n/e1hi4Exd/Q1Kh/+zmJgPOdRmJl1ujPxvVep8Fxu9q78jb+XL8UQb75vMq/CRWkSOIjilV7M1Pfw\npCQ0CiGEEM8Ao8nIuH/GsfzMcov1xsUbM7neZJwdnW3c2SOy6aTdAAlxBv6c9i93r5s/NOQaG4H/\nv9PQGVJPkWNCY1mZhqyt1oqF/ergcT+cxVNmEBF+0ep5nHTO1OjQmYA2HXCSyb2FEEIIYSt6o57R\nu0YTEhZisd7Wry3BtYJxcrBzLNDHwR/94cQKy/WS9RMn7Xb1sG1fgFFvYt3Mo9y8GGVW0yXcx//I\ndFwSIlNtv+bmzVfV3uB68XIs7FqR8FULObZ1Q5rn8fWvRuNeA8hbsFCm9p8ZJDQKIYQQT7FYQyzv\nbnuXXVd2Wax3q9CNUS+NwkGz84QqcZHwczcI22m5XqkjtP8enGw/Emoymtgw/ziXT941qzkaYvE/\n8h1uj0yKvr5EdWZXakvuvB5MfjGOHRNGEBd13+o58nh506BHP8q+XBtN0zL9PWQGCY1CCCHEUyoq\nIYohm4dw8OZBi/WBlQcysPJA+4eUdCftHgTNxtt8DkZIvA9025JTnD9k/mSzgzGByke/x/3B5ZRt\n95xz802VTuwpVJHSuge8GRvC4SWWJ/4G0DQHqrRoTa1Ob+LiZud7SdMhoTETaZrWBmhTunRpe7ci\nhBDiGXcn7g4DNg4g9I7lRRBGBozkrYpv2bgrCyLOwI8dLM/BCImTdtcaCnYKtn//cY7Qv66ZbdeU\nkUon5pE38tz/9y1YkalVOhKtc6VJ9H6ejzjI7TTmVyzoV4YmfQbzXKmckRskNGYipdRqYHVAQEBf\ne/cihBDi2XU9+jr9NvbjQqT5xNIOmgNBNYN4tcyrdujsEZf3w0+vQ+wd85qDE7T7Dip3sX1fSQ6G\nXOTQBsthtkLoj/jcThwZjXFyYValtmwoUR3f2Eu8FrGL3An3sTapoXMuN+q+0YMXm7bAwcExi7rP\nfBIahRBCiKfIxfsX6buhL9eizUfHnBycmFxvMk1LNLVDZ484HQLLeoLe/Elke07anezErqv8/cc5\ni7UyZ5ZR8OY+AI57+/JltTd44OJCy5sh+MVYXgEmWfna9WnwVh9y5/XK9J6zmoRGIYQQ4ilx6s4p\n+m/sz+2422a1XE65mNJgCrWL1LZDZ4/IppN2Jzt74AbbFocC5pfEfcPWUuzKNvSaIz9WaM7vpevx\nQtRxatzai7OVFWAAvAoVpnHvQZR40T8LO89aEhqFEEKIp8Dhm4cZtHkQUQnmU8K469yZ0WQG/gXs\nHFiUgl1fw+axlut2nLQ7Wdjei2ycfxqF+WXjope3UTJsDWHuz/FFta5Euzry+rXfyZ9gHtKTOTo5\nUb19J6q364iTs53nwPyPJDQKIYQQOdzuq7sZvnU4sRbWZ/Z29WZW01mU9y5vh84eYjLC+tGwd5bl\nuh0n7U524bcthITEY3J0Mas9d2MfZc7+xu9+9VhavhEvRR6g0rUTFsYi/6/4C/40CRyIV6EiWde0\nDUloFEIIIXKwTRc3MWrHKPQmvVmtYO6CzGk6B19PX9s39jCTEVYMhCO/WK7bcdJuAFN0NGfGT2fb\njQoYdXnM6vluHyf/pVV8UKsf8blMdLm6DDdTnNXjuXnmpWGPvpSrVc/+0xllIgmNQgghRA614uwK\nPt39KSZlMqv5evgyu+lsCuWx88oiJiP8MQCO/mq5bsdJuwFiDh3i3Eefs+e5LhhczAOjZ+Q5bkcd\n4Ntavahxfy9FI65aP5im4d+sJbU7d8c1t/mxcjoJjUIIIUQOtPjEYibtm2SxVt67PDObzCRfrnw2\n7uoRRgOsGABHl1mu23PS7oQEbs2YwbUFSzlQeTjxLuZPM+eOucY259tElShB+5urcMQ8nCcrUNKP\npn0GU7B02axs264kNAohhBA5iFKKmf/OZMa/MyzWqxSowvTG0/Fwts+l3hRGQ+I60sd+s1xvEgy1\nh9ll0u74s2e5Oup9Hpw6z2H/YcS6md9H6Wq6z5+5b/L8/SN43rO+/J9zrlzU7vwW/s1b5qg5F5+E\nhEZhd8n3exQvXpxTp07h6upqto+vry8XL15Er9fj5CQ/tkKIZ5NJmfhi3xcsDl1ssV67cG2mNJxC\nLqdcNu7sEUYD/NEPji23XH9lMtTob9ueAGUycXfxYm5++RUGg+LIi4N54F7cbD9H7nFEv52akZbn\naUxWtmZdGr7Vhzzedh7RtRE7r06efWmaNkjTtAuapsVpmnZA07S69u7paXfp0iW++eYbe7chhBDZ\nksFk4JO/PrEaGJuVaMa0RtNyQGD8wi6BUX/9OpcCA7nx+QSMegPHn+/NvbxlUu2jlAmj/iD37v1I\n8RjrgdHzuYJ0GB1Mm+HvPzOBESQ0WqRpWmdgKvA5UAXYDazTNM381xGRKby8vPD29mbixIlERETY\nux0hhMhWEowJjNw+kpXnVlqsdyjTgcn1JqNz1Nm4s0cYDfB7X+uBseWXUKOfbXsCIv9cw/m27Yj5\n+x8UGifLdSPC58VU+5gM10l4sAT9g23olPmT6AAOjk683KEzPb78jpL+9pt83F4kNFr2LrBQKTVH\nKRWqlBoKXAMG2rmvp5abmxsff/wxkZGRBAcHZ+g127ZtQ9M0goKCLNZ9fX3x9fVNtW3hwoVomsbC\nhQvZuHEjdevWJU+ePOTPn59evXpx7949AA4dOkTr1q3x8vIiT548tG3blrCwMLNzNGjQAE3TiI+P\nZ8yYMZQsWRIXFxf8/PwIDg4mISEhZd+7d+/i5uaGn58fSllekbRNmzZomsb+/fsz9D0QQjz9YvQx\nDNk8hE2XNlms93i+B0E1g3C09/10RgP83geO/2653vJLqN7Xti3du8eVd9/l6ogRmO4nrgV91q8D\n1wu+nLKPMsWhj9lMQtQSlOGm1WMVrlCJt76YRu3O3dE5m8/j+CzIkTeHaZrWEagP+AOVAXfgJ6XU\nm2m8pigwFmgB5CMxBK4AgpVSdx/azxmoBnz5yCE2ALUy8W1Y9d2ALbY4TaYaPLPRfz/G4MFMnz6d\nWbNm8fbbb1OmTJn0X/SEVq1axZ9//knr1q0ZMGAAu3fvZuHChYSFhTFhwgQaN25M3bp1CQwM5OjR\no6xevZrz589z5MgRHCw85depUyf27dtHx44d0el0rFy5kqCgIPbv38+qVavQNA0vLy+6dOnCggUL\n2LRpE02bpl77NTw8nHXr1lGtWjUCAgKy7L0LIXKOyPhIBm8ezL+3/rVYH1plKH1f6Gv/uQCNBlge\nCCdWWK7bITA+2PUX1z78EMPN/wfBi8WbE14s8d8rpRQm/Sn0MdtAWVj/Oole50bz3v3wb9jY/t9n\nO8uRoREYQ2JYfABcBtKc5l7TND8SLzEXAFYCJ4HqwDCghaZptZVSyWsA+QCOwI1HDnMDsN/K6c8A\nnU7HxIkTef3113n//ff5/Xcrv61mglWrVrF582bq168PgMlkonnz5mzatImWLVsye/ZsunXrlrJ/\nYGAg8+fPZ/Xq1bRr187seKGhoRw/fhwvr8QpG8aPH0/Dhg35888/Wbx4Md27dwdg0KBBLFiwgFmz\nZpmFxnnz5mE0Gunf3/b3+gghsp+I2Aj6b+zP6bunLdZHVx9N1wpdbdyVBUY9LO9jPTC2+gpe6mOz\ndkyxsdz88ivu/vRTqu2XC9flfKm2ifsY72KI2YzJcCnNY90sUpUPPn4XL6+8WdZvTpJTL0+/A5QF\nPMjYJeMZJAbGt5VS7ZVSHyilGgFTgHLA+CzrVDyWjh07UrNmTf744w927dqVZed54403UgIjgIOD\nQ0qwq1SpUqrACPDWW28BcPjwYYvH+/jjj1MCI4CrqysTJkwAYP78+SnbAwICCAgIYOXKlVy/fj1l\nu9FoZN68ebi7u/PGG2/8x3cnhMjprj64Ss/1PS0GRkfNkfF1xmejwJjGCGOrr20aGGOPHuNCh9fM\nAuONAtU4XaYTShnQx+4m4f6iNANjhHM+Tr3Ui6CJn0pgfEiODI1Kqa1KqTPK2o1hD0kaZWwGhAHf\nPVL+FIgGumualjtpWwRgBB6dtOk54Doiy3311VcAjBgxIsvOYenyb+HChQGoVs385uYiRRLXDb18\n+bLF4z0cQJPVqVMHR0dHDh06lGr7oEGDMBgMqcLk2rVruXz5Mm+++SZ58jx9qwgIITLufOR53lr3\nFhfvXzSr6Rx0fNXgK9r6tbVDZ48w6uG33nDC8sM5tJ4CLwXapBVlMHDru+8Ie+MNEi5cSFW77f08\nJ8r3wGgIJ+H+Ioxx/5D4z7y5BM2Jnd61uFxvAN8Oa08u56d73sXHlSND42NqmPR5g1Kp11lSSkUB\nfwFuwMtJ2xKAA0Dqa4eJX+/O2lYFQM2aNenYsSN79uzhl1+srFP6H3l6epptS57/Ma2aXm/5ibrn\nnjOfGNbJyQkfHx/u3089KWyXLl3w8vJizpw5mEyJP5KzZ88GkEvTQjzjTtw