LinearAlgebra.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Jupyter notebooks\n",
    "\n",
    "This is a [Jupyter](http://jupyter.org/) notebook using Python.  You can install Jupyter locally to edit and interact with this notebook.\n",
    "\n",
    "# Linear Algebra\n",
    "\n",
    "You have all seen basic linear algebra before, but this will summarize some different ways of thinking about the fundamental operations.\n",
    "\n",
    "Linear algebra is the study of linear transformations on vectors, which represent points in a finite dimensional space.  The matrix-vector product $y = A x$ is a linear combination of the columns of $A$.  The familiar definition,\n",
    "\n",
    "$$ y_i = \\sum_j A_{i,j} x_j $$\n",
    "\n",
    "can also be viewed as\n",
    "\n",
    "$$ y = \\Bigg[ A_{:,0} \\Bigg| A_{:,1} \\Bigg| \\dotsm \\Bigg] \\begin{bmatrix} x_0 \\\\ x_1 \\\\ \\vdots \\end{bmatrix}\n",
    "= \\Bigg[ A_{:,0} \\Bigg] x_0 + \\Bigg[ A_{:,1} \\Bigg] x_1 + \\dotsb . $$\n",
    "\n",
    "The notation $A_{i,j}$ corresponds to the Python syntax `A[i,j]` and the colon `:` means the entire range (row or column).  So $A_{:,j}$ is the $j$th column and $A_{i,:}$ is the $i$th row.  The corresponding Python syntax is `A[:,j]` and `A[i,:]`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 50, 130, 290])"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "plt.style.use('ggplot')\n",
    "\n",
    "def matmult1(A, x):\n",
    "    \"\"\"Entries of y are dot products of rows of A with x\"\"\"\n",
    "    y = np.zeros_like(A[:,0])\n",
    "    for i in range(len(A)):\n",
    "        row = A[i,:]\n",
    "        for j in range(len(row)):\n",
    "            y[i] += row[j] * x[j]\n",
    "    return y\n",
    "\n",
    "A = np.array([[1,2],[3,5],[7,11]])\n",
    "x = np.array([10,20])\n",
    "matmult1(A, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([[ 1,  2],\n",
       "        [ 3,  5],\n",
       "        [ 7, 11]]),\n",
       " array([10, 20]))"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A, x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 50, 130, 290])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def matmult2(A, x):\n",
    "    \"\"\"Same idea, but more compactly\"\"\"\n",
    "    y = np.zeros_like(A[:,0])\n",
    "    for i,row in enumerate(A):\n",
    "        y[i] = row.dot(x)\n",
    "    return y\n",
    "\n",
    "matmult2(A, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 50, 130, 290])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def matmult3(A, x):\n",
    "    \"\"\"y is a linear expansion of the columns of A\"\"\"\n",
    "    y = np.zeros_like(A[:,0])\n",
    "    for j,col in enumerate(A.T):\n",
    "        y += col * x[j]\n",
    "    return y\n",
    "\n",
    "matmult3(A, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 50, 130, 290])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# We will use this version\n",
    "A.dot(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 50, 130, 290])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Or this version (Python 3.5+)\n",
    "A @ x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Some common terminology\n",
    "\n",
    "* The **range** of $A$ is the space spanned by its columns.  This definition coincides with the range of a function $f(x)$ when $f(x) = A x$.\n",
    "* The (right) **nullspace** of $A$ is the space of vectors $x$ such that $A x = 0$.\n",
    "* The **rank** of $A$ is the dimension of its range.\n",
    "* A matrix has **full rank** if the nullspace of either $A$ or $A^T$ is empty (only the 0 vector).  Equivalently, if all the columns of $A$ (or $A^T$) are linearly independent.\n",
    "* A **nonsingular** (or **invertible**) matrix is a square matrix of full rank.  We call the inverse $A^{-1}$ and it satisfies $A^{-1} A = A A^{-1} = I$.\n",
    "\n",
    "$\\DeclareMathOperator{\\rank}{rank} \\DeclareMathOperator{\\null}{null} $\n",
