SecondOrderMethod.ipynb

{
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  {
   "cell_type": "code",
   "execution_count": 1,
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   "source": [
    "#math and linear algebra stuff\n",
    "import numpy as np\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": [
    "# Second order methods\n",
    "\n",
    "This notebook intends to give an extremely simple introduction to second order optimization methods."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Newton method\n",
    "\n",
    "Newton method is a really simple method originally designed in order to find iteratively the root of a function (where it is zero valued), we will see later some conditions that must be met in order to ensure the algorithm convergence.\n",
    "\n",
    "One of the first condition is simply to be sure that the function under consideration, for instance $f : \\underset{\\mathbb{R}^n}{x}\\mapsto \\underset{\\mathbb{R}}{f(x)}$, crosses the X axis of the graph at some point.\n",
    "\n",
    "The basic idea of newton method is to linearize the function at the current estimate, find the root of the linear approximation, and assign its value to the current estimate before starting a new iteration.\n",
    "\n",
    "The linear estimation $f_{lin}$ at point $a$, using Taylor serie gives:\n",
    "\n",
    "\\begin{align*}\n",
    "    f_{lin,a}(x) = f(a) + \\langle x-a, \\nabla f(a) \\rangle\n",
    "\\end{align*}\n",
    "\n",
    "Where we recall that $\\nabla f(a) =\n",
    "\\begin{pmatrix}\n",
    "\\frac{\\partial f(a))}{\\partial x_0} \\\\\n",
    "\\frac{\\partial f(a)}{\\partial x_1} \\\\\n",
    "\\vdots \\\\\n",
    "\\frac{\\partial f(a)}{\\partial x_{n-1}}\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Now, equating $f_{lin,a}(x)$ to zero leads to:\n",
    "\n",
    "\\begin{align*}\n",
    "    f_{lin,a}(x) &= 0\\\\\n",
    "    f(a) + \\langle \\nabla f(a),x-a\\rangle &= 0\\\\\n",
    "    \\langle \\nabla f(a),x\\rangle -\\langle \\nabla f(a),a\\rangle &= -f(a)\\\\\n",
    "    \\langle \\nabla f(a),x\\rangle &= \\langle \\nabla f(a),a\\rangle -f(a)\n",
    "\\end{align*}\n",
    "\n",
    "This simple linear equality defines an hyperplan onto which the current solution can be projected in order to get the next iterate.\n",
    "\n",
    "Here is a simple overview of the process for a 1D case:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
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   "outputs": [
    {
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8ORfKufriC7jhBqhY8b95x2OP89qy1xhebXiyLjcL5Ty5TUfkIiJyiYQEGDAABg+GxPV6\n4IqBPHnXkzx868PuBScXUI9cREQu8dVX0KcP/Pjjf4X858M/88jYR9jcdjO3Zrv1iu+XlNHlZyIi\ncs2shf79oVevC4/GuyzqQudHO6uIBxgV8hCg3pNzypUzypNzoZirRYvg5EmoW/e/ecv2LGPdH+vo\n/FjnFK0zFPMUKFTIRUTkAuePxtN4K0R8QjydFnRiaJWhZE6f2d3g5BLqkYuIyL+io+GFFyAmBtJ5\nT4f+6MeP+HTTpyyPWK4Ho/iYriMXEZFrMmAA9OjxXxE/dvYYbyx7g7mN56qIBygNrYcA9Z6cU66c\nUZ6cC6VcrV4N27dD06b/zesf3Z+n7nmKh2596JrWHUp5CjQ6IhcREcBzNN6tG2TI4JneeWgn4zaM\nY0vbLe4GJlekHrmIiLBxI1SvDj//DJm957M9Pf1pSt1Wih5le7gbXBhRj1xERFKkf3949dX/iviS\n3UvYdHAT0+tPdzcwuSr1yEOAek/OKVfOKE/OhUKutmyBFSugdWvP9LmEc3Rc0JGhVYaSKV2mVNlG\nKOQpUKmQi4iEuQEDoHNnuO46z/TYdWO5IfMN1Ctcz93AxBH1yEVEwlhMDJQrB7t3Q9ascPTsUe4d\neS/zm8yneJ7ibocXdgL2XuvGmDTGmHXGmK+807mMMQuNMT8ZYxYYY3L4Iw4REbnQgAHQsaOniAP0\nXd6X2gVrq4gHEX8NrXcAtiWa7g4sttbeCywFdErkNVDvyTnlyhnlyblgztXOnTB/Prz8smd6x6Ed\nTNo4if4V+6f6toI5T4HO54XcGJMXeAoYm2h2HWCi9/uJwNO+jkNERC40cCC0bw/Zs3umX1n4Cl0f\n70rurLndDUySxec9cmPM58AAIAfwirW2tjHmiLU2V6LXHLbWXp/Ee9UjFxHxgd274ZFHYNcuyJkT\nFv68kLZz27Kt3TYypsvodnhhK+B65MaYGsBBa+0G4EqBqVqLiPjRoEHQrp2niJ9LOEenBZ14u+rb\nKuJByNc3hHkcqG2MeQrIDGQzxkwGDhhjcltrDxpjbgH+vNwKIiIiyJ8/PwA5c+akWLFiVKhQAfiv\n5xLu0+fnBUo8gTy9YcMGOnbsGDDxBOr0xfuW2/EE8vT5eYESj5Ppffvgs8+i+PRTgAqMWTuGDL9k\nIMcfOaAQPtn+8OHD9fs7ienz3+/du5eU8tvlZ8aY8vw3tD4EOGStfcsY0w3IZa3tnsR7NLTuQFRU\n1L87h1yZcuWM8uRcMOaqTRu4/npPj/zImSMUGlWIRU0X8UDuB3y2zWDMkxtSMrTuViG/HpgB3A7s\nA5611h5N4j0q5CIiqeiXX6B4cfjpJ7jxRug4vyNnz53lw5ofuh2aEOCFPCVUyEVEUlfbtp6++KBB\nEPN3DGXGlWHbS9u4+bqb3Q5NCMCT3cQ/Evda5MqUK2eUJ+eCKVe//AIzZsArr3imX1n4Cj3K9PBL\nEQ+mPAUbPf1MRCRMDBoErVp5htTn75rPjkM7mNVwltthyTXS0LqISBhI3BvPkSuOBz98kMGVB1P7\n3tpuhyaJaGhdRESSlPho/MO1H3Jb9tuoVbCW22FJKlAhDwHqPTmnXDmjPDkXDLlK3Bs/dPoQ/aL7\n8e6T72JMsg78rkkw5ClYqUcuIhLiEh+N/++bN6l/X32K3lzU7bAklahHLiISwhL3xv+02yg/oTzb\nX9rOjVludDs0SUJKeuQ6IhcRCWHnj8ZvuMHy/JTO9CrbS0U8xKhHHgLUe3JOuXJGeXIukHOVuDf+\nza5v2HN0D+1KtnMllkDOU7BTIRcRCVEDB8L//Z/ncrPOCzrzTtV3yJA2g9thSSpTj1xEJATt3QsP\nPQQ7dsCnO0fwza5v+KbJN349U12STz1yEREBoH9/z/PGbea/GbBiAFERUSriIUpD6yFAvSfnlCtn\nlCfnAjFXu3ZBZCR07gy9l/WmYZGG3HfTfa7GFIh5ChU6IhcRCTH9+kH79vB73FY+3/Y521/a7nZI\n4kPqkYuIhJCffoIyZWDnTsuzXz9JzYI1+V+p/7kdljike62LiIS5vn2hY0dYcXAO+4/vp+3Dbd0O\nSXxMhTwEqPfknHLljPLkXCDlats2WLQI2rz0D68sfIV3nnyH9GnTux0WEFh5CjUq5CIiIeLNN+HV\nV2Hi9pHcc8M9VLu7mtshiR+oRy4iEgI2b4YqVeCHzX9RcsJ9rGixgkI3FnI7LEmmlPTIVchFRELA\nM8/A44/DjoJtyJQuE8OrDXc7JEkBnewWptR7ck65ckZ5ci4QcrVuHXz/PTxebxOzYmbRu3xvt0O6\nRCDkKVSpkIuIBLnXX4cePSw9ozvxRrk3yJU5l9shiR9paF1EJIitXAnPPQfD5kTy5orX2NBmA+nS\n6F5fwUo9chGRMFOxIjzbOJZhp4vwQY0PqHJXFbdDkmugHnmYUu/JOeXKGeXJOTdztXQp/PorHLv3\nPQrfVDigi7j2Kd/R+IuISBCyFnr1gs6vH+T1799iZcuVbockLtHQuohIEJo7F7p1g8cGtiJ7xmy8\n/eTbbockqUA9chGRMJCQAA89BE27bGDI79WIeTmGnJlyuh2WpAL1yMOUek/OKVfOKE/OuZGrmTMh\nTVrLV/90pE+FPkFRxLVP+Y4KuYhIEImPhzfegKc6z+LwmcO8WOJFt0MSl2loXUQkiEyeDB98fJYD\nz9zHx7U+ptKdldwOSVKReuQiIiHsn3+gUCGo3Gcwf2b4gchGkW6HJKlMPfIwpd6Tc8qVM8qTc/7M\n1SefQL4iB5h5YBjDqg7z23ZTg/Yp31EhFxEJAqdPQ//+kL1uL1oUa8Hd19/tdkgSIDS0LiISBN56\nCxZtWcfWYjWIeSmGHJlyuB2S+ICG1kVEQtDRozB0mOXoYx3pW6GvirhcQIU8BKj35Jxy5Yzy5Jw/\ncjVsGNzf8AvOpTnOC8Vf8Pn2fEH7lO+okIuIBLCDB2H0R2fYVaALw6sNJ22atG6HJAFGPXIRkQDW\noQOsyTyQPCV+5Mtnv3Q7HPExXUcuIhJC9u2DYmV/J81LD7Cm1WruzHWn2yGJj+lktzCl3pNzypUz\nypNzvsxVnz5wR4ue/N9DLwZ9Edc+5Tt6HrmISADavh0iV68lU5GF9Cwb43Y4EsA0tC4iEoDq1rNs\nKF6W12q0oGWJlm6HI36ioXURkRDwww8Qfegzctx4mohiEW6HIwFOhTwEqPfknHLljPLkXGrnylro\n0vMMVOnGe0+FzuVm2qd8R4VcRCSAzJ8PMdcP44l7HqFcvnJuhyNBQD1yEZEAkZAARR/7jf21HmTT\ny2vJnzO/2yGJn6lHLiISxKZOhb8e7MHLj7VWERfHVMhDgHpPzilXzihPzqVWrmJjoeuIVVBgCT3L\n9kiVdQYS7VO+o+vIRUQCwIcfWs6U78i71QeQNUNWt8ORIKIeuYiIy06cgLxPTSVv/XfZ/L9VpDEa\nLA1XKemR64hcRMRlg94+xbny3fmo7jQVcUk27TEhQL0n55QrZ5Qn5641VwcPwrurh/LE3aV5/I7H\nUyeoAKR9ynd0RC4i4qIu/X+Fku8z6ul1bociQUo9chERl+zYAff3aUL7JncxrEZft8ORAKDnkYuI\nBJEnmn3Pursa8FuPGJ2pLoBuCBO21HtyTrlyRnlyLqW5+m5lAt9l68g7NQaFRRHXPuU7KuQiIn5m\nLbzw7hTy5oUWDzVxOxwJchpaFxHxs+kzT9J0VSGWtfucMvkeczscCSAaWhcRCXDnzsHL04ZQJm95\nFXFJFSrkIUC9J+eUK2eUJ+eSm6uhH+3j2L2jmNh0sG8CClDap3xHhVxExE9OnYK+P3SjxX3tuSPn\n7W6HIyFCPXIRET95se93TPvnOf7qHUOW9FncDkcCkHrkIiIB6vc/EphwoAP9nxikIi6pSoU8BKj3\n5Jxy5Yzy5JzTXDUZMombb0xPx4qNfRtQgNI+5Tu617qIiI+t2XiS6HS9WPDcLIxJ1qipyFWpRy4i\n4mN3t+5Frny/sqbnJLdDccXGAxvJnjE7BXIVcDuUgKceuYhIgJk8Zw97rh/DjNaD3A7FFav2r6Lq\np1XZ8ucWt0MJWSrkIUC9J+eUK2eUJ+eulKv4eGj/VVca3N6BAjfc5r+gAkT0vmhqTavFuNrjyPZH\nNrfDCVkq5CIiPvL62GjOXL+acS++6nYofrdg1wLqz6jP9PrTqVGwhtvhhDT1yEVEfODEyXhu6FGS\n1yt24/W6Dd0Ox68iYyJp9XUrIhtFUvr20m6HE1RS0iPXWesiIj7Q7N0JZM+chdeeftbtUPxq2uZp\ndFrQifnPz6dEnhJuhxMWNLQeAtTPdE65ckZ5ci6pXO369Tizj7/O2PrDw+pys7HrxvLqoldZ3Gzx\nJUVc+5Tv+LSQG2MyGmNWGWPWG2O2GmMGeufnMsYsNMb8ZIxZYIzJ4cs4RET8qf6IgRTOUJWnH3nY\n7VD8ZsQPI+gX3Y+o5lEUvbmo2+GEFZ/3yI0xWay1p40xaYHvgFeA2sAha+0QY0w3IJe1tnsS71WP\nXESCytyVP1Prq1JsfXkThfPe6nY4fjFwxUDGrR/HkmZLyJczn9vhBLWAvI7cWnva+21G7/aOAHWA\nid75E4GnfR2HiIivWQstpnWlxvWdw6KIW2vpuaQnUzZPYUWLFSriLvF5ITfGpDHGrAcOAFHW2m1A\nbmvtQQBr7QHgZl/HEcrUe3JOuXJGeXIuca4GT4/iaKZ1TG3f2b2A/CTBJtBxfkfm75rP8ojl5MmW\n54qv1z7lOz4/a91amwAUN8ZkBxYYYyoAF4+XX3b8PCIigvz58wOQM2dOihUrRoUKFYD/doxwnz4v\nUOIJ5OkNGzYEVDyaDv7p8xYuXELvL1vzytNDyJY5U8DE54vp+IR4ag+qzd6je/mu33fkzJTzqu/f\nsGFDwMQfSNPnv9+7dy8p5dfryI0xrwNngJZABWvtQWPMLcAya23hJF6vHrmIBIVGQz9mwR+TOTRs\nOWnShO6Z6nHxcTSPbM6Bkwf46rmvyJohq9shhZSA65EbY248f0a6MSYzUAVYD3wFRHhf1hyY7cs4\nRER8ae8fx/j8rzcYU3d4SBfx2HOxNPi8AcdjjzO38VwV8QDh6x55HmCZt0f+A/CVtXYJ8BZQxRjz\nE1AJGOzjOELaxUN8cnnKlTPKk3NRUVHUf68/95ineLZs6N4A5XTcaWpPr036tOmZ2XAmmdNnTtb7\ntU/5jk975NbazcAle7a19jBQ2ZfbFhHxh7U//cY6O55NbUP36V7HY49Tc2pNCuQqwCe1PyFdGt0U\nNJDoXusiItcgT6enKXbjY3zTq5vbofjE4TOHqT6lOiVuKcGoGqNIY3RDUF8KuB65iEgoG/LFEv5O\nu4lpHTq4HYpPHDx5kAoTKlDujnKMrjFaRTxA6X8lBKj35Jxy5YzydHVn/znHGys7Uj9dBDmzZnI7\nnFS3//h+yk8ozzOFn2FIlSHXfM947VO+o0IuIpICEe+NJbO9gelv9XY7lFS3+8huyo0vx4slXqR3\nhd5h9eCXYKQeuYhIMu354yh3DS/E9BrzaVi+OKH0eyrm7xiqTK5CzzI9aVuyrdvhhJ2U9MhVyEVE\nkql491eItSfY9tZH53/xuh1Sqth4YCPVp1RnUKVBNC/W3O1wwpJOdgtT6j05p1w5ozxd3rxVO9jI\nRD5v18/tUFLVqv2rqPppVUZUG+GTIq59ynd0MaCIiEPWQvOpr1Dr1m4UyZfb7XBSTfS+aOrPqM+4\nOuOoWbCm2+FIMmloXUTEob5TF9J/XTsO99tK1swZAYJ+aH3BrgU0ndWUac9Mo9KdldwOJ+ylZGhd\nR+QiIg6cPnuOAWs60b3ksH+LeLCbtX0Wree0JrJRJKVvL+12OJJC6pGHAPWenFOunFGeLtXk3TFk\nM7fw5nN13A4lVUzdPJW2c9sy//n5fini2qd8R0fkIiJXsW3PYWYffZPIhotD4prqsevG0juqN4ub\nLabozUXdDkeukXrkIiJXcV/njmTIcpYN/T+8ZFmw9ciH/zCc4T8MZ1HTRdxzwz1uhyMXUY9cRCSV\nTVscw08ZprCr/Ta3Q7lmA6IHMGHjBJZHLCdfznxuhyOpRD3yEKDek3PKlTPKk0dCArSZ2ZmGebtT\nIPdNboerY4lVAAAgAElEQVSTYtZaei7pydQtU4mOiHaliGuf8h0dkYuIXEbn0d/wT9adjG8T6XYo\nKZZgE+g0vxMrflnB8ojl3JjlRrdDklSmHrmISBL+OhRHnn4PMLTqW3R6qvZlXxfIPfL4hHhafd2K\n7X9vZ16TeeTMlNPtkOQq1CMXEUklDYZ8wC1Z8tKxei23Q0mRuPg4mkU24+DJgyxsupCsGbK6HZL4\niHrkIUC9J+eUK2fCPU/f/niIaNOPqc3eDcrLzc6eO0v9z+tzPPY4cxvPDYgiHu77lC+pkIuIJGIt\nPDemD4/nbEC5QsF3jfXpuNPUnlabDGkzMKvhLDKnz+x2SOJj6pGLiCTy7qfb6LKtPPu7b+eW7Fc/\nMSyQeuTHY49Tc2pNCuQqwCe1PyFdGnVPg40eYyoicg1OnLD0iOpM2yK9HBXxQHL4zGEqT6pMkZuK\nML7OeBXxMKJCHgLUe3JOuXImXPP0wqB5ZMi9h7cbtXM7lGQ5ePIgT0x8gnL5yjG6xmjSmMD71R6u\n+5Q/BN7/toiIC7Zsj2Pm6c6Mqv0OGdJmcDscx/Yf30/5CeWpW6guQ6sMDcqT8+TaqEcuImHPWigU\nMRxzz3y29/omWcXQzR757iO7qTypMu1KtuPV0q+6EoOkLl1HLiKSAuM/+5ufbxvAjy8sD5oj2pi/\nY6gyuQo9y/Skbcm2bocjLtLQeghQ78k55cqZcMrTqVPQYVZvat/ZiAdvvc/tcBzZdHATFSdWpP8T\n/YOmiIfTPuVvOiIXkbDWceAW4gp+zseNt7sdiiOrf1tN7Wm1eb/6+zQo0sDtcCQAqEcuImErJsby\nwNCqvNGwNq9VbZ+idfizRx69L5r6M+ozvs54ahSs4Zdtin+lpEeuQi4iYclaKPHc1/xRpBu/9txI\n+rTpU7QefxXyBbsW0HRWU6bXn07FAhV9vj1xh24IE6bUe3JOuXImHPI0/fN/2Jb3FcY2eCfFRdxf\nImMiaTqrKZGNIoO2iIfDPuUWFXIRCTvHj0Ob8SMpkf8eahaq5nY4VzR181TazGnD/OfnU/r20m6H\nIwFIQ+siEnbavPIXE667jw3/W0GhGwtd07p8ObQ+dt1Yekf1ZsHzCyh6c/A9wEWST9eRi4hcxcaN\nMOmXN2jeuMk1F3Ffem/Ve7z9/dtENY/inhvucTscCWAaWg8B6j05p1w5E6p5SkiAZl02ke6BmQyu\n1tvtcC5r4IqBvL/6faIjokOmiIfqPhUIdEQuImHjk08sewt1YmDVN8iVOZfb4VzCWkuvpb2Y/dNs\noiOiyZMtj9shSRBQj1xEwsLff8PdNWZzY8NexHTckGqP+UytHrm1lo7zO7LilxUsbLqQG7ME12NU\nJXWoRy4ichldusfCk6/wQZ0PAu5Z3fEJ8bSZ04atf21lafOl5MyU0+2QJIioRx4C1HtyTrlyJtTy\ntHIlzPz9PR67uzBV7qridjgXiIuPo1lkM3Yf3c3CpgtDtoiH2j4VSALrz1IRkVQWFwct//cn9um3\nGFFjpdvhXCD2XCwNv2hIXEIcc56bQ+b0md0OSYKQeuQiEtIGD4bRv7bimVpZebfaO6m+/pT2yE/H\nnabuZ3XJnjE7U+pNIUPaDKkemwQf3WtdRCSR3buhxFMbSN+iGjs7xPhk2Dolhfx47HFqTq1JgVwF\n+KT2JwHXsxf36F7rYUq9J+eUK2dCIU/WQtt2lusbd6RfpT4B03s+fOYwlSdVpshNRRhfZ3zYFPFQ\n2KcC1VULuTGmgTEmm/f714wxM40xJXwfmohIys2YAdvtLK678TAvlnjR7XAAOHjyIE9MfIJy+cox\nusZo0hgdS8m1u+rQujFmk7X2AWNMGaA/MBR4w1pbyufBaWhdRFLg6FG474GzmJfvY1L9j6l0ZyWf\nbcvp0Pr+4/upPKkyjYo2onf53hiTrNFTCRO+GlqP9/5bA/jIWjsX0FkZIhKwevaE2xsMp+QdD/i0\niDu1+8huyo0vx4slXqRPhT4q4pKqnBTy34wxY4CGwDxjTEaH7xM/Ue/JOeXKmWDO0w8/wJcLDrDr\n5mEMqzrM7XCI+TuG8hPK06V0F14t/arb4bgmmPepQOfkLItngWrAMGvtUWNMHqCLb8MSEUm+uDho\n3Rrua9+Lh+9rwd3X3+1qPBsPbKT6lOoMqjSI5sWauxqLhK7L9siNMdmttceNMdcntdxae9inkaEe\nuYgkz+DB8NWadex+7Cl+evkncmTK4fNtXq5Hvmr/KmpPr83I6iNpUKSBz+OQ0JCq15EbY+ZYa2sa\nY/YAFki8YmutvTPloToMToVcRBzatQtKPWq5q195XnzkeVo91Mov202qkEfvi6b+jPqMrzOeGgVr\n+CUOCQ2perKbtbam998C1to7vf+e//J5ERfn1HtyTrlyJtjyZK1nSL1mly+INcdoWbyla7Es2LWA\n+jPqM+2ZaSriiQTbPhVMnFxH3vKi6bTGmN6+C0lEJHkmTIAjJ86wPGMXhj85nLRp0roSR2RMJE1n\nNSWyUWRAnC0v4cHJdeRTgZxAS+AGYDyw3Frr89MvNbQuIldz8CDcfz80HDmQ3+xaZjac6dftnx9a\nn7Z5Gp0WdGJek3mUyKN7ZknK+Oxe68aYhsAo4BTQ2Fr7XcpCTB4VchG5mkaN4Pr8vzPj+gdY/X+r\nuTOXfzt/xhg+/vFjekf1ZsHzCyh6c1G/bl9Ci09uCGOMuQfoAHwJ7AOaGmOypCxE8QX1npxTrpwJ\nljzNnQtr1sDxh3vyYokX/V7EASgM/aL7EdU8SkX8CoJlnwpGTq4j/xp4yVq7xHhuR9QZWAMU8Wlk\nIiJXcOIEtGsHXYevZcDuhcTUivF7DANXDITCEB0RTb6c+fy+fRFw1iPPbq09ftG8gtbaHT6NDA2t\ni8jldegAR49Zfi5XlhbFWtCyhP/OVLfW0mtpL2b/NJtt3bZhT+j3lKSOlAytX/WI3HtTmKLAfUCm\nRIt8XshFRJKycqXn6WZ9Z37G5k2niSgW4bdtJ9gEOs3vxIpfVrA8Yjk3vXST37YtkhQnPfLewPve\nryeAIUBtH8clyaDek3PKlTOBnKezZ6FlSxg24gz9V3VjeDX/XW4WnxBPq69bseb3NSxtvpQbs9zo\nl+2GgkDep4Kdk4ef1AcqAQestS2ABwHf3/dQRCQJ/ftD4cLwc+5hPHLbI5TLV84v242Lj6PprKbs\nPrKbhU0XkjNTTr9sV+RqnPTIV1trHzHG/IjniPwEsN1aW8jnwalHLiKJbNgAVavC/JW/UWXmA6z9\nv7UUyFXA59uNPRdLwy8aEpcQxxcNviBz+sz/LnP6PHIRJ3zSIwfWGmNyAh8DPwInge9TEJ+ISIqd\nOwcvvABvvQXDN/egzUNt/FLET8edpu5ndcmeMTszGswgQ9oMPt+mSHJcdWjdWtvOWnvUWvshUAVo\n7h1ilwCh3pNzypUzgZinYcPgxhuhcOVVLNmzhO5luvt8m8djj1Pt02rckvUWpj0zTUU8Bc4lnOPd\n799l0ORBbocSspwckf/LWrvXR3GIiFzWTz95CvmaNZbGCzsyoOIAsmXM5tNtHj5zmGqfVuOhPA8x\nqsYo0hgnpxRJYmt/X0urr1uRK3MuInJEuB1OyHJ0i1a3qEcuIgkJUL48PPss3FBhKu98/w6r/2+1\nTwvrwZMHqTK5ClXvqsrQKkPx3AsraeqRX+pE7AleX/Y607dMZ0iVITR9oOkVcyj/SdVbtBpj5hlj\n8l9rUCIi12L0aE8xb/7iKbov7s7wasN9WsT3H99P+QnlqVe43lWLuFxqdsxsiowuwvHY42xpt4Vm\nDzZTDn3sSp+G8cBCY0wvY0x6fwUkyReI/cxApVw5Eyh5+vln6NMHxo2Dd1YNpfTtpSlzRxmfbW/3\nkd2UG1+OlsVb0qdCHxWgZNh/fD91P6tL18VdmVR3EuPqjLvgOvtA2adC0WV75Nbaz40x3wCv4zlz\nfTKQkGj5O36IT0TCVEKC58YvPXpAllt+5f3Z77Ou1TqfbS/m7xiqTK5CjzI9aFeync+2E2riE+IZ\ntWYUfZf35eVHXmb6M9PJmC6j22GFlSv2yI0xGYDuQGPgMy4s5G/6PDj1yEXC1qhRMGUKrFgBzWY3\n4c6cd9KvYj+fbGvjgY1Un1KdQZUG0bxY82S9N5x75Ov/WE+rOa24Lv11fFjzQwrd6PPbi4S8VH0e\nuTGmGvAO8BXQ11p7+tpDTB4VcpHwtHs3PPIIfPcdHM7yPQ0+b0DMyzFkzZA11be1av8qak+vzcjq\nI2lQpEGy3x+OhfzkPyfpE9WHyZsmM7jSYCKKRagNkUpS+3nkvYAG1trubhRxcU69J+eUK2fczNP5\nIfXu3eGeggl0mN+BQZUG+aSIR++Lpta0WnxS+5MUFfFwNGfHHIqOLsrBUwfZ0nYLLYq3cFTE9dnz\nnSv1yMv6MxAREYAPP/Q8GKVTJ5iyaQrGGJo80CTVt7Ng1wKen/U805+ZTqU7K6X6+kPN7yd+p8P8\nDmw4sIGxtcdS+c7KbockXj69jtwYkxeYBOTG01//2Fr7njEmF56eez5gL/CstfZYEu/X0LpIGNmz\nB0qWhG+/hbx3nqTQyEJ83uBzHrv9sVTdTmRMJK2+bsWshrN4/I7Hr2ldoT60Hp8Qz5gfx9A7qjdt\nHmpDz7I9L7jXvKSuVO2RpwZjzC3ALdbaDcaYrHju1V4HaAEcstYOMcZ0A3JZay+536IKuUj4SEiA\nKlWgWjXo0gVeX/o6u4/uZkq9Kam6nambp9J5QWfmNZlHiTwlrnl9oVzINx3cRKuvW5EuTTo+qvUR\n9910n9shhbzU7pFfM2vtAWvtBu/3J4HtQF48xXyi92UTgad9GUeoU+/JOeXKGTfy9MEHcPq0Z0h9\n39F9jF47msGVBqfqNsauG0uXRV1Y3GxxqhTxUHU67jTdFnWj8qTKtCzekugW0ddcxPXZ851k3Wv9\nWnjvElcM+AHIba09CJ5ib4y52V9xiEjg2bnTc+OX776DdOmg2+JutH+kPbfnuD3VtjHihxG8+8O7\nRDWP4p4b7km19Yaa+bvm025uOx7N+yib224md9bcbockV+GXQu4dVv8C6GCtPWmMuXgc6rLjUhER\nEeTPnx+AnDlzUqxYMSpUqAD89xeepjWdnOnzAiWeQJyuUKGC37ZXtmwFIiKgUaMofv8d/syUjpW/\nriQiZwRRUVGpsr2BKwYy6vNRvF317X+LuPanC6dnfjOTkatHsjfnXkbXGE2m/ZnYvnY7uSvkTpX1\nn58XKD9voEyf/37v3r2klM8fmmKMSQfMAb6x1o7wztsOVLDWHvT20ZdZawsn8V71yEVC3NChMG8e\nLFkCmAQe+fgROj/Wmcb3N77mdVtr6bW0F7N/ms3ipovJky3PtQd8kWDvkSfYBMauG8trS1+jRbEW\n9K7Qmyzps7gdVtgKuB651zhg2/ki7vUVEOH9vjkw2w9xhKyLjwzk8pQrZ/yVp61bYcgQGD8e0qSB\nSRsnkT5tep4r+tw1rzvBJtBxfkfm75rP8ojlPiniwW7rn1spN74c4zeMZ3GzxbxV5S2fFXF99nzH\np0PrxpjHgSbAZmPMejxD6D2Bt4AZxpgXgH3As76MQ0QCT1wcNGsGgwZB/vyeu4X1WtqLmc/OvOa7\nhMUnxNN6Tmu2/72dpc2XkjNTztQJOkSciTvDgBUDGPPjGPpW6Evrh1vreetBTM8jFxFXvPkmrFoF\nc+eCMdBrSS9+Pf4rk+pOuqb1xsXH0SyyGX+e+pPZjWb75I5wiQXb0Pri3YtpO7ctxW4pxohqI7g1\n261uhySJpGRo3W9nrYuInPfjj57njK9f7ynie47sYcyPY9jYZuM1rTf2XCwNv2jIuYRzzHlujm5c\nkshfp/6i88LOrNi3glFPjaJGwRpuhySpRGMpIUC9J+eUK2d8maczZzxD6u++C7d6Dwa7Lu5Kh1Id\nuC37bSle7+m409SaVov0adMzs+FMFXEvay3j1o+j6AdFyX1dbra22+pKEddnz3d0RC4iftWjB9x/\nPzznPZ8tel80q39bzaSnUz6kfjz2ODWn1qRArgJ8UvsT0qXRrzbwPGO99ZzWnIk7w/wm8ymep7jb\nIYkPqEcuIn6zeDG0aAEbN8L113tOSiv5cUm6Pt6VRkUbpWidh88cptqn1Xgoz0OMqjHK7ydtBWKP\n/Oy5swxaMYhRa0bxRvk3eKnkS6RNk9btsMQB9chFJGAdOeIp4uPGeYo4wIQNE8iSPgsNizRM0Tr/\nPPUnVSZXoeqdVRlSZYieiQ0s27OMNnPbUOSmImxos4G82fO6HRI7Du0gfZr0FMhVwO1QQpJ65CFA\nvSfnlCtnfJGnl16CunU9D0YBz3D4a8teY3i14SkqwL8d/41y48tRr1A9FXHg0OlDtJjdguaRzRlS\neQgzG850vYj/ceIP2s5py+PjHufTrz51NZZQpkIuIj43bZrnDPXBiZ6BMnDFQJ6860kevvXhZK9v\nz5E9lB1flpbFW9K7Qu+wLuLWWiZtnESR0UXIkTEHW9ttpU6hOq7GdDz2OG8se4OiHxTlugzXEfNS\nDGXzlXU1plCmHrmI+NT+/VCihOc2rA97a/bPh3+m1NhSbGq7KdnXMcf8HUOVyVXoWaYnbUu29UHE\nyeNmj3znoZ20mduGI2eO8FGtj1L0R1Fq+if+H8asHcOAFQOoeldV+j3Rj3w587kaU7BRj1xEAkpC\nAkREwP/+918RB+iyqAudHu2U7CK+6eAmqn1ajUGVBtG8WPPUDTaIxJ6LZch3QxixagS9yvaifan2\nrp6pn2ATmLF1Br2W9qLgDQVZ8PwCHrzlQdfiCTcaWg8B6vs6p1w5k1p5eu89OHUKunf/b96yPctY\n98c6Oj/WOVnrWv3baqpOrsqIaiPCuoiv2LeC4mOKs+b3NaxrvY5Oj3VytYgv2b2ERz5+hLe/f5uP\na33MN02+SbKI67PnOzoiFxGf2LQJBgzw3IY1nfc3TXxCPJ0WdGJolaHJumFL9L5o6s+oz/g648P2\njmSHzxym26JufLPrG0ZUG0G9wvVcPTdgw4ENdF/cnV2HdzGw0kDq31df92t3iXrkIpLqzpzxDKV3\n6+a5i9t5H//4MZM3TWZ5xHLHRWjhzwt5fubzTK8/nYoFKvoo4pTzdY/cWsvUzVN5ddGr1C9cn/4V\n+5MjUw6fbe9q9h7dy+vLXmfRz4t4rdxrtHqoFRnSZnAtnlCjHrmIBISuXT13b2va9L95x84e442o\nN5jbeK7jIh4ZE0mrr1sR2SiS0reX9lG0gevnwz/Tbl47Dpw8QGTDSErlLeVaLIdOH2LAigFM3DiR\nl0u+zM72O8mWMZtr8ch/NA4SAtR7ck65cuZa8jRvHnz1FXz4oeeBKOf1j+7PU3c/RYk8JRytZ9rm\nabSd25b5z88PuyIeFx/H4G8HU2psKSoXqMza/1vrWhE/HXeawd8OptCoQpw9d5at7bby5hNvJruI\n67PnOzoiF5FUc/AgvPgiTJ8OORM9AnzX4V2M3zCeLe22OFrP2HVj6RPVh8VNF1Pk5iI+ijYwrfx1\nJa3ntCZv9rys+b81rt0N7VzCOSZumEjvqN48dvtjfPfCdxS8oaArsciVqUcuIqnCWqhZE4oV85zk\nltjT05/m0byP0r1M96TfnMiIH0bw7g/vsrjZYu6+/m4fRZt6UqtHfvTsUXos7sHsn2YzvNpwGtzX\nwJWT2ay1fL3ja3os6cFNWW7ircpvuTqkH27UIxcR14waBX/9BX36XDh/ye4lbDq4ien1p191HQNX\nDGT8hvEsj1geNjcSsdYyY+sMOi3oRJ1767DtpW3kzJTz6m/0ge9//Z6ui7ty9OxRhlYZSvW7q4f1\nXfOChXrkIUC9J+eUK2eSm6dNm+DNN2HKFEif/r/55xLO0XFBR4ZVHUamdJku+35rLT2X9GTK5ilE\nR0SHTRHfc2QPNabWoF90P7549gs+qPmBK0U85u8Y6n5Wl4ZfNKRl8ZZsaL2Bp+55KlWLuD57vqNC\nLiLX5PRpaNQI3n4b7rnnwmVj143lhsw3ULdQ3cu+31pLx/kdmb9rPssjlpMnWx4fR+y+uPg4hnw3\nhJIfl6RcvnKsb73elRP6fj/xO62/bk3Z8WUpnbc0P738ExHFIvTI0yCjHrmIXJNWrTzXjU+efOH8\no2ePUmhkIeY/P59itxRL8r3xCfG0mdOGrX9tZV6Tea4NKV+L5PbIV+1fRas5rch9XW4+qPEBd11/\nlw+jS9qxs8cYunIoH6z9gJbFW9KjTA9yZc7l9zjkUuqRi4hfff45LFsG69Zduqzf8n7Uvrf2ZYt4\nXHwczSObc+DkARY2XUjWDFl9HK27jscep9eSXnyx/Qvervo2zxV9zu/959hzsXy49kMGfjuQ6ndX\nZ33r9dyR4w6/xiCpT0PrIUC9J+eUK2ec5GnvXs8zxqdNg2wXXVK849AOJm6cSP+K/ZN8b+y5WBp8\n3oBjsceY23huSBdxay1fbvuS+0bd9+912I3vb+zXIp5gE5iyaQqFRhVi4e6FLG66mAlPT/BrEddn\nz3d0RC4iyRYXB40be27B+nAST858ZeErdHu8Gzdfd/Mly07HnabuZ3XJliEbMxrMCOnbe/5y7Bde\nnvcyuw7vYtoz01x5JvfCnxfSbXE3MqTNwPg646mQv4LfYxDfUo9cRJKtVy/48UfPXdzSXDSut/Dn\nhbSb246t7baSMV3GC5Ydjz1Ozak1yZ8zP+PqjHP1qV2pJake+bmEc7y36j0GrhhIx0c70qV0l0ty\n4Ws//v4j3Zd0Z+/RvQyqNIhnCj+jS8mCgHrkIuJzS5fC+PGwfv2lRfxcwjk6LejEsKrDLilch88c\npvqU6pS4pQSjaowK2Sdl/fj7j7Sa04qcmXKysuVKv98NbfeR3by29DWW7V3GG+Xe4MUSL5I+bfqr\nv1GCVmh+ksKMek/OKVfOXC5PBw96HoQycSLkzn3p8jFrx3BL1luoc2+dC9938iBPTHyCsneUZXSN\n0SFZxE/EnqDj/I7UmFqDDqU6sLjpYr8W8b9O/UWHbzpQ8uOSFLqxEDvb76RtybYBU8T12fOd0Ps0\niYhPxMfD889DixZQpcqlyw+fOcyby9/k3SffvWAId//x/ZSfUJ66heoytMrQkBzenR0zmyKji3As\n9hhb2m2h2YPN/PZznvrnFP2j+1NoVCHibTzb2m3jjfJvhPQJhHIh9chFxJH+/WHRIliyBNIl0ZTr\nOL8jZ8+d5cOaH/47b8+RPVSaVIm2D7ely+Nd/Bitf+w/vp/bW91OwTIFGVNzjF9PJIuLj2Pc+nG8\nufxNyuYry4CKA4Li3vRyZeqRi4hPLF/uuZf62rVJF/GYv2OYsnkK29ptu2BelclV6FGmB+1KtvNj\ntL4XnxDPqDWj6Lu8LxyEjW02XvEWtKnJWsusmFn0WNKD27LdxuxGsyl5W0m/bFsCk4bWQ4B6T84p\nV84kztOff0KTJp4T3G67LenXd17QmR5lenDTdTcBsPHARipOrEj/J/qHXBFf/8d6Hv3kUWZun8m3\nL3wLUfitiK/Yt4LS40rz5vI3GVFtBEuaLQmaIq7Pnu/oiFxELishwXNyW9OmUK1a0q/5Zuc37Dq8\ni8hGkYDnFqS1p9dmZPWRNCjSwI/R+tbJf07Se1lvPt38KYMrDSaiWITf+uBb/9xKjyU92HhwI/2f\n6E/j+xvrfujyL/XIReSyBg2CuXMhKirpIfW4+Dge+PABhlQeQq17axG9L5r6M+ozrs44ahas6fd4\nfWXujrm8NO8lyuYryztV3/l35AFS73nkSdl/fD+9l/Xmqx1f0f3x7rz0yEt+O/oXd6hHLiKpJjoa\nhg+/fF8c4IO1H3B79tupWbAmC3YtoOmspkx7ZhqV7qzk32B95PcTv9NhfgfW/7GesbXHUvnOyn7Z\n7tGzRxn87WA++vEjWj3Uip3tdwblA2XEP9QjDwHqPTmnXDkzc2YUzz3nuV789tuTfs2h04foH92f\nd558h8iYSJrOakpko8iQKOIJNoHRa0bz4IcPcu8N97K57Wa/FPGz587y9sq3Kfh+Qf469Reb2m5i\ncOXBIVHE9dnzHR2Ri8gFzp2Dvn2hZcvL98UB+kT1ocF9Ddh0cBOdF3Rm/vPzKZGnhP8C9ZHNBzfT\nak4r0pg0RDWPosjNRXy+zfiEeKZsnsLry17nwdwPsqz5Mr9sV0KDeuQicoFevWDVKliwANJe5nyq\nrX9upcLECvQs25NhK4ex4PkFFL25qH8DTWWn407Td3lfPln/Cf2f6M//PfR/ju5Ady09cmstC35e\nQLfF3ciSPgtDKg9x5cEqEjhS0iNXIReRf82dC23aeB6IcvOlDy4DPMWn2pRqZEmfhXV/rGNx08Xc\nc8M9/g00lS3YtYC2c9tSKm8p3n3yXW7Jeovj96a0kK/5bQ3dFnfjtxO/MajSIOoWqhuSd72T5ElJ\nIVePPASo9+SccnV5e/fCCy/A9OmwbVvUZV83b+c81v2+js0HNxMdER3URfzgyYM0/rIxbee2ZXSN\n0Ux7ZlqyinhK7Dq8i4ZfNKTO9Do0LNKQre22Uq9wvZAv4vrs+Y4KuYgQGwsNGkD37vD441d43blY\nmkU2I0v6LKxosYJ8OfP5L8hUlGAT+OjHj7j/g/u5I8cdbGm3hWp3X+GEgFTw56k/eXney5QaW4r7\nb76fne130vrh1iHxKFdxl4bWRYR27TxPNvviC7jcgaG1lvITyrPhwAZ2td/FzVkvM/Ye4Lb+uZXW\nc1pzLuEcH9X6iAdyP3BN67va0PrJf07y9sq3eW/1ezR9oCm9yva64Dp0kcR0HbmIJNukSbB4MaxZ\nc/kiHp8QT/PI5ny//3tWtFgRlEX8TNwZBqwYwJgfx9C3Ql9aPdTKp3dHi4uP4+N1H9Mvuh8VC1Rk\nzf+t4c5cd/psexK+NLQeAtR7ck65utD69fDKKzBrFuTI8d/8xHmKi4+j6aymRO2N4oViL/Bo3kf9\nH2xJQywAACAASURBVOg1WrJ7CQ98+AA7Du1gY5uNtC3Z1mdF3FrLF9u+oMjoIkTGRDK38Vym1JsS\n9kVcnz3f0RG5SJg6dAjq1fM81azIZS5Zjj0XS8MvGnL4zGFiz8UysNJA/wZ5jf469RedF3Zmxb4V\njHxqpM9vGxu1N4pui7sRFx/HqKdGUeWuJB7cLpLK1CMXCUPx8VCjhqeAv/120q85HXeaup/VJXuG\n7Bw+e5g699bhf6X+599AU8hay4QNE+i+pDtNH2hKnwp9yJohq0+2ZYxh04FNdF/Sne1/bad/xf40\nKtrI0TXoIhfTdeQi4sjrr8O338KiRUnfR/147HFqTq1J/pz5qVeoHj2X9mRjm42kT5ve/8Em009/\n/0TrOa05HXeaMTXHUDxPcZ9t65djv5AvIh83P34zPcv0pM3DbciYLqPPtiehT9eRhyn1npxTruCr\nrzz3UP/ss6SL+OEzh3mk1yMUuakIH9X6iK6Lu/LOk+8EfBGPPRdLn6g+lBlfhmcKP8P3Lb/3WRE/\ncuYIXRd1pfiY4rAPdry8gw6PdlARvwJ99nxHhVwkjOzcCS++CDNmJH3ntoMnD1JhQgUezP0go2uM\nZtTqUdx9/d0+v8b6WkXtjeKBDx9g08FNrG+9nval2vvkZLYzcWcY+t1QCo4syLGzx9jcdjOshxyZ\nclz9zSI+oqF1kTBx4gSUKgUdOkDr1pcu3398P5UnVea5os/xRvk3+Pv03xQeVZhvX/iWQjcW8n/A\nDhw6fYhXF73Kkt1LeL/6+9QpVMcn24lPiGfypsm8sewNHr71YQZWGvhvTnz5PHIJP7qOXESSlJAA\nTZtC2bJJF/HdR3ZTeVJl2pVsx6ulXwXg9WWv0+T+JgFZxK21TN40ma6LutKoaCO2tttKtozZfLKd\neTvn0X1Jd3JkzMH0+tMpfXvpVN+OyLVQIQ8BUVFRVKhQwe0wgkK45qpfP/j7b8+Q+sVi/o6hyuQq\n9CzTk7Yl2wLwycxPmPnzTGJejvFzpFe389BO2sxtw5EzR5jTeA4P3/qwT7azav8qui3uxp+n/mRw\n5cHUKlgr5O+H7kvh+tnzBxVykRAXGQljx3ru3JYhw4XLNh7YSPUp1RlUaRDNizUHPEehI1ePpHe9\n3lyf+XoXIk7aP/H/MOS7IQz/YTi9yvaifan2PrlP+c5DO+m5tCff//o9b1Z4k+bFmut+6BLQ1CMX\nCWHbtkH58jBvHpQseeGyVftXUXt6bUZWH0mDIg3+nR8ZE0mvpb3Y2GZjwBSwb3/5llZft+Lu6+9m\n5FMjuSPHHam+jQMnD9B3eV8+3/Y5nR/tTIdHO5AlfZarvk89cklN6pGLyL+OHIE6dWDYsEuL+PK9\ny6n/eX3G1xl/wd3OYs/F8urCVxldY3RAFPHDZw7TfXF35u2cx3vV3/PJM7tPxJ5g2MphjFwzkogH\nI4h5KYYbstyQqtsQ8SVdfhYCdH2mc+GSq/h4aNzYc/e25s0vXDZ/13zqf16f6c9Mv+SWpe+teo/C\nNxUmw68XjcH7mbWW/2/vvuNruv8Hjr+OPUPVqlGjdmLWqhkEMUqNiFGbIKjVCqL2nvGtTa3a1KZB\nRIa9twRVm5gRJDI/vz9u6kdFZNybO/J+Ph73Iffec859n8/jc73vOe9zPp+1F9diPd+atCnTGmTO\n7rDIMOaemEvR34pyM/Amp51OM7PhTEniBpJcvnvGYPyf3EIIvXNxgbAw3dH4+7b5bcNppxPbHLdR\n/esPJx4PeB3A1MNTOdL9CA8uPkjCaD/09/O/cd7jzKPXj9jmuI0q+arodftRKopNlzfh6ulKkWxF\ncP/RnXK5y+n1M4RISlIjF8LCLF8OkybB8eOQ7b1r1dZeXMvgvYPZ3X433+b59qP1nHY6kTlNZmY2\n/MTg6wYWHhnOjCMzmHl0Ji7VXRhYdaDeR5Pz/McTFw8XlFJMtZtKvcL1Er1NqZELfZIauRDJ3OHD\nuqNxH58Pk/jSM0sZ7TUaj04e2OS0+Wi9c4/OscN/h9FuNzty9wi9dvUin1U+TvY8SaEvCul1+xcC\nLuDi4cK1Z9eYVHcSDtYOMqmJsBjSky2A1J7izpLb6vZtcHCAVaugxHtjuLgdc2O8z3i8OnvFmMSV\nUgzaO4gxtmPImi4rkHTtFPg2kD67+tB6Y2t+rfUre9rv0WsSvx14m05bO9HgjwY0LtKYq32v4mjj\nKEncCCz5u2ds0puFsACvX0OzZjB0KNi/Nyz6RJ+JzDs5D58uPhT9smiM627128qz4Gf0qNAjiaLV\n/XjYeHkj1vOtUSiu9L1CG+s2eruY7VnwM4bsHUKFxRUomLUg1/pfo3+V/qRJadyL+IQwBKmRC2Hm\noqKgZUvIkQMWLwZN0yVKV09Xtvtvx6OjB19l/irGdd9GvKXUvFIs+X6JXurFcXEr8BZ99/TlduBt\nFn+/WK9DnoaEhzDn+BxmHJmBQykHRtuOJnem3HrbfkykRi70SWrkQiRDI0fC8+e64Vc1TXdV9iD3\nQfje8cW7izfZM2T/5Lpzjs2hTK4ySZLEwyPDcTvmxtTDUxny3RC2Om7V2xFyRFQEK8+tZLTXaKrk\nq8Lhbocpnr24XrYthKmTU+sWQGpPcWdpbbVihW5e8T//1A2/GhkVidNOJ04+OIlnZ89Yk/ij14+Y\nfmQ6MxrM+Og9fbfT8XvHqbSkEvtv7ud4j+MMrzlcL0lcKcUO/x2UXViWVRdWsbnNZv5s86ckcRNk\nad89UyJH5EKYKS8v3RXq3t660+rhkeF03taZgDcB7Ou4j0xpMsW6vusBV7qW60qRbEUMFmNQaBAj\nDozgz6t/MrPBTNrZtNNbHfzo3aMM9RjKi5AXTLWbSpOiTWRSE5EsSY1cCDN07ZpuStK1a6FePd3Q\nqo6bHYmIimCTwybSp04f6/pnHp6hydom+PX1I0u6LHqPTynFlqtbGOA+APsi9kyrP01vE7D4P/Vn\n+IHhnHxwknG24+hUthMpU6TUy7YTQmrkQp+kRi5EMvD0qW7o1YkTdUk8ODyYFhtaYJXWio0OGz97\nylopxUD3gYyzHWeQJH7n5R367enHjec3WNdqHTUL1NTLdh++esgYrzFs8dvCL9V+YU3LNZ/9wSJE\nciA1cgsgtae4M/e2Cg3VXaHesiX06KE7dd1oTSNyZ8rNulbr4lR33nxlM0GhQXQr3+2TyySknSKi\nIph9dDYVFlWgUp5KnO11Vi9JPCg0iJGeI7FZYINVWiv8+/kztPpQSeJmxty/e6ZMjsiFMBNKQc+e\nunr45Mm6mcHsV9tTMU9F5jaeG6dBTt5GvGWox1CWN1+u19PRpx+cxmmXE1nSZuFI9yMU+7JYorcZ\nGhHKwlMLmXRoEvZF7Dnb66xBpi8VwtxJjVwIMzF2LOzapbu47VVUAPX/qE/Dbxoyrf60OF/kNcl3\nEqcenGKL4xa9xPQq9BW/HvyV9ZfWM9VuKp3Kdkr0BWdRKooNlzbg6ulKyRwlmVxvMmVyldFLvIYg\nNXKhT1IjF8JCrVgBK1fCkSPwPOIedqvsaGfTjlG1R8U5cT549YBZR2dxoucJvcS03W87/f/qT73C\n9bjkfCnWW93iyuOmBy4eLqTUUrKs+TJsC9omPlAhLJzUyC2A1J7izhzbat8+3W1me/ZAcNqb1Fpe\nix4VejDadnS8jn5dPV3pUaEHhb8o/NllY2une0H3aLGhBUM9hrLyh5Usb7480Un87MOzNFzdkD67\n+zCs+jCO9zguSdzCmON3z1zIEbkQJuz8efjxR92AL2T3o/aK+oyoMYI+lfrEazunHpxi7429iZrd\nLDIqkvkn5zPOZxzOFZ1Z12od6VKlS/D2AP558Q+/HvyVA/8c4Ndav9KzQk+9T10qhKWTGrkQJuru\nXahWDWbOhOK1ztNoTSMm15tM53Kd47UdpRQ1l9eka7mudK/QPUGxnHt0DqedTqRPnZ5FTRdRInuJ\nz68Ui6fBT5noM5FVF1bxU+WfGFJtyGcHsDFVUiMX+pSQGrlBT61rmva7pmkBmqZdeO+1LzRN26dp\nmr+maXs1TdP/jaxCmLmXL6FxYxgwAApUO06D1Q2YYz8n3kkcYMPlDQSHB9OlXJd4r/sm7A2/7PuF\nhqsb0rtibw52PpioJB4cHswk30mUmFuCsMgwrjhfYbTtaLNN4kKYAkPXyJcDDf/z2jDAQylVHPAE\nhhs4Bosntae4M4e2+vde8dq1oVJrH75f9z3Lmi3Dwdoh3tsKCQ/BxcMFN3u3eN1u5uXlxZ7re7BZ\nYMPD1w+52Oci3cp3S/A83hFRESw5vYRivxXjfMB5jnY/yrwm88iVKVeCtifMjzl898yVQWvkSqlD\nmqYV+M/LzYHa0X+vBLzQJXchkr2oKOjcGaysoPFPe3HY1JH1rddTt1DdBG1vxpEZVM5bmVoFasV5\nnX9HULuX7R6Lmy6m/jf1E/TZoDutv91/O8MPDCd3ptxsddxKpbyVErw9IcTHDF4jj07kO5VSZaKf\nP1dKZXvv/Q+e/2ddqZGLZEMpGDQIzpyBfnO30W+vE9vabkvwfN33g+5TdmFZTjmdomDWgp9dPkpF\nsejUIkZ5jcKpghMja41M1Ohph+8cZqjHUF6FvmKq3VTsi9hb5KQmUiMX+mSu95HH+g3o0qULBQsW\nBCBr1qyUK1cOW1tb4P9P1chzeW4Jz3v18mL/fnBd/5Cf9g1iQuEJhP0dBvlJ0Pa6zumKfQb7d0k8\ntuUvBlzEcYYjKbQUeP3shXVO6wTvTy7rXAw/MJyjh47SrVw3JvSaQMoUKY3evoZ6/i9TiUeem9fz\nf/++desWCaaUMugDKABceO/5VSBX9N+5gauxrKvE5x08eNDYIZgNU22r5cuVKlBAqWkeS1SemXnU\nxYCLidresbvHVJ6ZedSr0FexLvcm7I1y2e+isk/LrhadWqQioyKVUglrp3sv76ke23uoHNNyqBmH\nZ6iQ8JCEhG525P+puDHV756pie5P8cqzSTEgjBb9+NcOoEv0352B7UkQgxAma/duGDYM2v9vDvMu\nTsCrsxc2OW0SvD2lFAP3DmRi3YmxXg2+98ZebObbcPvlbS72uYjTt04Jupjt5duXjDgwgjILy5At\nfTb8+/kzpNqQRN9jLoSIG4PWyDVNWwvYAl8CAcBoYBuwCd0Jw9tAG6VU4CfWV4aMTwhjO3YMvv8e\nWs2ZxIFny/Ho6EGBrP+9PjR+1l5cy+xjszne43iMiTngdQCD9g7i2L1jzG8yH/si9gn6nNCIUOaf\nnM+Uw1NoWrQpY2zHkD9L/kTFbo6kRi70KSE1chkQRggjuXQJ6tkpbMe5ciliOx4dPfgq81eJ2uab\nsDeUnFeSda3WUf3r6h+8F6Wi+P3M77h6utK1XFdG244mQ+oM8f6MKBXFuovrGHlwJKVzlmZSvUmJ\nOoNg7iSRC30yuQFhRNL470U34tNMpa1u3oSG9lGUHz6Q68od7y7eiU7iANOPTKda/mofJfErT65Q\ne0Vtfj/7O/s77mdq/amxJvFPtdO+v/fx7eJv+e3Eb6z8YSU72u1I1klcxJ2pfPcskSlctS5EsvLw\nIdg1iKRg/14EZbqCZwdPsqbLmujt3n15l99O/MYZpzPvXgsJD2Gi70QWnV7EWNux9Pq2V4LmIT/9\n4DTDDgzjzss7TK43mRYlWljkrWRCmCM5tS5EEnr+HGrZhpOydWe+LPCIHe126G140g5bOlA4a2HG\n1x0P6KYE7bO7D+Vzl8fN3o08mfPEe5s3X9xkpOdIvG55Mbr2aLqV7yaTmvyHnFoX+iQ1ciFM2OvX\nUK9BKM/qOlK8ZDib22xO1IAr7zt69ygOmxzw6+dHSHgIg/cNxue2D/Maz6Npsabx3t6TN0+Y4DOB\nNRfXMKDKAAZ/N5iMaTLqJVZLI4lc6JPUyJMpqT3FnbHaKjQUmrcO5n6tZpQrk4qtbbfqLYlHqSgG\n7h3IpLqT2Hh5IzYLbMiVMReXnS/HO4m/CXvDBJ8JFBlcBIXiat+r/Fr7V0niItHk/ynDkRq5EAYW\nHg4t2wVxoUxTGlYuyIoflpEqhf6+emsurCEkPITfz/5OcEQw7h3cKf9V+fjFGBnOsrPLGOczjloF\najG/yXw6NOqgtxiFEIYjp9aFMKDISHDo9BzPPPY41KzAou/nJ3gGsZg8C35GoTmFSKGlYFydcfSt\n1DdeF7Mppdjqt5URB0aQ1yovU+2mUjFPRb3FlxzIqXWhT+Y61roQFikqCjo4BbA/b3262zZgdqPp\ner3S2+uWF602tiJLuiwc6XYk3oOx+N72ZajHUELCQ5hjP4cG3zSQK9GFMENSI7cAUnuKu6RqK6Wg\n64B77PiyNv3tWuo1iT8LfkbX7V3p8GcHwiLD4p3ELz++TLN1zei4tSN9K/XlTK8zNCzS8IP4pE8J\nfZM+ZTiSyIXQM6Wg59CbbMhQi+H23ZnUYIxekrhSilXnV2E935osabNQNV9Vhnw3JM5J/F7QPbpv\n706dlXWoU7AO/v38+bHMj3o91S+ESHpSIxdCz/qO9mPJ2/pMbDScX2yd9bLN68+u03t3b16EvGDx\n94sJjQil3Z/t8Ovn99lhVgPfBjLl0BSWnFlCr2974VLdhSzpsuglLiE1cqFfUiMXwsicx55nSWgj\nZjSexIDaXRK9vbDIMKYdnobbMTdca7rSv0p/UmgpqLykMlPspsSaxN9GvGXeiXlMPTyV5sWbc6H3\nBfJa5U10TEII0yLn1CyA1J7izpBt1Wv8cRa/bcD87+foJYn73val3MJyHL9/nNNOpxn03SBSpUjF\nqvOrSJ0yNe1s2sW4XmRUJKvOr6L43OL43vHFq4sXS5otiVcSlz4l9E36lOHIEbkQetBzgg/Lg1uz\nrPkyOlWN/0hq73se8hyX/S78deMv5tjPoWXJlu9q7K/DXuPq6cpWx60f1d2VUrjfcMfFw4XMaTOz\npuUaanxdI1GxCCFMn9TIhUik7pP2svJVR9a2WkebivUSvB2lFOsurWPIviG0LNGSSfUmfVTLdj3g\nyt2gu6xqseqD10/eP4mLhwsPXz9kcr3JNC/eXG4lSyJSIxf6JDVyIZJY58nbWPPKiT/bbKN5+WoJ\n3s7fz//GeY8zj14/YpvjNqrkq/LRMrcCb7Ho9CLO9z7/7rUbz2/g6unKoTuH3k1qos9R44QQpk9q\n5BZAak9xp8+2aj95LWte9mZXW/cEJ/HwyHCmHJpClaVVsCtkx6mep2JM4gC/7P+FAVUGkNcqL4/f\nPKbfnn5UXVqVsrnKcq3fNZy+ddJbEpc+JfRN+pThyE93IRKgxfil7Hw9GvcOHtiVsUnQNo7ePYrT\nLifyW+XnlNMpCmYt+MllfW77cOL+CeY3ns9Yr7H8duI3OpbpiF8/P7JnyJ7AvRBCWAKpkQsRD0pB\nozFzOBAyC69uHlQvUTTe2wh8G8hwj+Fs99+Om70bDqUcYq1nR0ZFUnFxRcp/VZ6/bvxF3UJ1mVBn\nAoW+KJSYXRF6IjVyoU9SIxfCgJQC218ncSx0Gcd6+/DtNwXiub5i05VNDNo7iGbFmnGl7xWypsv6\n2XX6/dUPv2d+5MyYkz3t98R7ZjMhhGWTGrkFkNpT3CW0raKiFFVGjOBEyBrO9PeNdxK/FXiLpuua\nMs57HJscNrGg6YLPJnGvW15UWlKJpWeWMqPBDPZ23JtkSVz6lNA36VOGI0fkQnxGRGQU37oO4kaY\nLxeGeFE0b444rxseGY7bMTemHp7K4O8Gs9VxK2lSpol1nYsBFxl2YBhXn1zFJqcN1jms6Vupb2J3\nQwhhoaRGLkQswsIjsRnei0eRV7g4bA8FcsV+FP2+E/dP4LTTiZwZc7KgyQK+yfZNrMvfeXmHUQdH\n8deNvxhRYwQNvmlAzeU1udDnAnky50nsrggDkRq50CepkQuhR29Cwik5ojOvecSr+UcpMDtuSTwo\nNAjXA65svrqZGfVn0L50+1gvZnse8pzJvpNZdm4ZzhWdud7/OlZprWi1sRWDvxssSVwIESupkVsA\nqT3FXVzb6kVQKIWHORCmveTm+N0Q9vl1lFJsuboF6/nWhESEcNn5Mh3KdPhkEg8JD2Ha4WkUn1uc\nV2GvuNTnEuPrjscqrRVet7w48/AMg6oOisfe6Y/0KaFv0qcMR47IhfiPh0+DKTWuBVZpM3N1wkYy\npI29pg260+L99vTj+vPrrGm5hloFan1y2X8nNRntNZqKeSri29WXEtlLfPD+QPeBTLObRvrU6fWy\nT0IIyyU1ciHe88+DIMpMaUreDIW4MOF30qTS/db9VB00IiqC347/xkTfiQyoMoCh1YeSNlXaGLet\nlGLP9T0MOzCMLGmzMK3+NKrl/3hEuCWnl/DHhT/w7uIt46WbAamRC32SGrkQiXDxxnMq/8+eklm+\n5eTYeaRMEXvl6fSD0zjtciJL2iwc6X6EYl8W++Syx+8dZ6jHUJ68ecIUuyl8X+z7GJP0y7cvGeU1\nit3td0sSF0LEidTILYDUnuLuU23lcyaAb3+zpUruWpweNz/WJP4q9BWD3AfRZG0Tfqr8Ewc6Hfhk\nEr/27BqtN7am1cZWdCrTiQt9LtCseLNPJukJPhNoXKQxFb6qEO990yfpU0LfpE8ZjhyRi2Rv+8F7\ntNpuR7Nv2vFn/1GxHgnv8N9Bvz39qFe4HpecL31ynPNHrx8xznscGy9v5OdqP7OqxSoypM4Qaxw3\nnt9g+bnlXHK+lKj9EUIkL1IjF8na71tu4nTIjq42zizt9vMnl9OsNFoubcnFgIssarqIOoXqxLjc\nq9BXTD8ynXkn59GlbBdG1BzBlxm+jFMsP6z/gar5qjKsxrAE7YswDqmRC32SGrkQ8TBlqR+u/vUZ\nUnUE09r0iXGZyKhIFpxaAL3BOoc1a1quIV2qdB8tFxYZxuLTi5ngM4H639TntNPpWGcz+68DNw9w\nIeAC61uvT+juCCGSKamRWwCpPcWdl5cXSsGAyecZeb0uE+pO+GQSP/foHNWWVWPj5Y2wHMbVGfdR\nEo9SUWy4tIFS80qx69ou9v64lz9a/BGvJB4RFcHAvQOZ0WBGjD8SjEH6lNA36VOGI0fkIlmJjITW\nA46zI2Mz5jadS++aDh8t8ybsDWO8xrDy/Eom15tM1/JdSdkt5UfLef7jydD9QwFY/P1i6haqm6CY\nlp5Zypfpv6RFiRYJWl8IkbxJjVwkG2/eQP0e3pwq7MAah+U4lGvy0TJ7ru/BebczNb6uwayGs8iZ\nMSfwYR30/KPzuHi4cP35dSbVnYSDtQMptISd3Ap8G0jxucXZ++NeyuUul/CdE0YjNXKhT1IjF+IT\nHj2CWt3cuVOxI7s6bqBB0Q+Pnh++esgA9wGceXiGJd8vof439T/axu3A2/x68Ff2/b2PkbVG4vSt\n02dnMvuc8d7jaVasmSRxIUSCSY3cAkjtKXZ+flDGcSv3q3RiuvXoD5J4lIpiwckFlFlYhqLZinKx\nz8WPkviz4GfQACosrkChrIW41v8a/Sr3S3QSv/bsGivPr2RC3QmJ2o4hSJ8S+iZ9ynDkiFxYNC8v\naD5yLTQcjG8Pd4L8g969dzHgIk67nEihpcCrsxfWOa0/WDckPIQ5x+cw8+hMSA2XnS+TO1NuvcU2\nZN8QXKq7kCtTLr1tUwiR/EiNXFispUth8OolpGk4Bq/ue7HJaQNAcHgw473Hs/TsUibUmUDPb3t+\nUOOOiIpg5bmVjPEeQ9V8VZlYdyLFsxfXax1039/7cN7tzGXny58cm12YB6mRC32SGrkQ6K5Md3GB\nlX5zyNxkFl7dvCj6ZVEA9t7Yi/MeZyrnrczFPhc/OMJWSrHz2k6GHxhO9gzZ2eywmSr5qug9voio\nCAbvHcyMBjMkiQshEk0SuQXw8vLC1tbW2GGYhNevoX17uPTFJDLbLeNgFx8KZC1AwOsABu0dhOdB\nT5YPWE6joo0+WO/o3aMM9RhK4NtAptlNo3HRxgabtGTRqUXkypSL5sWbG2T7+iB9Suib9CnDkYvd\nhMW4exeq11DcKTqCdJXXcLi7L/mz5GfJ6SWUXlCa/Fb5WfHDig+SuN9TP1puaInjZke6l+/OuV7n\naFKsicGS+IuQF4zzGcfshrNldjMhhF5IjVxYhEOHwKGNokjfgbzJ7su+jvt4/OYxvXb1IjwynMXf\nL6ZMrjLvln/46iFjvMawxW8LQ6sNpV/lfqRPnf6T29dXHXSg+0DeRrxlYdOFid6WMA1SIxf6lJAa\nuRyRC7O3aBG0aBVJ2ZFOROY+ye72u3E75kbtFbVpZ9OOw90Ov0viQaFBjPQcic0CG6zSWuHfz59f\nqv8SaxLXF7+nfqy5uIbxdcYb/LOEEMmHJHILkFzvzwwLgz59YPaccKpO7UR4ppuMqDmCWitq4f/M\nn/O9z+NcyZmUKVISGhHKnGNzKDCwAPeC7nG211mmN5hOtvTZkizeIfuGMLzGcHJkzJFkn5lQybVP\nCcORPmU4crGbMEsBAdC6NVhlC6XICEfeRr0hd8bc9N3Tl3mN59G0WFNAN+DL+kvrGek5kpI5SjKz\n/ky6/dAtyeN1v+HO9WfX2eq4Nck/Wwhh2aRGLszOqVPQqhW06xTM6aI/EBj6gjsv7/Bj6R8ZW2cs\nmdJkAmD/3/tx8XAhdcrUTLWbim1B2wR/ZmLqoOGR4ZRZWIZpdtP4vvj3CY5BmCapkQt9kvvIhUVT\nSjfIy4gRMGteEHMC63Lv8T3yWeXDvYM75b8qD8DZh2dx8XDhn8B/mFxvMq1KtjLqFeILTy0kv1X+\nd2cJhBBCn6RGbgGSQ+0pJAS6dwc3N9iy7yHD7pXk8pPLDK8xnOM9jlP+q/L88+IfOmzpQOO1jWlR\nogVXnK/QulTrD5J4UrfVs+BnjPcZz6yGs8zqdrPk0KdE0pI+ZTiSyIXJu3kTqlfXJXPXP7Zgt7sg\nGVJl4Fq/awyoOoDnIc8Z6D6QiksqUixbMa73v06fSn1InTK1sUNnjNcYHEo5vBseVggh9E1qQbYC\nawAAHgxJREFU5MKk7d4N3brBwOHPOJfXmS1+W2hVshXrWq0jODwYt2NuzD42m7Y2bfm11q8Gm4Ak\nIXXQy48vY7vSlqt9r5I9Q3aDxCWMT2rkQp+kRi4sRng4jBwJa9Yqes79g1m3hhB2I4zRtUczrMYw\nlpxZwljvsdT4ugZHux99N5a6qVBKMXjfYFxrukoSF0IYlJxatwCWVnu6cwdq14Zj165TeEx9tjye\nQkotJZPrTcYmpw2lF5Rm3aV1bHPcxobWG+KVxJOqrfZc38PtwNv0rdQ3ST5P3yytTwnjkz5lOHJE\nLkzK9u3Qs3cYFfpP41QqNzrn68zai5fpUaEHay6t4VXoK2Y1mIV9EXuTvXgsLDKMwfsGM7vhbJOo\n0wshLJvUyIVJCAuDoUNh/VFf0jv0wibvN/Qo34NuO7pR+IvCPH7zmPF1xtOhdAdSpkiZ5PHFpw7q\ndswN9xvu/NXhL5P9sSH0R2rkQp+kRi7Mkr8/tOn8nJeVXUjh8BczGs9BKUXbzW1Jmyotba3b0rdy\nX9KlSmfsUD/rafBTJvpOxLuLtyRxIUSSkBq5BTDX2pNSsHChomK3tdxqYk0T+7Qc7XGYTVc20WZz\nG34o8QP/DPiHIdWG6C2JG7qtRh0cRTubdpTKUcqgn2No5tqnhOmSPmU4ckQujOLJE2jn/DcncjqT\n1+ERS1ps5NSDU5RZWIawiDD+bPMnLUq2MHaY8XLp8SU2X9nM1b5XjR2KECIZkRq5SHJ73MNp+9sM\nIirNxLXOL+TNkosxXmPIlj4bd17eYV/HfVT4qoKxw/zA5+qgSikarG5As2LN6F+lfxJGJoxNauRC\nn6RGLkzamzfQeeRRtkc5Ua5BXvrVnsnsY7NJnzo9jtaOrL64Gq8uXmY5CtrOazu5H3Sf3hV7GzsU\nIUQyIzVyC2AOtSd3r0DyOPVhT6ZWuDRpi1XWcCYdmsSo2qNwLOXI+svr8eps+CRuiLYKiwxjyL4h\nzGo4y2JuNzOHPiXMi/Qpw5FELgwqJETRbPhGmuyxpkylQOxtqrLs4jzaWLfhUp9L+D31Y+7Jufh0\n8TG50dni6rfjv1Hsy2LYF7E3dihCiGRIauTCYHb43KLDmr6Q7W/qlCrNkYcHGVh1IIOqDiJD6gy4\nerqy3X87Hh09+CrzV8YON1afqoM+fvOYUvNKcajbIUpkL2GEyISxSY1c6JPUyIVJCHodQbNJbnhH\nTcamSAUeqqcUyp6H31teJUfGHCilGOg+EN87vnh38TbrschHHRzFj2V+lCQuhDAaObVuAUyp9jR/\n+wmyu37LGfU72bOlwqZwdo73OM6cRnPIkTEHkVGROO104uSDk3h29kzyJK7PtroQcIGtflsZVXuU\n3rZpKkypTwnLIH3KcOSIXOjFnYAg7GeMwC/1GrLmTE+F/KWYareab/N8+26Z8MhwOm/rzKPXj9jX\ncR+Z0mQyYsSJ8+9ZhVG1RpEtfTZjhyOESMakRi4SJSpK4bJiKzP9e5M2LRTKmQu3RjOpX7j+B0OU\nhkaE4rjZkfCocDY7bCZ96vRGjDr+/lsH3ea3DVdPV873Pk+qFPJ7ODmTGrnQJ6mRiyTle+EOzVd0\nIjDDKVTkG5a2Xk270u1IoX1YsQkOD6bFhhZYpbVio8NG0qRMY6SI9SM0IpSf9/3M/CbzJYkLIYxO\nauQWIKlrT8EhkdiOHU2tTcV4neUUkxqMATfoUKbDR0k8KDQI+9X25M6Um3Wt1hk9ieujreYcn0PJ\nHCVp8E2DxAdkoqSeKfRN+pThSCIX8TJzsxdWY/PioybRoVRXnrjcZ1itnyHy42WfhzzHbpUd1jms\nWd58uUUcvQa8DmDa4WnMbDDT2KEIIQQgNXIRR6evPqXhspY8y3gI68w1ce+xhnxZ8r17/791woDX\nATRY3YD6heszvf50s5/S89/967mjJ5nTZmZWw1nGDkmYCKmRC32SGrnQu5dBkdhNH8KpFHPJkikP\nB9ofom7RarGucy/oHnar7Ghn045RtUeZfRL/19mHZ9lxbQf+/fyNHYoQQrxjtFPrmqbZa5rmp2na\nNU3TXIwVhyXQZ+0pMjKStWvXMm7ceOp0diJrnfScfrmAsdVmEzj6TqxJPDIyErclbpRuV5pyj8rx\neNNjbt26pbfY9CExbTXwr4E0eduEuTPmsnLlSvr27cs///yjv+BMiNQz4+ff782ECRMsvm8klPQp\nwzFKItc0LQUwF2gIWAPtNE2TobGM6MmTJ5w8eRJvb2+epc/H+BPz8ApcSoXa1TnW2ZdaaUszYsSI\nT64L8MfOP5gROIMWVVtgX8QeBwcH7t69i7e3N9OmTUvK3dGrf/fvrt9d5v48l8KFC6OUwsHBgdy5\nc7Njxw4ePnxo5CiFMfzbN7y9vWnduvUHfePu3bt4eHiwaNEiI0cpLJ2xjsgrA9eVUreVUuHAeqC5\nkWIxe7a2tolaf926DRQoUALbliOpt2IIP521I/Wzt+wesZvTsw7y8N5DihYt+u4/rZjWBeh6qBvf\nZ25GxscZqV27Nra2tly9epWqVavy9OlT3rx5k6g49SG+bbVu3Qa+LlwcgPs7n7J9606OHj36bv+C\ngoJYsWKFxdVIE9unkoP3+37Tpg78+efWD/rGjh07sLOzIzQ0lLt37xo5WuOTPmU4xkrkeYH3e/a9\n6NdEEnvy5Ald+/1KSL0mBHf2hZRXSbEnLRVS2tC4SiN8fX1p3rw5r169olKlSh+t2727MyHZ/qd7\nYUt/Vgxax7lz5yhUqBC+vr706tWL1KlTExkZScaMGY2whwn37/69LdcRgDC/hXTr1vuD/cuVKxfl\nypUzcqQiqb3r+yEHAQgJ+e2jvhESEgJApkyZCAgIMGa4wsIZ5ap1TdNaAQ2VUk7Rz38EKiulfvrP\ncpZ1mGNqUmUE62LQ+DqocDgaDoejICIe28gFdAKmGyhGY0sH/ASYb2VACGFm4nvVurGOyO8DX7/3\nPF/0ax9RSsnjM4+DBw/Ga/knz8JoNHou2uCMZGz6FN6GwiY38I6EiPOkT5+Nx48fv1v+8OHD+Pj4\n4OrqSmho6LvX159aD5012LwYUMDH665YsYLu3bvTs2dPIiMjzaqtHj9+TMrGaeBs50/uX0BAAO3b\nt+ePP/4w+r4Zs08lt8fjx49Jnz4bcB44GGPf8PX1xdPTEzc3N6PHawoP6VNxeySIkQJNCdwACgBp\ngHNAyRiWU0J/goKiVNvxG1SKgUVUlhElVeaJWdVwj+FqxepVKn36bMrKqrxKnz6bWrt2/We3tfXq\nVpVjWg41+vex8V7XXJx+cFplGZ9FpcuS1SL3TyTO2rXrLbbvC+OJznvxyqlGGxBG0zR7YA66swK/\nK6WmxLCMMlZ8luTlSxjy20FWPnAhQ9ZX5MiukSvrFyxqugibnDaAruZ369YtChYsSI4cOWLd3tqL\naxm8dzB7OuyhwlcV4rWuuVBKUWtFLTqW6UiLr1tY3P4J/bDEvi+MKyEDwhjtPnKllLtSqrhSqmhM\nSVzE3afuz3zyBHqOvECOgY1Y/7YbNcrlJnXWJ/xSawC+XX3fJXGAHDlyUKlSpc/+Z7T0zFJ+2f8L\nHp08qPBVhXitawriei/r5iubeRX6iu7lu5vV/umL3PMbNzly5ODNmzfJqm8klPQpw5GR3SzQvXsw\natZt1jz4lZTF9uLQoCVHH/uTK1sG1rW/RO5MuRO03TnH5jDr2Cy8OntR9Muieo7adISEh/DL/l9Y\n3nw5KVOkNHY4QggRKxlr3YKcOQOT3Z6xM3AyWvnl/Fi6M4FRdzn18BTzG8+nUdFGCd72JN9JLD+3\nHI+OHhTIWkCPUZueSb6TOPXgFFsctxg7FCFEMpOQU+uSyM1cVBTs3g3T3UI4n+5/RFaeQWublljn\nLsL0I9PpWq4ro2qPImOahN3DrZTC1dOV7f7b8ejowVeZv9LzHpiWB68eUGZBGU70PEHhLwobOxwh\nRDJjVjVykTgvX8LcuVCiVCQ9pg7lct1i1PnxOOvbLedm0FU2X93M/o77mVp/aoKTeJSKYqD7QNxv\nuOPdxdsikvjn6nSunq70qNAj2SdxqWfGnbRV3Eg7GY4kcjNz/jz06gUFCio2nN1JePcy5Kz2F392\n+IPSuazpur0rjtaOHOl2hLK5yyb4cyKjInHa6cSph6fw7OxJ9gzZ9bgXpunUg1PsvbGXETVjHlNe\nCCFMkZxaNwMhIfDnn7BgAdy5A42cjnIhpwuvIp8xud5kMqTKQJ89fSiXuxxz7OeQJ3OeRH1eeGQ4\nnbd1JuBNANvbbidTmkx62hPTpZSi5vKadC3Xle4Vuhs7HCFEMiXzkVsQpeDECVi+HDZtgkqVoH1/\nfw4wgr8enmBs5bE0LtKYoR5D8b7tzbzG82harGmiP/dtxFscNzsSGRXJrna7SJ86vR72xvRtuLyB\n4PBgupTrYuxQhBAiXuTUuol59AhmzAAbG+jQAfLnh71HHlKwX2/G3K1B1fxV8O/rj4ZG2UVlyZEh\nBwutF+oliQeHB9NsXTPSpEzDFsctFpnEY6rThYSH4OLhgpu9m9xuFk3qmXEnbRU30k6GI4ncBLx4\nAUuXQr16UKIEXL4MCxfC6UtBvP3uVxputyFTmkz49/OnWfFmNF7bmPmn5uPewZ2ZDWfqJeEGhQZh\nv9qerzJ/xbpW60iTMo0e9sw8zDgyg8p5K1OrQC1jhyKEEPEmNXIjefUKdu2CdevA2xvs7KBdO2jS\nBFKmCWPhqYVM9J2IfRF7xtmOI1emXEw5NIW5J+YyqvYo+lbqq7ejx+chz7Ffbc+3X33LvCbzSKEl\nn99394PuU3ZhWU45naJg1oLGDkcIkcxJjdzEBQTAzp2wdSv4+kKNGtC2LaxeDVZWutu9NlzagKun\nKyWyl2B/x/2UyVUGr1teNFjdgJLZS3K211nyZ8mvv5heB9BgdQPqF67P9PrT0bR49R+zN/zAcHp9\n20uSuBDCbCWfQy8jUEp3mnzGDKhZE4oXBw8P6NgR7t6FPXugUyddEve46UGlJZWYfWw2vzf7nT0d\n9pA3c166bu9Kx60dmWo3lW1tt8WYxBNae7oXdI/aK2rTokSLZJPE32+r4/eOc+CfAwyrMcx4AZko\nqWfGnbRV3Eg7GY4cketZYKAuWbu7w969kCoV2NvDiBFQty6kTfvh8mcfnmXYgWHcfHGTSXUn0bpU\nawBWnV/F0P1DcbR25LLzZazSWuk1zpsvbmK3yg7nSs78XO1nvW7bHCilGLh3IBPrTiRz2szGDkcI\nIRJMauSJ9OoVHD4MXl66WvelS7qjb3t73aNoUYjpQPdW4C1Geo7kwD8H+LXWr/Ss0JPUKVNz/dl1\n+uzuw7OQZyxuuphKeSvpPWa/p37U/6M+I2qMoE+lPnrfvjlYe3Ets47O4kTPE8nqmgAhhGmTsdaT\nwIMHcPw4HD2qS9yXL0PFimBrC7VrQ9WqkD6Wi8ifBj9los9EVl1YRf/K/Rny3RAyp81MWGQY0w9P\nZ/ax2QyvMZwBVQeQKoX+T5icf3SeRmsaMbneZDqX66z37ZuDN2FvKDmvJGtbraXG1zWMHY4QQrwj\nF7vp2fPncOGCbmCW48d1j7dvoUoVXcKePh0qV4Z06T6/reDwYNyOuTHr6CwcrR254nyFXJlyAXDo\nziGcdjpR+IvCnHY6He/Zxby8vLC1tf3scsfvHafZ+mbMbTQXB2uHeH2GpfDy8sILL6rlryZJPBZx\n7VNC2iqupJ0MRxI5uiFQb9yAK1d0ifv8ed3j5UsoXVp3xN2qFUybBoULx3yq/FMioiJYfnY5Y7zH\nUD1/dY52P/puLu8XIS9w8XBh9/XdzLGfQ6uSrQx2wZnPbR9ab2zN8ubLaVKsiUE+wxw8fv2Y3678\nxhmnM8YORQgh9CJZnFpXCp4+1Y1Tfvcu3L4N16/DtWvg76+7LaxwYd1gLGXL6h5lykDBgpAigeVT\npRTb/bcz/MBwcmXMxbT606ict/K799ZfWs+QfUNoUaIFk+pNIku6LInez0/Ze2MvHbd2ZH3r9dQt\nVNdgn2MOOmzpQOGshRlfd7yxQxFCiI9YdI1cKd3c2+Hh8Pq17vHq1f//GxgIT558/Lh/X5e8M2TQ\nDXf69de6R9GiUKyY7pawr7/WXV2uL4fvHMbFw4Wg0CCm2k3Fvoj9uyPtmy9u4rzbmYevH7Ko6SKq\n5quqvw+OwTa/bTjtdGJb221Uy1/NoJ9l6o7ePYrDJgf8+vkli4lghBDmxyJr5BkyQESELoGnSKFL\nuJky6R6ZM///v1ZWkDMn5MihO7KuWVP3d968ugSeMWFTcsfL1SdXGX5gOGcfnWV8nfF0KN3h3ehr\n4ZHhzDw6kxlHZjC0+lAGVR1E6pSp9fK5n6o9rb24lsF7B+P+ozsVvqqgl88yV1EqioF7B9LJqpMk\n8TiQembcSVvFjbST4Zh8In/6VJe8U6VK+GluQ3vw6gFjvMawzW8bQ6sPZX3r9aRL9f9XwB29exSn\nXU7ks8rHyZ4nKfRFIYPHtPTMUsZ4jeFApwNY57Q2+OeZujUX1gBg942dkSMRQgj9MptT66bo5duX\nTDs8jYWnF9KjfA+G1RjGF+m/ePd+4NtAhnsMZ7v/dmY3nE0b6zZJMnranGNzmH1sNh6dPCiSrYjB\nP8/UvQ57TYm5JdjksInv8n9n7HCEEOKTEnJq3USPcU1baEQobsfcKDa3GA9fP+Rcr3NMrT/1XRJX\nSrHx8kas51ujUFzpewVHG8ckSeKTfCcx9+RcvLt4SxKPNu3wNGoXrC1JXAhhkSSRx0OUimLNhTWU\nmFcCj5seHOh0gGXNl30w/vmtwFs0XdeUcd7j2OSwiYVNF5I1XVaDxuXl5YVSihEHRrDm4hp8uvjE\n+150S3U78DbzTs5jSr0pgIz3HFfSTnEnbRU30k6GY/I1clOx7+99uHi4kDZlWlb+sPKjuasjoiJw\nO+bGlENTGPzdYLY6bk2yOb2VUgx0H4jvHV+8u3iTPUP2JPlcc+Di4UL/yv31OmOcEEKYEqmRf8aZ\nh2dw8XDhduBtJtebTMuSLT86RX7i/gmcdjqRI2MOFjRZkKSntCOjIum9qzeXn1xmT4c9Bj/6NyeH\n7xym7Z9t8e/nT4bUGYwdjhBCfJZF3n5mLDdf3GSk50gO3jrIqFqj6FGhx0e3iwWFBjHScySbrmxi\nRv0ZtC/dPkmnAg2PDKfL9i48fPWQfR33yW1V74lSUQxwH8CUelMkiQshLJrUyP/jyZsnDPhrAJWX\nVKZk9pJc73+dPpX6fJDElVJsubqFUvNKERwezGXny3Qo0yFJk3hoRChtNrch8G0gLnldJIn/x6rz\nq0idMjXtS7f/4HWp08WNtFPcSVvFjbST4cgRebQ3YW+YfWw2bsfcaF+6PVf6XiFnxpwfLXf35V36\n/dWPa8+usbbV2o9q5UkhODyYFhtaYJXWig2tN3DE90iSx2DKXoe9xtXTlS1ttiTpjyshhDCGZF8j\nD48MZ9nZZYz1HkutArWYWHci32T75qPlIqIimHtiLhN8JjCgygCGVh9K2lRpDRpbTIJCg2i6timF\nvijE781+N8hUp+bO9YArd4Lu8EeLP4wdihBCxIvUyONBKcVWv60MPzCcfFb52NFuBxXzVIxx2dMP\nTuO0y4ksabNwpPsRin1ZLImj1Xke8pxGaxpRIXcF5jWZRwpNKiP/dSvwFotOL+J87/PGDkUIIZJE\nsswEvrd9qbasGuO8x/E/+//h0dEjxiT+Ouw1g/cOpvHaxvxU+ScOdDpgtCQe8DqAOivrUOvrWsxv\nMv+DJC61p/83dP9QBlQZQF6rvDG+L20VN9JOcSdtFTfSToaTrBL55ceXabauGR23dsS5ojNnep2h\nYZGGMdZRd/rvxHq+Nc9DnnOpzyU6l+tstHrrvaB71F5Rm5YlWjKt/rSP4jh37pxR4jI1Prd9OH7/\nOEOqDfnkMtJWcSPtFHfSVnEj7WQ4yeLU+r2ge4w+OJqd13YyvMZwNjps/GBSk/fdD7rPT+4/cTHg\nIsubLzf6/N03X9zEbpUdzpWc+bnazzEuExgYmMRRmZ7IqEgGug9kqt3UWG83k7aKG2mnuJO2ihtp\nJ8Ox6CPywLeBDPMYRtmFZcmZMSfX+l9j0HeDYkzikVGRzDsxj3KLymGTw4YLfS4YPYn7PfWj9ora\n/FLtl08mcaGz4twKMqTOgKO1o7FDEUKIJGWRR+RvI94y78Q8ph6eSrPizbjQ+8Ina6YA5x+dx2mX\nE2lTpsWniw8lc5RMwmhjdiHgAvar7ZlcbzKdy3WOddlbt24lTVAmKig0iJEHR7Kz3c7Plj+Se1vF\nlbRT3ElbxY20k+GY/O1nxo5BCCGESErxvf3MpBO5EEIIIWJn0TVyIYQQwtJJIhdCCCHMmCRyIYQQ\nwoyZZCLXNM1e0zQ/TdOuaZrmYux4TJWmafk0TfPUNO2ypmkXNU37ydgxmTJN01JomnZG07Qdxo7F\nlGmalkXTtE2apl2N7ltVjB2TKdI0bXh0+1zQNG2NpmlpjB2TqdA07XdN0wI0Tbvw3mtfaJq2T9M0\nf03T9mqalsWYMZqCT7TTtOjv3jlN0/7UNM3qc9sxuUSuaVoKYC7QELAG2mmaVsK4UZmsCGCwUsoa\n+A7oK20VqwHAFWMHYQbmAHuUUiWBssBVI8djcjRNKwD0BMorpcqgu5W3rXGjMinL0f0f/r5hgIdS\nqjjgCQxP8qhMT0zttA+wVkqVA64Th3YyuUQOVAauK6VuK6XCgfVAcyPHZJKUUo+UUuei/36N7j/c\nT98wn4xpmpYPaAwsNXYspiz6139NpdRyAKVUhFIqyMhhmaIgIAzIqGlaKiAD8MC4IZkOpdQh4MV/\nXm4OrIz+eyXwQ5IGZYJiaiellIdSKir66TEg3+e2Y4qJPC9w973n95Dk9FmaphUEygHHjRuJyZoN\n/ALI/ZaxKwQ81TRteXQZYrGmaemNHZSpUUq9AGYCd4D7QKBSysO4UZm8nEqpANAdhAA5jRyPOegG\n/PW5hUwxkYt40jQtE7AZGBB9ZC7eo2laEyAg+uyFFv0QMUsFVADmKaUqAMHoTomK92iaVhgYBBQA\n8gCZNE1rb9yozI78qI6FpmmuQLhSau3nljXFRH4f+Pq95/miXxMxiD6ttxn4Qym13djxmKjqQDNN\n024C64A6mqatMnJMpuoecFcpdSr6+WZ0iV18qCJwWCn1XCkVCWwBqhk5JlMXoGlaLgBN03IDj40c\nj8nSNK0LulJgnH4cmmIiPwkU0TStQPRVoG0Bucr405YBV5RSc4wdiKlSSo1QSn2tlCqMrj95KqU6\nGTsuUxR96vOupmnFol+qh1wgGBN/oKqmaek03QD/9ZCLAv/rv2e/dgBdov/uDMiBh84H7aRpmj26\nMmAzpVRoXDZgcpOmKKUiNU3rh+7KvRTA70op+YLEQNO06kAH4KKmaWfRnaoaoZRyN25kwsz9BKzR\nNC01cBPoauR4TI5S6nz0WZ3TQCRwFlhs3KhMh6ZpawFb4EtN0+4Ao4EpwCZN07oBt4E2xovQNHyi\nnUYAaYD90ZNAHVNKOce6HRlrXQghhDBfpnhqXQghhBBxJIlcCCGEMGOSyIUQQggzJolcCCGEMGOS\nyIUQQggzJolcCCGEMGOSyIWwUNHT3N7UNC1r9PMvop9//bl147DtQ4mPUAihD3IfuRAWTNO0n4Gi\nSqlemqYtAv5WSk0zdlxCCP2RI3IhLJsbUEXTtAHoxgKfGdNCmqZt1TTtpKZpFzVN6xH92teapl3T\nNC2bpuOjaZpd9Huvov/NrWmad/RMaReiRxsUQiQhOSIXwsJpmtYAcAfslFKen1gmq1IqUNO0dOjm\nO6illHoRPZymPXAC+EYp1Sd6+SCllJWmaYOBtEqpydFjjmdQSr1Jkh0TQgByRC5EctAYeACUjmWZ\ngZqmnQOOoZtxsCiAUmoZYAX0An6OYb2TQFdN00YBZSSJC5H0JJELYcE0TSuHbmauqsDgf6eR/M8y\ntYG6QBWlVDngHJAu+r306BI7QKb/rquU8gVqoZtqeIWmaT8aYj+EEJ8miVwIyzYfGKCUugdMI+Ya\neRbghVIqVNO0EuiS/r+mAquBUcDS917XQFdHBx4rpX6Pfl/mLhciiUkiF8JCaZrWE7j9Xl18AVBC\n07Sa/1nUHUitadplYBJwNHr9WkBFYKpSah0Qqmla5+h1/r24xhY4r2naGXTTUs4x1P4IIWImF7sJ\nIYQQZkyOyIUQQggzJolcCCGEMGOSyIUQQggzJolcCCGEMGOSyIUQQggzJolcCCGEMGOSyIUQQggz\n9n/nwI3XXGP9ZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x3f185d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x=np.linspace(-5,10,100)\n",
    "f = lambda x : 0.5*x**2-3\n",
    "f1 = lambda x : x\n",
    "def ftayl(a):\n",
    "    def fa(x):\n",
    "        return f(a)+(x-a)*f1(a)\n",
    "    return fa\n",
    "\n",
    "fig=plt.figure(0,figsize=(8,8))\n",
    "ax=fig.add_subplot(111)\n",
    "plt.title(\"Overview of newton method\")\n",
    "plt.xlabel('X axis')\n",
    "plt.ylabel('Y axis')\n",
    "ax.grid(True, which='both')\n",
    "ax.axhline(y=0, color='k')\n",
    "ax.axvline(x=0, color='k')\n",
    "plt.xlim(0,12)\n",
    "plt.ylim(-5,50)\n",
    "\n",
    "#Plot original function\n",
    "plt.plot(x,f(x))\n",
    "\n",
    "#Initial estimate\n",
    "xi = 8\n",
    "\n",
    "for i in range(4):\n",
    "    #Plot estimate\n",
    "    plt.scatter(xi,0)\n",
    "    plt.plot([xi,xi],[0,f(xi)],'k')\n",
    "    ax.annotate('$x_'+str(i)+'$',(xi,0))\n",
    "    #plot linearization in xi\n",
    "    plt.plot(x,ftayl(xi)(x),'g')\n",
    "    #Compute next estimate\n",
    "    xi = xi-f(xi)/f1(xi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Extending Newton method for convex optimization\n",
    "\n",
    "We are generally not really interested in finding the root of an objective function, plus, there is very few chances that we would ever be able to prove it exists.\n",
    "However, when manipulating functions that are convex and 2 time differentiables, we know, thanks to Fermat theorem, that its derivative vanishes at the optimal point. So finding the root of the derivative of our objective seems to be a much more interesting challenge.\n",
    "\n",
    "We will replace $f(x)$ by $\\nabla f(x)$ in our previous equations to obtain the desired equation update:\n",
    "\n",
    "\\begin{align*}\n",
    "    f1_{lin,a}(x) &= 0\\\\\n",
    "    \\nabla f(a) + H_f(a) (x-a) &= 0\n",
    "\\end{align*}\n",
    "\n",
    "Here we recall that $H_f(a)$ the Hessian of $f$ in $a$ can be defined as  $ H_f(a) = \\nabla \\nabla f(a) =\n",
    "\\begin{pmatrix}\n",
    "\\frac{\\partial f(a))}{\\partial x_0 \\partial x_0} & \\frac{\\partial f(a))}{\\partial x_1 \\partial x_0} & \\cdots & \\frac{\\partial f(a))}{\\partial x_{n-1} \\partial x_0} \\\\\n",
    "\\frac{\\partial f(a))}{\\partial x_0 \\partial x_1} & \\frac{\\partial f(a))}{\\partial x_1 \\partial x_1} & \\cdots & \\frac{\\partial f(a))}{\\partial x_{n-1} \\partial x_1} \\\\\n",
    "\\vdots & \\vdots & \\vdots & \\vdots \\\\\n",
    "\\frac{\\partial f(a))}{\\partial x_0 \\partial x_{n-1}} & \\frac{\\partial f(a))}{\\partial x_1 \\partial x_{n-1}} & \\cdots & \\frac{\\partial f(a))}{\\partial x_{n-1} \\partial x_{n-1}} \\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "The equation becomes:\n",
    "\n",
    "\\begin{align*}\n",
    "    \\nabla f(a) + H_f(a) x - H_f(a) a &= 0 \\\\\n",
    "    H_f(a) x &= H_f(a) a - \\nabla f(a) \\\\\n",
    "    x &= H_f^{-1} (H_f(a) a - \\nabla f(a)) \\\\\n",
    "    x &= a - H_f^{-1} \\nabla f(a)\n",
    "\\end{align*}\n",
    "\n",