+Qc91PbkR8+gdUpDL\nKRczmsygcfHGdujsEcmBMXSV5XrrbyCgt01aib9wgbBu3YiYNh0MhlS1SI+SHKnQlfiY9egfLEeZ\n7lk9zlm3kvxU9A28qjdhTs/quOokMD4qp97T+DjKJX22fFMInCFxJLIssDlp29fAj5qm7SUxVA4A\nCgMzrZ1E07R+QD9InKT6v8iMh0pyugkTJrBy5UpGjx7Nq6++anGf5AdSDI/8JZHs3r175M1rm8sK\nN27cMPvvbjAYiIiIwMMj9aoMuXLlomfPnkyZMoUNGzZQsWJF1q1bR40aNahcubJN+hVCZD8Hbhxg\nyOYhPNA/MKt5OHvwfZPveTH/ixZeaWNGPfzWC0JXW663/gYCemV5G0op7v38Mzcmf4GKjTWrR+Uu\nzP7y9YiPXgIq3upx7ju5sz1fHcLcfGlSoQDfdauKi5MERkuehZHG5GGjSCv15O0p6UIp9QswnMQH\nbg4DdYCWSinzawX/f81spVSAUiogf/78/73rZ1zp0qUZNGgQFy5cYNq0aRb3Sb6HMDw83Kx29uxZ\nIiOt/SfPfNu3bzfbtmvXLoxGI1WqVDGrDRw4EE3TmDVrljwAI4Rgfdh6+m/sbzEw+uTyYUGLBdkj\nMBoSYFlP64GxzVSbBEb9zZuE9+/P9eCxFgPjTc8C7C5bnPj4XVYDoxEH9ntW4acinQlz86XZ888x\no1s1CYxpeBZC4xNRSs1QSvkqpVyUUtWUUjvs3dOz5pNPPiFv3ryMHz+eBw/M/yItX748Hh4erFy5\nkpsPTakQGxvL22+/bctWGTduHHfvpszcRFxcHKNHjwagVy/zv0DLlClD48aN+fPPP5k5cyZ58+al\nS5cuNutXCJE9KKX4/t/vGbl9JPFG83BTJE8RfmjxA2W9ytqhu0cYEhJHGE/+abne5luo1jPL27i/\nPoQLbdoSvWOnWU3voHGkeGH2+3pgNN2yeowrLoX4ucjr/O39MgYHHa9UKsh33ari7CSxKC3PwuXp\n5OEm8xvVUm+3fqODsAtvb28+/PBDRo0aZbGu0+kYNmwY48aNo0qVKrz66qsYDAY2btxI4cKFUx5s\nsYUKFSpQsWLFVPM0njt3jlatWqU8lf2oQYMGsWnTJm7cuMHQoUPJlcvOS38JIWwqzhDHJ399wrqw\ndRbrpTxLMbvpbJ7LbX7PtM0ljzCeWmO53nYaVH0rS1swRkVx47PPiFxpfh+lAq575uZE8YLEOyRv\nMRfn4MpO75qczFMOkuZcbP1iIaZ09kfnKIExPc/Cd+hU0mdrv6YlzyBt7Z5HYUdvv/222aouDwsO\nDmbChAm4uroye/Zs1q5dy2uvvUZISAg6ne2W0/r111/p3bs3q1evZvr06ZhMJoKCgli+fLnVyWDb\ntm2Lj48PIA/ACPGsuRVzi17re1kNjM/ne56FLRbmgMCo2SQwRu/Zy/l27SwGxmhnJ/aXLMQh3+TA\naNnpPBX4segbnHQvnxIY2/kX5hsJjBmmZWDWmmxN07QGwFasrAiTNOXOWRKn3PF7+AlqTdPcSVwZ\nRgMKKKWiM6OngIAAlZFl4EJDQ6lQoUJmnFLYSYMGDdi+fbvVZQHTcv78eUqXLk3t2rXZudP8Mout\nyM+hELYVejuUoVuGWnxCGqBJ8SaMrzMeN52bjTuzIEOB0fLVlMxgio/n1pRvuPPDD/DI37NGDS7k\nzxz9c7oAACAASURBVMvZgt6Y0lipRTnmY6NPPU65FUy1vUPVInzRsTKODs/2Ki8AmqYdUEqluxTZ\nU395Wil1TtO0DSQ+IT0YePipimAgNzArswKjEBn15ZdfopRiyJAh9m5FCGEjmy9uZvSu0cQazB/e\nAOj7Ql+GVBmCg5YNRr4MCbCsB5xaa6GoQbvpUMXq6r3/WVxoKFdHjSL+zFmz2u3crhwrmp9oV+c0\njuCElqsmv+V7nmu61N/PTgFFmdDhRQmMjylHhkZN09oD7ZO+TP7VoaamaQuT/hyhlHp4ZuhBJM6x\n+K2maY2BUKAGiXM4ngY+yqS+2gBtSpcunRmHE0+hS5cusWTJEs6cOcOCBQuoXLkyr7/+ur3bEkJk\nMaUU847NY+rBqRbrOgcdwbWCaePXxsadWWGIh197wGlLl881aPcdVOlmofbfKaOR2/Pmc2vaNHhk\nbtx4J0dOFsrHFW/3NI/hoPNDy9WA3z1cuaZLNUUzb1Qvxvj2L+AggfGx5cjQCPgDPR7ZVirpA+Ai\nkBIak0YbA4CxQAugJYmXpacCwUqpu2QCpdRqYHVAQIBtV2UXOcb58+cZPXo0bm5uNG3alO+//z5l\nvkkhxNMpwZhA0O4gVp+3PE2Nt6s3UxtOxb+Av407s8IQD7++BafXWyhq0H4G+GfNEoYJ4eFcff8D\nYg8eTLVdAeHe7pwslA9DWlPiaO7o3BqiOfuxyi2BS48ExjdfLs7YtpUkMD6hHBkalVJBQNBjviYc\nyPrJo8QzZdu2bY+1f4MGDZ7o/kchRM50O/Y2w7cO5/Aty+vWl/Eqw/RG0ymcx3azPaTJEA+/dIcz\nIRaKWRcYlVJELl/Ojc8n8D/2zjs6qmrtw8+ZkklvkAQCofcaIBCUDiKgghQF9eK1ofd+ioWLCqLX\ndlVEQWliAQsKqKggVZoU6R2k9xYSQnrP1P39MQECczKTCZMG+1mLxeK8++z9zpDM/M7eb7Hl5l5n\ny/T24mDNMNL9vJ3MoKA1tEPn0xFF8WKFj4kTXtcLxsfvrMNb/ZsVmZwocU2lFI0SiUQikVR0jqcd\n5/k/nyc+J17V3q1mNyZ0nYCf3q+MPSsCl4Lxc4h+2PPLpqSQ8N83yV679vrrGoUTEaGcDQtCOBF6\nijYSvW8vNDp7Y4313mYOGK7vLf1U57q8cW9TKRhvEikaJRKJRCLxMBsubODVv14l15Kran+i+RO8\n2PZFtJoK0n3EnA/zH4UTq1SMCgz6Alp7vgFB1tq1JLzxX6ypqVevCSAx0I/DNaqS7+VEpigGdD5d\n0Xq1uCoGtxvM7PS+vrXsv7rWY2y/JlIwegApGiUSiUQi8RBCCL4//D2Tdk1CqBSY1ml0vNnxTQY1\nHFQO3hWBOR9+Hg4nVzvaFA0M/AJaD/PoktbsHBI/HE/Gr79ddz1Xr+NwjapcDnK++6rxaobepyuK\n5lpZov1eFv66QTA+270+r/RpLAWjh5CiUSKRSCQSD2C2mnlv+3ssOLFA1R5sCObT7p8SU81lObyy\nw5wPP/8DTq5xtCkaGPQltBrq0SVz9+wh/tUxmOPirl6zFdRcPBERgs1JcqBGVwWdd080+qjrrh/T\nW1ntY7ZXXS7ghZ4NGNW7kRSMHkSKRolEIpFIbpK0/DRGrR/F7sTdqvZ6QfWY3nM6UYFRqvZyoYwF\nozCZSJr+GSmzZoHtWpJKakHNxWwnNRe1ej163zsQShsU5foj/bM6K8t8TYhC2nDUXY148a6GSDyL\nFI0eRNZplEgkktuPU+mnGPnnSOKy41TtnSI78XG3jwnwcl5bsExxKRi/glaeqyFrPH2a+JdfIf/w\n4avXTFoNRyOrEBca6PTeqOZtyMnqSG6WDzfuGcZrbfzuZ8JayPBKn8Y810N+D5cGskCcBxFCLBFC\nPBMUFFTerkgkEomkDNh8cTPDlw8vUjAObzqc6b2mVzzB+NMjZSIYhRCk/fQzZwYPuSoYr9Rc3NCk\nllPB6B9ahb7/9ypm273kZvk42JM1Nn7zM2IuJBjH9msiBWMpIncaJRKJRCJxEyEE847O46OdH2ET\nNge7TtHxWuxrDG3s2XjAm8acZxeMp9Y62hQNDJ4JLR/wyFKW1FQS3vjvdaV0sry9OFizKml+jiLw\nqhuKhrb39Cem/0Os+OoY6ZcyHcZkKDZ+8TeSX2jr6417mzKiSz2HsRLPIUWjRCKRSCRuYLaZmbBj\nAj8f+1nVHuAVwKfdPyW2emwZe+YCcx78+DCcXudo87BgzN64ifhxr2FNSgYg3cfAmfBgEoL8wEli\nSrUGjbhrxHNUrVmXpZ/tJ/GMo2DMUQS/+JvILiQY3+rfjCc61fWI75KikaJRIpFIJJJikmHMYPSG\n0WxP2K5qrx1Ym+k9p1MnqE7ZOuYKp4JRC0NmQoshN72MzWgk6ZNPSJ39PQJICvDldHgwqf5F7ywC\nGHz96PLIY7Ts1QfQsGrWQeKOOnb4NSL41c9ImvZaOaN372/OP++oc9O+S1wjYxol5Y6iKG6VRPjl\nl1/o27cv4eHh6PV6qlSpQrNmzRg+fDizZ88u8r41a9YwbNgwatWqhbe3NyEhIXTo0IH33nuPjIwM\n1Xu+++67q/5d+WMwGKhduzb/+Mc/2L9/v9uvVyKRVE7OZpxl+PLhRQrG2OqxzL1nbsUTjKZc+PEh\nJ4JxlkcEY/6x45x9cCjJ339PXEgAGxtFsatedZeCsWnn7jzx6Re07n0PiqJhw7xjnNqT5DDOjGCB\nv4nLumuC8YNBLaVgLEPkTqOkUvHMM88wc+ZMfHx8uPfee6lbty5CCI4ePcqSJUtYv349jz322HX3\nGI1GRowYwZw5c/Dx8aFfv340atSI7Oxs1q5dy3//+1+mT5/OokWLiI1VP05q3bo1AwcOBCAzM5PN\nmzczb948fvvtN/788086depU6q9dIpGUH9sTtvOf9f8h0+R4XAowtNFQxsaORa/Rl7FnLrgiGM9s\ncLQpWnjga2h+c4XGhRCk/TCHi59M4lyAD2eb1saody0vQqrXoNdT/0ftltFXr237/TSHNzm2XbQh\nWOJnIk5njx9VFJgwuBVD21egEka3AVI0SioNmzZtYubMmdSsWZOtW7dSs2bN6+xms5n169c73Pd/\n//d/zJkzh7Zt2/L7778TFXXtQ0YIwWeffcaLL75I37592bNnD3XrOsbFREdH8/bbb1937d///jdf\nfvklb7zxBuvWqTzBSySSW4L5x+Yzfvt4LMLiYNMoGsa0H8PDTR6ueEWkTbnw4zA485ejzUOC0ZKU\nxMmxYzl8+hgXGkRi0bo+wPQLDqHtPffT9p770emviey9q86zZ+U51Xv+8DVzSn9NMH78QGseaFdT\ndayk9JCiUVJp2LJlCwBDhgxxEIwAer2e3r17X3dt06ZNfPvtt4SEhLB06VKqV69+nV1RFEaOHElc\nXBwTJkxg1KhR/P7778Xy56mnnuLLL79k586dJXxFEomkImOxWZi0axJzjsxRtfvr/ZnYbSKdalTA\nkwaXgvEbaD7wppY4+8vPbJs9i3hfL0R4sMvxoTWiiOk/iKade1wnFgEOb45ny4KTqvf96WPisJcV\nAI0Ck4a2ZlAbKRjLA4+JRkVRqgDPAUII8T9PzXs7MmnYfeXtgtuM/nlpqa9RpUoVAE6cOFHse2bO\nnAnA008/7SAYCzNmzBgmT57M4sWLiY+PJzIysthr6PUV7DhKIpHcNFmmLF756xU2X9ysaq/pX5Pp\nvaZTP7h+GXtWDFwJxge/hWb3l2hqIQRnd+9gy4wpXMrJBD+Dy3tqNm1BTP/B1GsTg6LSIvD03iTW\nzzmqeu8Wg5k9Brtg1GoUPh0WzYDWxf98lngWT+40VgXexl63U4pGicfp27cvQUFBLF++nAEDBvDQ\nQw/Rvn17GjRoUOSx0KZNmwC46667nM4dEhJCu3bt2LJlC5s2bWLoUNe11b766isAOnfu7OYrkUgk\nFZkLmRcYuXYkpzNOq9rbRbTj0+6fEuIdUsaeFQNTDswbBmc3Oto0OvsOYwkEo9Vi4fjWjez49UeS\nLznGHN6Iomho2OEOYvoPpnrDxkWOizuaysqvDyKEo22vl4XN3vaQAJ1GYcpDbbi3VdEP/5LSx5Oi\nMRl4F7tovC2RbQRLlxo1arBw4UKefvpplixZwpIlSwAICAjgzjvvZPjw4Tz88MNotdf6kiYkJABc\nF8dYFFfGxMU5dnbYt2/f1ZjGzMxMNm7cyK5du4iMjGTSpEk3+9IkEkkFYdelXYxaP4p0Y7qqfVCD\nQfy343/RayvgCYNLwfgtNBvg3pR5uRxYu4rdyxeRleyY0XwjOp2O5j37EHPvQIKrORd4iWczWf75\nAWwWR9lwRG9hjY8ZFLtgnP5IW/q2qOaW7xLP4zHRKIRIwb7TeNsihFgCLImJiXm6vH25VenRowfH\njx9n8+bNbNiwgb1797J582ZWrlzJypUrmT17NkuXLsVgcH1kUhT5+fkO1/bv3+9QXqdWrVps3LiR\nWrVqlXgtiURScVh4YiHvbnsXi80x4UVBYXTMaP7Z7J8VL+EFXAvGB7+Dpv2LPV12Wip7/1jM/jV/\nYMzJcTneoGiIvmcAbQc+iG+g61a6aZdyWDptP2aj1cF2Wmdlua9dMOq1CjP+0Y7ezSKK7buk9JB1\nGiWVDo1GQ5cuXXjjjTf47bffSEhIYOXKlVSrVo01a9bw+eefXx1brZr9yfTChQsu570yJiwszMH2\n2GOPIYTAZrNx6dIl3n//fS5cuED//v3Jzc310CuTSCTlgdVmZdKuSby55U1Vweir82Vaz2k81vyx\niisY5w71iGBMiTvPyi+mMGvkk+xY9KtLwehrMnNHq/Y8893PdP7niGIJxqzUfBZP2Ud+jtnBdlFr\nZZGfCZsCXloNXz4qBWNFotg7jYqi+ABhwCUhhEnFbgAigMtCCMetGkmxKYukklsJRVG4++67ee+9\n9xgxYgRr167lpZdeAuzxhmfOnGHNmjUOmdWFSUtLY/fu3QC0a9fO6VoRERGMGzeOtLQ0Jk6cyBtv\nvMEnn3zi2RclkUjKhBxzDmP/Gsv6uPWq9ki/SKb1mkajkEZl61hxMWbDvKFwTiVhR6ODB2dDU+fJ\nlUII4o4cZNeSBZzeU7xqEME5+TTy8iNmwif4NCr+e5OXZWLxlH1kpxkdbEkaG7/5mbAo4KXT8NWj\n7ejeOLzYc0tKH3d2Gt8EjgH+Rdj9gKPAuJt1SiIpCQEBAYD9A/AKzzzzDACzZs0iMTGxyHsnTpyI\n0WikcePGtG3btljrvfnmm4SFhTF9+nTOnDlzE55LJJLyID47nkf/eLRIwRgdFs28e+dVbME498Ei\nBKMehn7vVDDabFaObd3EvNf/w/x3XiuWYAzPyKHjyYvc27k3nX76xS3BaMqzsGTaftITHU9n0jU2\nfvE3YtSAQafh68dipGCsgLgjGvsBa4QQqWrGgutrgMpXL0ZSKVixYgULFizAbHY80sjOzmby5MkA\ndO3a9er1zp078/jjj5Oamsp9992nmuTyxRdfMGHCBLRaLVOmTCm2PwEBAYwZMwaz2exQ+FsikVRs\n9l3ex8PLHuZEmnoJr/71+jOrzyyq+FQpY8+KyRXBeH6Lo+2KYGxyr+qtZmM+e1cu5ZuX/sXSyR9y\n6ZTzMmYamyAqJZOuR8/TMcdK9LTPqDZ2DBovr2K7m5dtYtGUfSSdz3KwZSuC+X4mcjTgo9fy7ePt\n6dLQMUxIUv64kwhTB/jTxZjjgKw/IikRjz/+eJG2GTNmcPToUUaNGkVISAhdunShYcOG6HQ64uLi\nWLZsGenp6cTGxjJy5Mjr7v3iiy8wm83MnTuXxo0b069fPxo2bEhOTg7r1q3j4MGDaDQapk6dSp8+\nfdzy+dlnn2XixInMmTOHsWPH0rRp05K8dIlEUoYsObWEt7a8hdnm+AAK8GLbF3mqxVMVM34RCgTj\nA3B+q6NNo4dhP0Djfg6m3MwM9q5Yyr5Vy8jPUm+HWBi9xUqtlEzqJGdgsFgJ6H0X1d59F12Ie6WG\nMlPyWDJVfYcxXxH86m8kQyvw9dLyzePt6Vivggp1iVuiUQ/YXIwRgHfJ3ZHczsyePbtI2+TJkxk+\nfDiBgYGsXr2a/fv389dff5GdnU1wcDDR0dE8+OCDjBgxAq8bnn4NBgNz5szhscceY+bMmWzZsoXF\nixdf3bFs0qQJc+bMcRrLWBQ+Pj6MGzeOF1544WpijkQiqZjYhI3pe6cz88BMVbuPzofxncfTq3av\nMvbMDYxZBTuMxReMaQkX2b3sdw6t/xOL2SElwQEfo5m6yRnUTM1EZxMoPj5Ue3scQUOGuC2kUy5m\ns2TqPnIyHNc1I1jgZyJJK/Dz0vLdkx1oXyfUrfklZYsi1Cpqqg1UlANAnhCig5MxOwF/IcRtvd0S\nExMjdu3a5XLckSNH5M5UORIfH0+HDh3IyMhg9erVdOzYsbxdKhfkz6HkdiDXnMvrm15nzfk1qvYI\n3wim9ZxG0yoV+HfBmAVzHoAL2xxtWi8Y+gM07nv1UvzxI+xcvICTu7ahWj37BgJzjdRLSqdaevbV\n2DXvFi2I/PgjDHXruu1u/Il0ls34G1OeY0a6GcHvfibO6m34G3TMfrID7WpXwGLptwmKouwWQsS4\nGufOTuNiYKyiKK8KIT5SWXAs0BZwsEkkFZHIyEiWLl1Kly5d6NevH2vXrqVNmzbl7ZZEIvEwl3Iu\n8cLaFziSekTV3rJqS6b0mEKYbwWOoyumYBQ2G6d272DnkgXEHztcrKnDMnOodzmd0Jx8ru4jKgpV\nnnmGsJHPoZSgVerpfUmsmnUIq8XxgDJfse8wXtTZCPDW8cNTsURHue5dLSl/3BGNE4F/AOMVRRkK\nrAIuAjWAPkA0cB4pGiWViOjoaJYuXcq6devYsmUL0dHRFTeOSSKRuM3B5IO8sPYFkvLUu5n0rdOX\n/3X6H966ChxZlZ9pj2G8sN3RpvWCYXOw1OnB4TUr2LV0IWkJF11OqdgEkelZ1EvKICD/+qNjXfXq\n1PhoAr7t25fI3UMbL7Jh3jHVzc2sghjGZK0gyEfPnKdiaVnTdW1HScWg2KJRCJGmKEp3YB7QEfuu\nooCrDyZbgOFCiDRPOymRlCbdunWjW7du5e2GRCLxMCvOruCNTW9gtDrWBAR4NvpZ/t3q3xX7QTE/\nE+YMgbgdjjatF3n9v2b/gTT2TnyS3Az11oeF0QmolZRGnaQMvC2O3VgC7+lHtbffRhsY6LarQgh2\n/3GW7YvVS5ClaGz86m8iUyMI