    "If $A \\in \\mathbb{R}^{m\\times m}$, which of these doesn't belong?\n",
    "1. $A$ has an inverse $A^{-1}$\n",
    "2. $\\rank (A) = m$\n",
    "3. $\\null(A) = \\{0\\}$\n",
    "4. $A A^T = A^T A$\n",
    "5. $\\det(A) \\ne 0$\n",
    "6. $A x = 0$ implies that $x = 0$\n",
    "\n",
    "When we write $x = A^{-1} y$, we mean that $x$ is the unique vector such that $A x = y$.\n",
    "(It is rare that we explicitly compute a matrix $A^{-1}$, though [it's not as \"bad\"](https://arxiv.org/abs/1201.6035) as people may have told you.)\n",
    "A vector $y$ is equivalent to $\\sum_i e_i y_i$ where $e_i$ are columns of the identity.\n",
    "Meanwhile, $x = A^{-1} y$ means that we are expressing that same vector $y$ in the basis of the columns of $A$, i.e., $\\sum_i A_{:,i} x_i$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[2 3]\n",
      " [0 4]]\n",
      "[[13 12]\n",
      " [12 16]]\n",
      "[[ 4  6]\n",
      " [ 6 25]]\n"
     ]
    }
   ],
   "source": [
    "B = np.array([[2, 3],[0, 4]])\n",
    "print(B)\n",
    "print(B @ B.T)\n",
    "print(B.T @ B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.5   -0.375]\n",
      " [ 0.     0.25 ]]\n"
     ]
    }
   ],
   "source": [
    "Binv = np.linalg.inv(B)\n",
    "print(Binv)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1. 0.]\n",
      " [0. 1.]]\n",
      "[[1. 0.]\n",
      " [0. 1.]]\n",
      "[[1. 0.]\n",
      " [0. 1.]]\n"
     ]
    }
   ],
   "source": [
    "print(Binv @ B)\n",
    "print(B @ Binv)\n",
    "print(np.linalg.solve(B, B))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Vandermonde matrices\n",
    "\n",
    "A Vandermonde matrix is one whose columns are functions evaluated at discrete points."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "shape = (50, 4)\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-1,1)\n",
    "A = np.array([x**3, x**2, x, 1+0*x]).T\n",
    "print('shape =', A.shape)  # This is a tall matrix with 4 columns\n",
    "\n",
    "lines = plt.plot(x, A, '.')\n",
    "plt.legend(lines, range(4))\n",
    "plt.ylim((-1.1, 1.1));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* This type of matrix is very common; we can also create it with `numpy.vander`.\n",
    "\n",
    "We can evaluate polynomials using matrix-vector multiplication.\n",
    "For example,\n",
    "$$ 5x^3 - 3x = \\Bigg[ x^3 \\Bigg|\\, x^2 \\Bigg|\\, x \\,\\Bigg|\\, 1 \\Bigg] \\begin{bmatrix}5 \\\\ 0 \\\\ -3 \\\\ 0 \\end{bmatrix} . $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = np.array([5, 0, -3, 0])\n",
    "plt.plot(x, A @ p)\n",
    "plt.plot(x, np.polyval(p, x), 'o');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now suppose we know the value of a polynomial at a few points.\n",
    "We can use the Vandermonde matrix to find a polynomial through those points."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x1 = np.array([-0.9, 0.1, 0.5, 0.8])  # points where we know values\n",
    "y = np.array([1, 2.4, -0.2, 1.3])     # values at those points\n",
    "\n",
    "plt.plot(x1, y, 'o');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "B = [[-0.729  0.81  -0.9    1.   ]\n",
      " [ 0.001  0.01   0.1    1.   ]\n",
      " [ 0.125  0.25   0.5    1.   ]\n",
      " [ 0.512  0.64   0.8    1.   ]]\n",
      "p = [12.98319328 -1.74789916 -9.47605042  3.35210084]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "B = np.vander(x1)                     # Vandermonde matrix at the known points\n",
    "p = np.linalg.solve(B, y)             # Compute the polynomial coefficients\n",
    "plt.plot(x, A @ p)                    # Plot the polynomial evaluated at all points\n",
    "print('B =', B)\n",
    "print('p =', p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Evidently $$p(x) = 12.983 x^3 - 1.748 x^2 - 9.476 x + 3.352$$\n",
    "is the unique cubic polynomial that interpolates those points.\n",
    "Applying $B^{-1}$ converted from the values at the marked points to the polynomial coefficients."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Inner products and orthogonality\n",