    "We can notice that we have to compute $H_f^{-1}$, the inverse of $H_f$. Although $H_f$ is a symmetric matrix, this should be considered as a very difficult task in the general case, especially when dealing with high dimensional problems."
   ]
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   "source": [
    "## Second order method in the linear case\n",
    "For a simple quadratic problem such as the least square minimization, it is very simple to prove that the Newton method converges in only one step, and is equivalent to compute the pseudo inverse. Let's define the following problem:\n",
    "\n",
    "* $f(x) = \\|Ax-b\\|_2^2$\n",
    "* $\\nabla f(x) = A^t A x - A^t b$\n",
    "* $H_f(x) = A^t A$\n",
    "\n",
    "The newton method can be used on the gradient, because the probem is convex, and two time differentiable. Assuming an initial solution vector $x_0$, the first iterate reads:\n",
    "\n",
    "\\begin{align}\n",
    "    x_1 &= x_0 - H_f^{-1} \\nabla f(x_0)\\\\\n",
    "    &= x_0 - (A^t A)^{-1} (A^t A x_0 - A^t b)\\\\\n",
    "    &= x_0 - (A^t A)^{-1}A^t A x_0 + (A^t A)^{-1}A^t b\\\\\n",
    "    &= (A^t A)^{-1}A^t b \\\\\n",
    "    &= A^+\n",
    "\\end{align}"
   ]
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    "## Low cost Hessian estimation\n",
    "\n",
    "Newton method is then useless for our least square estimate problem, because it amounts to the pseudo inverse, however, it is interesting to notice that one can use a \"low cost\" estimation of $H_f^{-1}$ instead of computing the full inverse.\n",
    "Let's take a closer look at the algorithme update:\n",
    "\n",
    "\\begin{align*}\n",
    "    x^{k+1} &= x^{k} - H_f^{-1} \\nabla f(x^{k})\n",
    "\\end{align*}\n",
    "\n",
    "We can reformulate this update as a fixed point method:\n",
    "\n",
    "\\begin{align*}\n",
    "    x^{k} &= x^{k} - H_f^{-1} \\nabla f(x^{k})\\\\\n",
    "    x^{k} &= T x\n",
    "\\end{align*}\n",
    "\n",
    "We recall that, from the Picard fixed point theorem, this class of algorithm converges if the operator $T : x \\rightarrow x - H_f^{-1} \\nabla f(x)$ is a strict contraction. In our least square case, this reduces to an operator norm condition:\n",
    "\n",
    "The method converges if it exists $\\rho \\in [0,1[$ such that\n",
    "\n",
    "\\begin{align*}\n",
    "\t  \\forall (x,x') \\in \\mathbb{R}^n \\times \\mathbb{R}^n, x \\neq x' :\\quad &\\|Tx - Tx'\\| \\leq \\rho \\|x-x'\\| \\\\\n",
    "\t  &\\|x - H_f^{-1} \\nabla f(x) - x' + H_f^{-1} \\nabla f(x')\\| \\leq \\rho \\|x-x'\\| \\\\\n",
    "\t  &\\|x - x' + H_f^{-1} \\nabla f (x'-x)\\| \\leq \\rho \\|x-x'\\| \\\\\n",
    "\t  &\\|(Id - H_f^{-1} \\nabla f) (x-x')\\| \\leq \\rho \\|x-x'\\| \\\\\n",
    "\t  &\\|(Id - H_f^{-1} \\nabla f)\\| \\|(x - x')\\| \\leq \\rho \\|x-x'\\| \\\\\n",
    "\t  &\\|Id - H_f^{-1} \\nabla f\\| \\leq \\rho\n",
    "\\end{align*}\n",
    "\n",
    "An extremely simple approximation of $H_f^{-1}$ that is easily invertible, and satisfy the above condition is $H_f^{-1}=\\frac{1}{\\|\\nabla f\\|} Id$.\n",
    "This method is a simple instance of the gradient descent."
   ]
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   "source": [
    "## Barzilai-Borwein and quasi Newton method\n",
    "\n",
    "The fact that $H_f^{-1}$ is difficult to compute, give rise to a variety of optimization methods, where this matrix is estimated. Those method are known as quasi-Newton methods.\n",
    "\n",
    "In the general case, we recall that the second order Taylor expansion of a function $f$ as defined above reads\n",
    "\n",
    "\\begin{align*}\n",
    "    f(x^{k}+\\Delta x)\\approx f(x^{k})+ \\langle \\nabla f(x^{k}),\\Delta x \\rangle +{\\frac {1}{2}}\\Delta x^t H_f \\Delta x\n",
    "\\end{align*}\n",
    "\n",
    "Which derivative with respect to $\\Delta x$ gives\n",
    "\n",
    "\\begin{align*}\n",
    "    \\nabla f(x^{k}+\\Delta x)\\approx \\nabla f(x^{k}) + H_f \\Delta x\n",
    "\\end{align*}\n",
    "\n",
    "When looking for a $\\Delta x$ such that this expression vanishes, we have the Quasi Newton relationship or secant equation:\n",
    "\n",
    "\\begin{align*}\n",
    "    0 &= \\nabla f(x^{k}) + H_f\\Delta x\\\\\n",
    "    H_f\\Delta x &= -\\nabla f(x^{k})\\\\\n",
    "    \\Delta x &= -H_f^{-1} \\nabla f(x^{k})\n",
    "\\end{align*}"
   ]
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   "cell_type": "markdown",
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   "source": [
    "### Barzilai-Borwein method\n",
    "\n",
    "In 1988, in \"Two-Point Steap Size Gradient Methods\", Barzilai and Borwein derived a two point approximation for the secant equation seen earlier that reads\n",
    "\n",
    "\\begin{align*}\n",
    "    H_f (x^{k+1} - x_k) &= \\nabla f(x^{k+1}) -\\nabla f(x^{k}) \\\\\n",
    "    x^{k+1} - x_k &= -H_f^{-1} ( \\nabla f(x^{k+1}) -\\nabla f(x^{k}) ) \\\\\n",
    "    \\Delta x &= -H_f^{-1} \\Delta \\nabla f(x)\n",
    "\\end{align*}\n",
    "\n",
    "With $\\Delta x = x^{k}-x^{k-1}$ and $\\Delta \\nabla f(x) = \\nabla f(x^{k}) -\\nabla f(x^{k-1})$.\n",
    "\n",
    "Barzilai-Borwein idea is simply to assume that $H_f^{-1}$ writes $\\lambda Id$, like in the gradient descent method, and then looks for the $\\lambda$ that is close to satisfy their two point Quasi Newton relationship in the least square sense:\n",
    "\n",
    "\\begin{align*}\n",
    "    \\lambda^\\star = \\underset{\\lambda \\in \\mathbb{R}}{argmin} \\quad \\| \\Delta x +\\lambda Id \\Delta \\nabla f(x) \\|_2^2\n",
    "\\end{align*}\n",
    "\n",
    "Minimizing this simple problem gives:\n",
    "\n",
    "\\begin{align*}\n",
    "    \\lambda^\\star = - \\frac{\\langle \\Delta \\nabla f(x), \\Delta x \\rangle}{\\langle \\Delta \\nabla f(x), \\Delta \\nabla f(x) \\rangle}\n",
    "\\end{align*}\n",
    "\n",
    "The newton method updates, that previously read\n",
    "\n",
    "\\begin{align*}\n",
    "    x_{new} &= x + \\Delta x \\\\\n",
    "    x^{k+1} &= x^{k} - H_f^{-1} \\nabla f(x^{k})\n",
    "\\end{align*}\n",
    "\n",
    "Now reads\n",
    "\n",
    "\\begin{align*}\n",
    "    \\Delta x &= x^k - x^{k-1} \\\\\n",
    "    \\Delta \\nabla f(x) &= \\nabla f(x^{k}) - \\nabla f(x^{k-1}) \\\\\n",
    "    x^{k+1} &= x^{k} - \\frac{\\langle \\Delta \\nabla f(x), \\Delta x \\rangle}{\\langle \\Delta \\nabla f(x), \\Delta \\nabla f(x) \\rangle} \\nabla f(x^{k})\n",
    "\\end{align*}\n"
   ]
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