8dXzw1OxtKghBWNlwq02gkKIs8CdiqK0xS4cg4F0YJsQYo/n3ZNI\nJBKJxD2EEHyx/wtm7J+hajdoDbzX+T361umraq8wOBGMGdYAdgc9xoFPZmMxqoviwvj6+FLr/CWi\nLiWjtzluAWr8/Kj25n8JHDCgRCLaZhNs+vk4Bzao73LGa20s8DOSp4HwAANzR8TSMCLA7XUk5UuJ\nek8XCEQpEiUSiURSoci35PPfzf9lxdkVqvYwnzCm9pxKi6otytgzN8nPKBCM1xfcTszzZ2daLY5n\nVkWI/S6nqVojivrZRkI2bkVTRC6MT5s2RH78EV41a5bIVavZxupvD3Nqz2VV+2mdlcV+JswK1Aj2\nYe6IWOpU9SvRWpLypUSiUVEUP6AR9kxplWaXEolEIpGULUm5Sbyw9gUOphxUtTcNbcrUnlOp5let\njD1zk/wM+GEwXLRX4RACzuaEsDOlJhdyrySMOM+GrtWiNS0bNkPz+UysRXXD0mqp+uz/UfVf/0LR\nlUgOYMqzsPyLA1w8ph6ZdkhvYYWvGZsCdav6MWdELDWCfUq0lqT8ceunRFGUmsAUoD+gxf5Tqyuw\ndQa+Ap4VQqz3rJu3LkKIih1PI7mlKW7JLYmkonMk5QjPr32exFx1gXRXrbt4v/P7+Op9y9gzNykk\nGK1C4UhGGLtSa5JidL0zp2g0NL6jC+363oeyaCmp77yPtYjfcX1UFJEfTcD3JipG5GaaWDJtH8kX\nslXtOwxmNnhbQIHGEQH8MKID4QEVOOFI4pJii0ZFUaoD24EI7OV3woE7Cg3ZXnBtGLDecy7eumi1\nWsxms0MxaomkrDCbzWi12vJ2QyK5Kf489yevbXqNPEueqv3plk8zss1INIo7nXPLgfwM+GEQxvP7\n2J9ek72pkWRbDC5v0xu8adnzbtrdOxBDVjYXX34Z42H18kIAQQMHEvHGG2j9S35EnJGUy+Kp+8lM\nUn/P13ub2eltr8/YskYQ3z/ZgRA/+V1X2XFnp/Et7KKwtxBinaIob1FINAohzIqibAQ6edjHW5aA\ngAAyMzOpWrVqebsiuU3JzMwkIEAGo0sqJ0IIvj74NVP2qPeM12v0vHPnO/Sv37+MPXMDcz6c3QjH\nV5D190p2n9dxIL0DJpvrr2e/4BDa9O1P6973YPDzI/3nnznz4QREfr7qeE1gINXffovAe+65KZeT\nzmexZNo+8rIc2zBaEazwNXPYy56Z3b5OCF8/3p5Ab/drPUoqHu6IxnuAxUKIdU7GnAe63JxLtw+h\noaGcP38egMDAQPR6vTyqlpQ6QgjMZjOZmZmkpaVRq1at8nZJInEbo9XI21veZunppar2UO9QpvSY\nQnR4dBl7VgwyE+DESji+Ek6vJzPXyo7kKA5m1MIqXO+GhkbWJKb/YJp26YFOr8eSmkrcK6+Sva7o\nr2ff9u2J/GgC+urVb8r1uKOpLP/iAOZ8x3I9JgSL/Uyc0dsLenduUJWv/tkOX6+SxUtKKh7u/E9G\nACdcjDEDMiWqmBgMBmrVqkVqaipnz57FanX8JZRISgOtVktAQAC1atXCYHB9/CWRVCRS8lJ4ad1L\n7Evap2pvGNKQ6T2nE+kfWcaeFYHNBgl77SLx+ApIsGc9p5u82Z4cxeGMcGy4Fos1m7Ygpv8g6rVp\nj6Kxj8/euJH418ZhTU5Wv0mnI+zFF6jy5JMoNxmKcmJXImu+O4zN4hgnmasIFvgZSdDZbXc1jWD6\nI23w1svwl1sJd0RjKhDlYkwj4FLJ3ancKIrSH+jfoEGDYt9jMBioXr061W/y6U8ikUhuB46nHef5\nP58nPide1d69Znc+7Pohfvpy3r8wZsHp9XaReHwV5FwrR5Nq9GF7ShRHMsIROD9dUhQNDTvcQUz/\nwVRv2PjqdZvRyOWJk0j74Yci7/WqU4fIiRPxadH8pl/OgfVx/PXzcdWk7UzFxi/+JlK1duN9rarz\n6bBo9NoKHkMqcRt3RONmYICiKNWEEA7CUFGUhkBfYI6nnKtsCCGWAEtiYmKeLm9fJBKJ5FZjw4UN\nvPrXq+RaclXtTzR/ghfbvohWU067W6ln4MQqu1A8uwms17fnSzb6sj05imOZYS7Fok6vp3mPu4m5\ndyDB1a7fVMg/dpz4l1/GeKLow7/goUOJGDsGje/NZYsLIdix5Ay7lp9VtSdpbPzqbyS7QB8OjanJ\n+MGt0GpkqNWtiDui8WPgfmCDoigvAb5wtWZjV+BTwAZM8rSTEolEIrl9EULw/eHvmbRrEkJlq0un\n0fFmxzcZ1HBQ2Tpmtdi7tRxfYT96TjqqOuxyvh/bk6M4nlUVXIhFLx8f2vYbQJt+A/ANvL7FnrDZ\nSJszh8sTJyFMJtX7tcHBVH//PQJ69SrRSyqMzWpjw7xjHN6coGqP01pZ4GfCWCAYH7+zDm/e1wyN\nFIy3LO70nt6uKMq/gM+BwpHHmQV/W4AnhRCHPOifRCKRSG5jzFYz/9v2PxaeXKhqDzYE82n3T4mp\nFlM2DuWmwsk/7YksJ1ZDftH9nhPz/NmaHMWpbNcVMrz9/Gl77/206dsfbz9/B7v58mUSxr1OzqZN\nRc7hd+edVB8/Hn1EePFeixMsJiurvj7Emf3qsZIndFaW+pmwFOjDZ7vX55U+jWUy5y2Ou72nvyko\nq/Ms9t7TVYAMYBswXQhxzPMuSiQSieR2JC0/jVHrR7E7cbeqvX5Qfab1mkZUgKtw+5tACEg+Dsf+\nsO8mXtgGwub0lvi8ALYl1eJMTqjL6X0CAml33yCi774XQxFHyVlr15Lw+htY09S7rih6PeEvjybk\n0UevJsjcDMZcM8tm/E3CyQxV+99eFlb5mBEF+vCVPo15rkfxY/kllRe38+CFECeAUaXgi0QikUgk\nAJxKP8XIP0cSlx2nau9UoxMfd/2YAK9SqDNqMdpjEq9kO6efK9ZtcbmBbEuuxbmcEJdjfYOCad9/\nMK1734PeW71Lii0vj8QJE0j/6eci5zE0bEDkxIl4N25c5Bh3yEk3snjqPlLjc1TtWw1mNhV0eQF4\n875mPNm5rkfWllR8ZPEkiUQikVQoNl3cxCsbXiHbrN6ebnjT4YyOGY1O48GvsKzEa0ksp9aBWV00\n3YgQcCE3iG3JtQr1hS4a/5BQ2g8YQstefdAbim6pl3foEPEvv4LpzJkix4QMH074y6PRFCE63SXt\nUg5Lpu4nK9WxOLhA8KePmb0Ge2k4RYEPB7dkWHtZ5/V2wp02gm2wd4CZK4TIKLjmB8zAniCTC0wQ\nQqiX5pdIJBKJxAlCCOYdncdHOz/CpnIErFN0vBb7GkMbD/XEYvZ6iVd2E+P3uH37uZxgtqbUJT7X\nMQbxRgKqhNHh/gdo0aM3OietY4XNRuo333B5ylQwO3ZcAdBWqULkB+/j362bWz47I/FMJkun7yc/\nR73LyzJfM8cKurxoNQqfDG3N/dE1PLa+pHLgzmPaGKCLEGJGoWvjgUeBbOzxjZ8oinJECLHKgz5K\nJBKJ5BbHbDPz4fYPmX98vqo90CuQT7p/Qmz12JIvYsqB0xvsIvHEKshSzwp2hhBwmsZsS67FpWT1\ndn2FCQqPoMPAoTTv1hOtznkrPfOlS8SPGUvu9u1FjvHv3p3q77+HrkoVt30vinOHUljx5QEsJkeh\nbkLwu5+JcwVdXry0GqY/0oa7m1fz2PqSyoM7ojEGuNqjSFEUPfAYsAPoDoQCe4EXACkaJRKJyxbG\nuAAAIABJREFURFIsMowZjN4wmu0J6mKpTmAdpvWcRp2gOu5Pnn6+YDdxJZz5C6xG9+dQNIiaHTmp\na8O2v1O4HHcRcC4YQ6pHEjtoGE06dUOrc/5VazOZSJ09m5TPv8CWq16DUjEYiBg7huCHHvJohvKx\n7ZdYO/sINptjKaMcRfCbn5HEgi4v3noNXz0aQ9dGYR5bX1K5cEc0hgOFI5JjgADgSyFEPhCvKMoi\n7AW+JRKJRCJxydmMszy/9nnOZp5VtcdWj2VSt0kEGYJU7Q7YrBC361rtxMslrALnHQQNemNrcDcn\n0gPZtmwZyef/dnlbaI0oOg4eRuM7uqBx0bZPCEH22rUkTvgI8/nzRY4zNG1KjY8/wuBGt7HisG/N\neTb/elLVlq6x8YufifSCLi/+Bh3fPN6eDnVdZ4RLbl3cEY3ihvGdC65tKHQtCbglHkEURekKvAy0\nAyKBJ4QQ35WrUxKJRHILsS1hG/9Z/x+yTFmq9mGNhzGmwxj0GufHuuSlw6m1dpF4YhXkpZbMobAm\n0KgPNOyDrUYMx7ZvZdv3P5N68YLLW6vWqkPHwcNoGHsnmmJ0pMk/fpzE8ePJ3brN6bjQJ58k7KUX\n0TiJg3QXIQRbF5xi72p1oXq5oMtLTkH1niAfPd8/2YHWUa4TfSS3Nu6IxvPYazNe4X4gTghxutC1\nSEC9kFTlwx84CHxf8EcikUgkHmL+sfl8sP0DrMLqYNMoGsa0H8MjTR8peoLkkwW7iSvg/FawWdx3\nQqOHul