    "\n",
    "The **inner product**\n",
    "$$ x^T y = \\sum_i x_i y_i $$\n",
    "of vectors (or columns of a matrix) tell us about their magnitude and about the angle.\n",
    "The **norm** is induced by the inner product,\n",
    "$$ \\lVert x \\rVert = \\sqrt{x^T x} $$\n",
    "and the angle $\\theta$ is defined by\n",
    "$$ \\cos \\theta = \\frac{x^T y}{\\lVert x \\rVert \\, \\lVert y \\rVert} . $$\n",
    "Inner products are **bilinear**, which means that they satisfy some convenient algebraic properties\n",
    "$$ \\begin{split}\n",
    "(x + y)^T z &= x^T z + y^T z \\\\\n",
    "x^T (y + z) &= x^T y + x^T z \\\\\n",
    "(\\alpha x)^T (\\beta y) &= \\alpha \\beta x^T y \\\\\n",
    "\\end{split} . $$\n",
    "The pairwise inner products between two sets of vectors can be expressed by collecting the sets as columns in matrices and writing $A = X^T Y$ where $A_{i,j} = x_i^T y_j$.\n",
    "It follows from this definition that\n",
    "$$ (X^T Y)^T = Y^T X .$$\n",
    "\n",
    "### Orthogonal matrices\n",
    "\n",
    "If $x^T y = 0$ then we say $x$ and $y$ are **orthogonal** (or \"$x$ is orthogonal to $y$\").\n",
    "A vector is said to be **normalized** if $\\lVert x \\rVert = 1$.\n",
    "If $x$ is orthogonal to $y$ and $\\lVert x \\rVert = \\lVert y \\rVert = 1$ then we say $x$ and $y$ are **orthonormal**.\n",
    "A square matrix with orthonormal columns is said to be an **orthogonal matrix**.\n",
    "We typically use $Q$ or $U$ and $V$ for matrices that are known/constructed to be orthogonal.\n",
    "Orthogonal matrices are always full rank -- the columns are linearly independent.\n",
    "The inverse of a orthogonal matrix is its transpose:\n",
    "$$ Q^T Q = Q Q^T = I . $$\n",
    "Orthogonal matrices are a powerful building block for robust numerical algorithms."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Make some polynomials\n",
    "a0 = A @ np.array([0, 0, 0, .5]) # .5\n",
    "a1 = A @ np.array([0, 0, 1, 0])  # x\n",
    "a2 = A @ np.array([0, 1, 0, 0])  # x^2\n",
    "\n",
    "plt.plot(x, np.array([a0, a1, a2]).T, '.');\n",
    "plt.legend(('$1/2$', '$x$', '$x^2$'));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Inner products of even and odd functions\n",
    "Which of the above functions are even and which are odd?  Note that\n",
    "* the integral of an odd function over a symmetric interval such as $[-1, 1]$ is zero; the same holds for a sum of the function evaluated at a symmetric set of points, and\n",
    "* the product of even and odd functions is odd.\n",
    "\n",
    "Together, this explains that even functions are orthogonal to odd functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-4.718447854656915e-16, 2.453227608198226, -3.552713678800501e-15)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q0 = a0 / np.linalg.norm(a0)\n",
    "\n",
    "a1 @ q0, a2 @ q0, a2 @ a1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### What is the constant component of $a_2$?\n",
    "\n",
    "We see above that $a_2$ is not orthogonal to $q_0$.  However, we can measure the component of $a_2$ in direction $q_0$ by $q_0^T a_2$ and can write that as a vector (e.g., to plot it or add/subtract it) via $q_0 q_0^T a_2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, q0 * (q0 @ a2), '.');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's project that away to create a vector $x$ that is orthogonal to $q_0$.  We can use the orthogonal projection $y = (I - q_0 q_0^T) a_2 = a_2 - q_0 (q_0^T a_2)$.  We can also normalize the result and collect the vectors as columns in a matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.00000000e+00 -1.32894490e-16 -1.77240772e-16]\n",
      " [-1.32894490e-16  1.00000000e+00 -1.93779647e-16]\n",
      " [-1.77240772e-16 -1.93779647e-16  1.00000000e+00]]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y = a2 - q0 * (q0 @ a2)\n",
    "\n",