2gUV9oeDeE1sVqsXB08wa2T36etAT1/taFCa9Tn45DhtEgpmOxaiRa0tJInjadtJ9+AlvR\n9R514eFEfjgevzvvdOslucJqtbHuh6Mc2+bQDRiA8zorC/1MmApOwKv6e/HDU7E0rR7oUT8klRN3\nRON84B1FUX7FHsxxBzD5hjFNgVMe8q1cEUIsB5YDKIryXfl6I5FIJLcGFpuFibsmMvfIXFW7v96f\nSd0mcWeNG8SSxWQXh1eynVNL+FXjFw6N7rYLxXrdwWCv82i1mDn050p2/D6fjMuJLqep1qARHQc/\nRL227YsVYyjMZtJ++pmk6dOxZagXzQZAqyVk2DDCXngebbBnd/bMRisrZx7k3MEUVfsxvZVlvias\nBS+nepA3c0fEUi/MdXa45PbAHdH4KfZ4xcEF/94HvHvFqChKXaA99oxqt1EU5QGgGxANtMYeLzlX\nCDHcyT01C3zoiz17OwH4HXhHCHGr7HhKJBLJLUGWKYtXNrzC5vjNqvaa/jX5rNdn1AuuZ7+Qk2xv\n1Xd8hf342Zipep9Lqre2i8RGfaB6Gyi0I2gxmTi4bjU7Fv9KVnKSy6kiGzfjjiEPUbtVm2InpGRv\n3kzi+PGYTjoXur53dCTitdfwbtSoWPO6Q362maWf7SfxjPp7uNfLwp+FurzUCvVl7ohYokLVu9RI\nbk/c6T2dDXRSFKVFwaXDQlxXSEtgF5S7SujLG9jFYjb2hJsmzgYrilIf2II9QWcRcBToALwI9FUU\npZMQQv1xSiKRSCRlyoXMC4xcO5LTGadV7TERMXza7ROCM+Lgr4n2HcW4ndi/WtxE5wP1exTEJ94N\ngZEOQ8wmIwfWrGDn4t/ITnMdAxnVrCUdhzxMVPOWxRaLpnPnSJzwEdlr1zodp4+KImLsGPx79iyV\n3s1ZqfksmbqPtEvqmdmbvM1sNVzr8tIg3J+5I2KJCPRM0XDJrUNJ2ggeLOL6WeDsTfgyCrtYPIl9\nx3Gd8+HMwC4YXxBCTLtyUVGUTwrmeh/4d6Hr7wGvu5izhxBivdueSyQSiaRIdl3axaj1o0g3pqva\nB4d34A2zP/rPO0HmxZItEhRlF4mN+kKdzqD3UR1mzs9n/+rl7FyygNwMdX8KU7tVGzoOHkbNpi1c\njr2CNTublC++IGX290UW6AbQ+PpS9dn/I+Sf//RookthUuKzWTJ1PznpjqWGbAjW+JjZb7gWV9qs\neiA/PNWBKv6GUvFHUrmpMG0EhRCFa0A6HVuwy3g3dpH62Q3mt4BngEcVRRkthLjSC2oyMMeFG0XX\nPJBIJBKJ2yw8sZB3t72LRSVRRQFeTsvm0TO/4v7+mgJRHa4JxfBm9t52RWDKy2XvymXsXrqQvCzX\nx9x128TQcfBDRDZyeuh1HcJmI2PhQi5/OhlrcrLTsUGDBxM+6iV0YaVXcCThZDrLZvyNMdfxvbcg\nWOpr4oTXtQPDNrWC+e6JDgT5uMhWl9y2VBjR6CY9Cv5edcMROUKILEVRNmMXlR2BPwuuJwPOf4sl\nEolE4hGsNiuT90zmu0Pfqdp9bTY+vpxM1zzXXVWuYgiEBr3sIrFBb/Bz3RUlPyebvSuWsGfZIvJz\n1HtZF6Z+TEc6Dh5GtfoNi+8XkLtnD4nvf0D+Ied1IX2io4l4fRw+LVu6Nb+7nPk7mZUzD2I1O2Zo\nGxEs8DcRp7tm61gvlFmPtcffUFllgaQsqKw/HY0L/j5ehP0EdtHYiALR6C6KovgDVyqpaoBaiqJE\nA6lCCIcdSUVRnsG+w0mtWrKBu0QiuX3JMecwZv1oNhSR8BJptjAtMYlGTo5ur1KlwbUkllp3gLZ4\nu2B52VnsWb6IPcsXY8pTj+W7iqLQqMOdxA4eRnidesWa/wrmhAQuT5xE5rJlTsfpqlUjfPRoAu+7\nt1TiFgtzeHM86+ceQ6h0eclWBL/6G0nSXrP1aBzG58Pb4a13XV9ScntTWUXjldYARdUtuHL9ZuoV\nXNc2EXin4M9s4PEbBwshvgK+AoiJiSlB5LZEIpFUclJPE3/oV0ae/pkTmFSHROcbmZyYRJWiahRq\ndFD7zoLaiX2gqntdUHIzM9i9dCF7Vy7DnJ/nfLCi0OTOrsQOGkrVqNpurWPLyyPl629ImTULkV/0\nbqliMFDlqaeoMuIpNL6lm4kshGDPynNs+1092ShVY+NXPxMZhQRjvxbVmPJQG7x0rmtMSiSVVTSW\nOgUJMaX7OCiRSCSVGasZLmy/2rJvX9ZZXowII7WI9nkDsrJ5KzkVh5QP3yr2LOdGfaB+T3sLPzfJ\nSU9j55IF7F+9HIvReX9pRaOhaefuxA4aSmhkTbfWEUKQ9ccfJH48EUtCgtOxAf36EvHyy+hr1HBr\njZIgbIJNv5zg73VxqvYErY3f/IzkFdKGg9vW4KMhrdBppWCUFI/KKhqv7CQW9cly5brr1DiJRCKR\nFJ/cVDi5xi4UT66BfPvH8RJ/X96qHoFZ5ehVEYIX09J5MiPr2pN4RMtrSSw12kIxWu+pkZWazM7F\nv3FgzUosZvXdzStotFqade1F7MAHCa5W3e218g4eInH8ePJ273Y6ztCsKdXGjcM3JsbtNUqC1WLj\nz+8Oc2LXZVX7GZ2VRX4mzIX+a/4RW4v/3d8CjUbujUiKT2UVjccK/i6qAuqVCOaiYh4lEolEUhyE\ngKSjV3cTubAdCuUf2oBpIUHMClZ/hvex2RiflEIvk7AfN1+pnRgcdVNuZSZdZseiXzm4bhVWi/MW\nghqtjpY9e9N+wAMEhUe4vZYlOZnLkyeT8dsC+/tRBNrQUMJGvUTw4MEoRey2ehpTvoU/vjhA3FH1\nfhaH9Rb+8DVjK6QNn+laj9f6NSn12ErJrYfHRKOiKF0BC7DtxozmUuBKrOHdiqJoCq+nKEoA0AnI\nBZx3gpdIJBKJI+Z8OLfpWsu+dPVqZLmKwriwKvzppx6rF2EVTAvpSNM7H4S6XcHr5mP60hMvseP3\n+Rza8Cc2q2Pf6sJo9Xpa9epL+wFDCKhS1e21bCYTaT/8QPKMz7Hl5BQ9UK8n9NFHqfp//0YbEOD2\nOiUlN9PE0un7STqfpWrfZbCwztt8XaDVS3c15MVeDaVglJQIT+40rsdeuv+soigfAd8KIZyfFZQQ\nIcQpRVFWYc+Qfg6YVsj8DuAHfFmoRqNEIpFInJF1CU6ssgvFU+vA7Pzj85JWywsRYRwxqBelbhlY\nnyl3f0WYX7hH3EuNv8iO3+dzeOM6RFFJNAXovAy07t2PmP6D8Q8JdXstIQTZ69aTOOFDzOecl+/1\n796d8DGvYqhb1+11bobM5DwWT9lHRpJ6ss8GbzM7CnV5AXj9nqY83dW97HCJpDCKcLLV7tZEirIe\n+49ncyAUuCSEcOzdVPT9A4GBBf+sBvQBTgMbC64lCyFeLjT+xjaCR4BY7DUcjwN3lnUbQUVR+gP9\nGzRo8PSJEydKbZ35uy7Qu2kEIX6l00FAIpHcBthskLCvQCiugPi9xb71gJcXL0SEkaxTP4LtV6cf\n73Z6F2/dzbehS4k7z7YFP3Nsy0ZcHWLpDd5E972PmHsH4htUsuIZxpMnSRz/ITmb1csFXcGrXj0i\nXhuLf5cuJVrnZki6kMXSafvJzXTcl7EhWOlj5qDh+l3Y9wa2YHhH9zLEJbcPiqLsFkK4DML1mGi8\nYfFWQFchxHQ37nkbezeXojgnhKhzwz1RwLtAX6AKkAAsBN4RQqgHeJQBMTExYteukrbgds6ifRd5\n8ad9RIX6MOuf7WlcreyOQiQSSSXHmA2n19tF4olVkJ3o9hQr/Hx5I6wqxiJON5+Lfo5/tfrXTR9/\nJp07w7YFP3N8+2ancYQAXj6+tO3Xn7b33I9PQGCJ1rOmp5P02QzS5s0DJ8femsBAwkaOJOThh1D0\nZd855eKxNJZ//jemfEcfzQgW+5k4rb8mrjUKfPxAa4a0cy9LXHJ7Ua6i8XantETjgbgMHvhiC0aL\n/QPBz0vLJ8Oi6dO8msfXkkgktwhp567FJp7dCNYSRA0pWkStjnxeJYzP0/aoDjFoDbzX+T361ul7\nU+4mnj7JtgU/cXKn65B0g58f7e4ZSJt+/fH28y/ResJiIW3+fJKnTMWaUVTpX0CjIXjYUMJeeAFd\nSEiJ1rpZTu25zKpvDmGzOH5v5ymCBX4m4gt1edFrFaY+1IZ+Ld3PFJfcXhRXNFbW7OnbjqQsI8/8\nsOuqYATIMVn51w+7GXVXI57v2UCWTpBIJGC1QNzOa9nOSUdKNo9PiL1VX6M+5NfpzBt7JrLy7ErV\noWE+YUztOZUWVVuU2O2EE8fYtuAnTu/Z6XKsd0AgMfcOJLrPfRhuomB2ztatJH4wHqOLcCLf2Fgi\nxr2Gd+PGTseVJgf/usiGH4/ZMwduIFOx8au/iZRCRbsNOg1fDG9HjyaeiSmVSKAEorHg6PkRoCng\nJ4S4q+B6HaADsLo8j4ZvVc6n5pJrUj8y+XTNcY5eymTS0Nb4esnnAInktiMvDU7+aReJJ1fb/10S\nwppeq51Ysz1odSTlJvHC2hc4mHJQ9ZamoU2Z1nMaEX7ul7IBiDt6iG2//cS5v13HVPoGBRPTfzCt\ne/fDy9unROsBmC5c4PJHH5G1eo3TcfqaNQl/9RUCevcut2xjIQQ7l55h57KzqvZkjV0wZmmuCUZf\nLy2zHovhzvruZ4xLJM5wS2EoivIuMA57L2a4/plHA/wIvMT12cwSD9CudgiLnuvEiO93cfJytoP9\nj4OXOJOcw8x/xhAVWrqtqiQSSTkjBCSfuLabeH4rCOflZ1TRekGdLgW9ne+GkDrXmY+kHGHk2pFc\nzlUvGt27dm/e6/Qevnr3PnOEEMQdPsDW337iwqG/XY73Cwmlw4AhtOzVB72h5Mk11uwcUr78ktTv\nvkM46Xut+PpS9V//IvTxx9AYDCVe72ax2QR//XiMQxvjVe0XtVYW+JnIL9TQJcBbx3dPdKBd7fI5\nQpfc2hQ7plFRlIeAecBKYAwwDBgrhNAWGrMdyBRC9C4FXysNpZkIk5VvZtTP+1hzRP1DPNTPixn/\naEvHelVKZX2JRFJOWExwbvO1+MS0MyWbxz+ioGVfX6jXHQzqsYBrzq1h3KZx5FnUS7o80+oZnot+\nDo1S/BZ0QgjO/b2XbQt+4uLRwy7HB1QJo8P9D9CiR290XiWvFiFsNjIWLebyJ5OwJiU7HRt0//2E\n/ec/6CPK91jXYray+pvDnN6bpGo/pbOy2M+EpdAGaKifF98/2YEWNdxvwyi5vSmNmMYXgJPA/UII\nk6Iog1TGHAG6uzGnxE0CvPV89WgMk1Yf47N1pxzsqTkmhs/azlsDmvOoLK8gkVRuspOulcQ5tQ5M\n6kWcXVI92i4SG/eFaq1BU7TQE0Iw68Aspu6dqmr30njxTqd3uK/efcVeXgjBmb272PbbTyScPOZy\nfGBYBLGDHqR5t15odTeXoZy7dy+JH4wn/8ABp+O8W7ei2uuv49Oq1U2t5wmMeRaWz/ib+BPqnXAP\n6i2svKHLS3iAgbkjYmkYIStqSEoPd0RjS+A7FwW744GSBbbcAhSq01iq62g0Cq/0aUKTaoG88ut+\n8s3X1y6z2AT//f0gRxIyebt/c7x0shm9RFIpEAIuHbi2m3hxN6qZD67Q+0L9ntda9gUUr8KC0Wrk\n7S1vs/T0UlV7qHcoU3pMITo8uljzCZuNk7u3s+23n7h8xvEh90aCq1UndtAwmnbujlZ3c/HZ5kuX\nuDzpEzKXLHE6ThceTvjLowm87z4UJ2K6rMjJMLJk2n5S4hzDkAC2G8z85X190e4awT7MezqW2lX8\nyshLye2KO7+VCvY2o86IAPJL7k7lRgixBFgSExPzdFms1791JHWr+vHM97uIz3B82+dtP8/JxGxm\nDG9LVf/yi8uRSCROMOXCmb+uxSdmqcevuSSoln0nsVEfqN0Z9O7F/iXnJfPSupfYn7Rf1d4opBHT\nek4j0t91zwZhs3F8+xa2L/iJpPNnXY4PjaxJx8HDaHxnVzQ32bPZlp9PyjffkDJzFiJP/WgdQPHy\nIvTJJ6j69NNo/CqG2EpPzGXx1H1kpah/ja71NrHb+/rY1XpV/ZgzIpbI4JInBkkkxcUd0XgCuLMo\no6IoGqAzcOhmnZIUnxY1glg0sjPPzt3NzrOOGZM7zqZy//TNfPXPdjSPlHEuEkmFICOuYDdxJZzZ\nAJYSPGsrGoiKvZbtHNYESpjheyz1GM+vfZ6EnARVe/eo7kzoMsFlwovNZuXY1k1sX/AzKXHO2+8B\nVI2qTezgYTTq2AmN5ubEohCCrJUrufzRx5jjnQvvgD59CH/lZbxqVpyC1xeOprJq1iHysx0TdKwI\n/vA1c8TresHYOCKAOSNiCQuQmwKSssEd0TgfeE9RlNFCiEkq9nFAA2CKRzyTFJuwAANzR3TkrcUH\n+XHHBQf7xfQ8Hvh8KxMfbM29rWSRV4mkzLFZ7UfNx1fA8VWQ6Dy+rkgMQdDwLrtIbHAX+LrfV/lG\n1l9Yz5i/xpBryVW1P9HiCV5s8yJaJ6LOZrVyZNN6ti+cT1rCRZdrhtWpxx2DH6JB+44eORLOP3KE\nxPc/INdFAqKhcWMixo3DL7bDTa/pKYQQ7F11nm2/n1JtfGNCsMjPxFn99Qd9rWoGMfuJDrKdrKRM\ncSd72gfYDLQGdmEPtGkPfAp0AWKAbUA3IYSlVLytJJRm9rQzhBD8sO0c7yw5jNWm/v/6fM8GjLqr\nkSwELpGUNvkZcGqtfTfxxCrITSnZPFUbXdtNjIoFrWda1wkhmH1oNp/s/gShEjep0+h46463GNhg\nYJFzWC1mDm1Yy45Fv5CReMnlmtXqN6TjkIep17a9R+oeWlJSSJoylfRffnHaalAbEkLYSy8R/MAQ\nlJs8/vYkxjwLa2cf4fQ+9QzpXEXwm5+RS7rrX1uHOqF8/XgMAd5l38ZQcmtSKm0EFUUJwr6T+A+g\n8G+eDZgLjBRClDC979ahvETjFbaeSuHZubtJy1WvQ9a7WQSfDovG3yALgUskHiXlVMFu4go4twVs\nJXh+1uihTie7SGx4N1Sp73E3zVYz/9v2PxaeXKhqDzYEM7nHZNpFtFO1W8xmDq5bzY5Fv5CVrC54\nChPZqCl3DHmI2q3bekQsCpOJ1LnzSP7sM2zZ6gkjAOh0hP7jH1R97lm0gSXrSV1apFzM5o8vD5Bx\nWT3uMkOx8Yu/iTTt9d/RXRpW5atHY/DxqjjiV1L5KdXe04qihGLfZawCZAA7hBCuPzluE8pbNAJc\nSM3l6e93cfSSuoZvFOHPzH/GyGw7ieRmsJrthbWvZDunnCzZPL5VC3YT+0C9HuBdegInLT+NUetH\nsTtxt6q9flB9pvWaRlRAlIPNbDJy4M+V7Fz8G9mprndOazZrwR1DHiaqeSuPdVTJ3rCBxPEfYjp7\n1uk4v65diBg7FkO9eh5Z15Oc2JnI2h+OYDGp55ae01lZ6msi94aT+97NIpj+SBsMOikYJZ6lVEWj\nxDkVQTQC5Bgt/Gf+PlYeSlS1B/nomfGPtnRqIFtNSSTFJifF3qrv+Ap76z5jZsnmqdayoBNLX4hs\n67R2oqc4lX6KkX+OJC47TtXeuUZnPu76Mf5e1xf8Nufns3/1cnYuWUBuhnrtwMLUahnNHYMfomaz\nkveivhHj6dMkjv+QnI0bnY7zqluXiLFj8O/WzWNrewqr1cbW306xf61j7PkVthvMbPS2IG7Q2ANa\nRzJpaGv02vIvCyS59ZCisRypKKIR7G2opq49weQ1J1TtWo3CG/c25fE765Rbb1WJpEIjBFw+fK0k\nzoUdlKh2os7b3oGlUR9o2AeCanjYUedsuriJVza8QrZZ/Th3eNPhvBzz8nUJL6a8XPatWs6uJQvI\ny3ItjutGt6PjkIeIbNTUY35bMzJInjGD1LnzwFL0cb8mIICqzz1L6COPoNxE95jSIifDyMqZB0k4\nmaFqt2hgmbeR416Ou48PtY/i/UEt0cpYdEkpURodYVAUxRd4CogGagJqUbhCCNHLnXklpYdGo/DS\nXY1oUi2A/8zfT67p+pINVpvgnSWHOZKQyf8GtpDHHhIJgDkfzm68JhQzit4ZckpgjWtJLHW6gFfZ\n94UXQjDv6Dw+2vkRNuEoSHSKjnEdx/FgowevXjPm5rD3jyXsXr6I/GzXYer1Y2LpOGgY1Ro08pzf\nVivpv/xK0pQpWNMcy4ldRVEIfvBBwl58AV2Vitk+Nf5EOitnHiQ3U703RppOsMDHSKrW8WHkiU51\nePO+ZvKhXlIhKLZoVBSlFbAKCOO6WvQOyK3LCkjfFtWpXcWPp7/fRVyaY+D1/F1xnLyczRePtiM8\nwL2iwBLJLUFmApwoqJ14ej2Y1UvQOEeBmjHXhGJEixLXTvQEZpuZD7d/yPzj81XtgV6BfNL9E2Kr\nxwKQl53FnuWL2PvHEoy5OS7nbxh7Jx0HP0R4Hc/GDeZs207i+PEYjzlvOejbvj0R417Du6nndjY9\niRCCv9fGsfm3k4giKloc1VtY4WvGrPJjMrJHA0bf3UgKRkmFwZ2SO2uBbsBbwPfARSGE1fldtxeF\n2gg+feKE+nFweZOaY+LZubvZdjpV1V49yJsvH21Hq5rBZeyZRFLG2GyQsPdaEkuCeicUl3gFQINe\nBdnOvcGvYsQIZxgzGL1+NNsvbVe11wmsw/Re06kdWJvczAx2L13I3pXLMOcX3UUFAEWh8R1d6Dho\nKFVr1fGoz6a4OC5/9DFZq1Y5HaePjCT81VcJ6HN3hRVUpnwL6+Yc5eSuy6p2G7De28Rug9VhG0ar\nUXitXxNGdKl4STySWxOPxzQqipILLBVCDL1Z5251KlJMoxpmq43/LT3M91vPqdoNOg0fPdCK+6PL\nNuZKIil1jFn2XcQrRbZz1L/QXRJaDxr1s+8o1roDdBUrhu5sxllGrh3JuUz13/GO1TsysdtEdHk2\ndi5ZwP7Vy7EYjU7nVBQNTTt3o8OgoVSp4ZhZfTPYcnJI/momqd9+izCpH+ECKD4+VH3maUKfeAKN\nd8U9EUm7lMMfXx4kLUF9tzZXI1jkayJO5xguUD3Im6kPt6F9nZsv3C6RFJfSiGnMBtQ/gSSVCr1W\nw7v3t6Bp9UDeXHQQs/X6BwejxcaLP+3jcEImr/ZpIoOvJZWb1DP24trHV8DZTWAtWpQUiUZnF4dX\nsp2rNvC8nx5iW8I2/rP+P2SZ1GMRhzUexnMNRrDnx5/5e80KLGbn74dGq6VZ1550GPggIdVc9512\nB2GzkblkCZcnTsKS5LxqW+CA/oSPHo0+IsKjPnia03uTWDP7MOZ89YO4i1ori/xM5KgkQfdqEs7E\nB1vLLi+SCos7onEtEFtajkjKnoc71KJBuD///mE3KTmOXxxfbjjN8UtZTHm4DYGy84CksmC1wIXt\n15JYkp3HxRWJT6i9uHajPlC/J/hU/JCN+cfm88H2D7CqRA5pFS2jG42k+iEzs6f9G6uTTGQAjVZH\nix530eH+BwkK97xQy9u/n0sffED+/r+djvNu2ZKIca/h26aNx33wJDarje2LT7NnZdE9t3d7WVjv\nY8Z2w3O4TqMwtl8Tnupct8Iet0sk4N7xdD1gOzAJmCBkrZ4iqejH0zdyMT2PZ77fxaF49ZIa9cL8\nmPXPGOqF+avaJZJyJzfVXjPx+Ao4uQbyXdcSVCW8+bUklpox4KTfckXCYrMwcddE5h6Zq2qPMAXx\nWHZ3kncexGZ1Lha1ej0te/ah/YAhBFYN87iv5sTLJH0yiYxFi537EVaV8NGjCRowwCP9qUuT3EwT\nq74+xMVj6lneZgQrfc0c8XIU8zVDfJj+SFuioyr+Q4nk1qW02gg2BrYA6cA+7N1gbkQIIZ4q9qS3\nIJVNNALkmay88ut+lv6doGoP8NYx7eE2dG8cXsaeSSQqCAFJxwrVTtwGKuVkXKI1QN2u17qxBNfy\nvK+lTJYpi1c2vMLm+M0OtoAcHXeci6TGeS3C5vz90XkZaN27LzH9h+Af4vl4OpvRSOq335H81VeI\n3KIz0xW9ntAnnqDKM8+g9a/4Hasunclg5VcHyU5TjwlN09j43c9Esko5nb7NqzHhgVYE+ciTHEn5\nUhqJMDWB1UBjF0OFEKJyPJ6XEpVRNIK9PMSM9aeYuOoYaj8WGgXG9mvC013qySMUSdljMdpjEq9k\nO6eXMMTav9q13cR63cCr4guToriQeYGRa0dyOuP0ddeDsvS0OhVEvQQ/FBcf8XqDN9EmWJ8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Uf/qVFRXMeaZelUHFPP/NYpguXuBop0p4/oNQo8cmkU914ciVaeeEgkqrSlEaa9njzS3Fs2wtil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\nEwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f1d366f5550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(10,8))\n",
    "plt.plot(n, chol_time, linewidth=5, label=\"Cholesky\")\n",
    "plt.plot(n, qr_time, linewidth=5, label=\"QR\")\n",
    "plt.plot(n, svd_time, linewidth=5, label=\"SVD\")\n",
    "plt.plot(n, cg_time, linewidth=5, label=\"CG\")\n",
    "plt.plot(n, np_time, linewidth=5, label=\"Numpy\")\n",
    "plt.plot(n, lsqr_time, linewidth=5, label=\"LSQR\")\n",
    "plt.legend(loc=\"best\", fontsize=20)\n",
    "plt.xscale(\"log\")\n",
    "plt.yscale(\"log\")\n",
    "plt.xlabel(r\"Dimension\", fontsize=20)\n",
    "plt.ylabel(r\"Time, sec.\", fontsize=20)\n",
    "plt.xticks(fontsize = 20)\n",
    "_ = plt.yticks(fontsize = 20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Nonlinear least squares problem (J. Nocedal, S. Wright Numerical Optimization, Ch. 10)\n",
    "\n",
    "**Q:** what if we want model measures of a nonlinear function $f(x, w)$?\n",
    "\n",
    "**A:** we can use methods of nonlinear optimization!\n",
    "\n",
    "**BUT!** We have not analytical solution. Therefore, we have to use iterative methods!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Gauss-Newton method\n",
    "\n",
    "$$\n",
    "S = \\frac{1}{2}\\| f(X, w) - y\\|^2_2 = \\frac{1}{2}\\|r(w)\\|_2^2 \\to \\min_w\n",
    "$$\n",
    "$$\n",
    "S' = \\sum_{i=1}^m r_i(w)r_i'(w) = J^{\\top}(w)r(w), \n",
    "$$\n",
    "where $J$ is Jacobi matrix of the residual $r(w)$\n",
    "\\begin{equation*}\n",
    "\\begin{split}\n",
    "S''(w) = & \\sum_{i=1}^m r_i'(w)r_i'(w) + \\sum_{i=1}^m r_i(w)r_i''(w) \\\\\n",
    "= & J^{\\top}(w)J(w) + \\sum_{i=1}^m r_i(w)r_i''(w)\n",
    "\\end{split}\n",
    "\\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Newton method\n",
    "$$\n",
    "S''(w_k)h_{k+1} = -J^{\\top}(w_k)r(w_k)\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\left(J^{\\top}(w_k)J(w_k) + \\sum_{i=1}^m r_i(w_k)r_i''(w_k)\\right) h_{k+1} = -J^{\\top}(w_k)r(w_k)\n",
    "$$\n",
    "\n",
    "**Q:** what is changed if we add \"Gauss\" in the algorithm name?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Gauss-Newton method\n",
    "$$\n",
    "\\left(J^{\\top}(w_k)J(w_k)\\right) h_{k+1} = -J^{\\top}(w_k)r(w_k)\n",
    "$$\n",
    "\n",
    "**Note:** step size in this method is defined by line search method according to  any of the previuosly described rules."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Convergence theorem\n",
    "\n",
    "**Theorem.** Assume the residuals $r_i(w)$ are bounded, its gradients are Lipschitz, and the Jacobi matrix $J$ is full-rank. Then\n",
    "$$\n",
    "\\lim_{k \\to \\infty} J^{\\top}(w_k)r_k = 0,\n",
    "$$\n",
    "if the step size is chosen according to Armijo and strong Wolf conditions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Speed of convergence\n",
    "\n",
    "$$\n",
    "\\|w_{k+1} - w^* \\|_2 \\leq \\| (J^{\\top}J(w^*))^{-1}H(w^*)\\| \\|w_k - w^* \\|_2 + O(\\|w_k - w^* \\|^2_2)\n",
    "$$\n",
    "\n",
    "- It depends on the ratio between $J^{\\top}J$ и $H(w_k) = \\sum\\limits_{i=1}^m r_i(w_k)r_i''(w_k)$\n",
    "- The smaller $\\| (J^{\\top}J(w^*))^{-1}H(w^*) \\|$, the faster convergence\n",
    "- If $H(w^*) = 0$, then convergence is locally quadratic"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### The case fo the large residuals\n",
    "\n",
    "- In that case we can not discard $H(w_k)$\n",
    "- It indicates that the chosen model $f(X, w)$ is not adequate for given data\n",
    "- It requires *hybrid* algorithms, that work as Gauss Newton method in the case of small residuals and as Newton or Quasi-Newton methods in the case of large residuals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Pro & contra\n",
    "\n",
    "Pro\n",
    "\n",
    "- computation of the hessian $r''(w)$ is not required\n",
    "- from Jacobi matrix to hessian approximation\n",
    "- hessian approximation often is accurate\n",
    "- in the case of matrix Jacobi has full rank, it is guaranteed that obtained direction is descent direction\n",
    "- it has interpretation as linearization of the function $f(x, w)$ near stationary point\n",
    "\n",
    "Contra\n",
    "\n",
    "- hessian approximation may be inaccurate \n",
    "- if matrix $J^{\\top}J$ is close to singular, the solution is not stable, and even convergence is not guaranteed"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Levenberg-Marquardt method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### What problems do we have now?\n",
    "\n",
    "- Newton method gives **locally** but **quadratic** convergence\n",
    "- Singularity of hessian or its approximation leads to unstability of the solution\n",
    "- Gradient descent converges to stationary point from **any** starting guess, but only **linear**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### How to solve these problems? At least partially..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "fragment"
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   "source": [
    "**Idea:** separate spectrum of the hessian from zero by auxilliary item in the form $\\lambda I$\n",
    "\n",
    "Levenberg-Marquardt method:\n",
    "$$\n",
    "(f''(x_k) + \\lambda_k I)h_k = -f'(x_k), \\qquad \\lambda_k > 0\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Why is it a good idea?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- In the case of $\\lambda_k \\to 0$ the method works as Newton method\n",
    "- In the case of $\\lambda_k \\to \\infty$ the method works as gradient descent\n",
    "- The item $\\lambda_k I$ is an estimate of $H(w_k)$ in the Gauss-Newton method\n",
    "- If the estimate of the hessian $J^{\\top}J$ is sparse, then addition of $\\lambda_k I$ does not violate sparsity of the hessian estimate"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### But we still have one problem...."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "There are a lot of strategies of choosing $\\lambda_k$. The general ide a is similar to backtracking:\n",
    "\n",
    "- set some initial value\n",
    "- if the function deacresing is large enough, then the method is in the region where quadratic approximation is good. Therefore, we can decrease parameter\n",
    "- if decreasing is not enough large, then  increase $\\lambda_k$, resolve the problem, get new direction $h_k$ and check it"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Convergence\n",
    "\n",
    "- Proofs of convergence is difficult because of various strategies of defining $\\lambda_k$\n",
    "- It is guaranteed convergence to the stationary point if the curvature of the function is modeled adequately in every point"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Pro & Contra\n",
    "\n",
    "Pro\n",
    "\n",
    "- комбинация достоинств ранее изученных методов\n",
    "- автоматическая адаптация к градиентному методу и методу Ньютона\n",
    "- улучшенная версия метода Гаусса-Ньютона\n",
    "- сохранение разреженности оценки гессиана\n",
    "- стандарт де-факто для решения нелинейной задачи наименьших квадратов [[1]](https://en.wikipedia.org/wiki/Levenberg–Marquardt_algorithm#Implementations)\n",
    "\n",
    "Contra\n",
    "\n",
    "- нет универсального рецепта по выбору параметра $\\lambda$\n",
    "- скорость сходимости непостоянна и зависит от адекватности выбора $\\lambda_k$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Recap\n",
    "\n",
    "1. Least squares problem\n",
    "2. Algorithms for the linear case\n",
    "3. Algorithms for the nonlinear case"
   ]
  }
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