    "q1 = a1 / np.linalg.norm(a1)\n",
    "q2 = y / np.linalg.norm(y)\n",
    "\n",
    "Q = np.array([q0, q1, q2]).T\n",
    "print(Q.T @ Q)\n",
    "plt.plot(x, Q, '.');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([[   5,  110],\n",
       "        [ 110, 2500]]),\n",
       " array([[   1,   60],\n",
       "        [  60, 1600]]))"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = np.array([[1, 2], [30, 40]])\n",
    "z @ z.T, z * z.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([[1, 2],\n",
       "        [2, 4]]),\n",
       " array([[1, 2],\n",
       "        [2, 4]]))"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = np.array([1, 2])\n",
    "np.outer(z, z), z[:, None] * z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Gram-Schmidt Orthogonalization\n",
    "\n",
    "Given a collection of vectors (columns of a matrix), we can find an orthogonal basis by applying the above procedure one column at a time and saving the result.  Let's think of the first two columns,\n",
    "$$ \\Bigg[ a_0 \\, \\Bigg| \\, a_1 \\Bigg] = \\Bigg[ q_0 \\,\\Bigg|\\, q_1 \\Bigg]\n",
    "\\begin{bmatrix} r_{00} & r_{01} \\\\ 0 & r_{11} \\end{bmatrix} . $$\n",
    "#### Column 0\n",
    "The equation for column 0 reads\n",
    "$$ a_0 = q_0 r_{00} $$\n",
    "and we require that $\\lVert q_0 \\rVert = 1$, thus\n",
    "$$ r_{00} = \\lVert a_0 \\rVert $$\n",
    "and\n",
    "$$ q_0 = a_0 / r_{00} . $$\n",
    "#### Column 1\n",
    "This equation reads\n",
    "$$ a_1 = q_0 r_{01} + q_1 r_{11} $$\n",
    "where $a_1$ and $q_0$ are known and we will require that $q_0^T q_1 = 0$.\n",
    "We can find the part of $a_1$ that is orthogonal to $q_0$ via\n",
    "$$ (I - q_0 q_0^T) a_1 = a_1 - q_0 \\underbrace{q_0^T a_1}_{r_{01}} $$\n",
    "leaving a sub-problem equivalent to that of column 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.00000000e+00 -4.66599631e-18 -5.83579544e-17  1.54642560e-16]\n",
      " [-4.66599631e-18  1.00000000e+00 -1.34356916e-17  1.43379430e-16]\n",
      " [-5.83579544e-17 -1.34356916e-17  1.00000000e+00 -1.54044017e-16]\n",
      " [ 1.54642560e-16  1.43379430e-16 -1.54044017e-16  1.00000000e+00]]\n",
      "[[ 2.83562932e+00 -6.10622664e-16  3.81827281e+00 -1.33226763e-15]\n",
      " [ 0.00000000e+00  3.29047206e+00  1.11022302e-16  5.27186935e+00]\n",
      " [ 0.00000000e+00  0.00000000e+00  1.66365006e+00  1.52655666e-16]\n",
      " [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  4.71247213e+00]]\n",
      "5.754298069240366e-16\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def gram_schmidt_naive(A):\n",
    "    \"\"\"Compute a QR factorization of A using the Gram-Schmidt algorithm\"\"\"\n",
    "    Q = np.zeros_like(A)\n",
    "    R = np.zeros((A.shape[1], A.shape[1]))\n",
    "    for i in range(len(Q.T)):\n",
    "        v = A[:,i].copy()\n",
    "        for j in range(i):\n",
    "            r = Q[:,j] @ v\n",
    "            R[j,i] = r\n",
    "            v -= Q[:,j] * r # \"modified Gram-Schmidt\"\n",
    "        R[i,i] = np.linalg.norm(v)\n",
    "        Q[:,i] = v / R[i,i]\n",
    "    return Q, R\n",
    "\n",
    "A = np.vander(x, 4)\n",
    "Q, R = gram_schmidt_naive(A)\n",
    "print(Q.T @ Q)\n",
    "print(R)\n",
    "print(np.linalg.norm(Q @ R - A))\n",
    "plt.plot(x, Q);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(50, 4)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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R26azTaS1SWTuIiMp9psltbe3l/7+/iGubF2og4ghk4DUD8S/teHgvFMdQKvV8uSTT/Lss88iyzJ33XUXeXl57Ny5E4AtW7bw6quv4vV6+cUvfjF4zXPPPTfVqSdFxJCFPtg47vMXLDHT3Rmh6rifO7cmojcM/eO7A00cvPI8CYYMNuT/CbqrvSmHIzdxBfcv/iveufB93m36W+4s/Iu4d7hRUVEZmQanjxePdHCgaQBJVliVl8C2BaksSDOP6Jq+kVBQ5kyln8RkLfMWDf/db2yM6s21cEwUBV24g2DCUgJXM/8/MFE6AOXl5ZSXlw/53ZYtWwb//fTTT/P000/HYqopIxqzMXmrECQfinbs5CqdTmDZqmjUzoXTAcpWXXfFeEKdHGj+ewxaK5sK/hyjLmGUkaIU2leyIf+POXjlefY2/oC7Cr81LTU0VFRUoiiKQlWnn9dq3FR1+DBqBbaUJPHIglSybIYJjXP2ZICIqLDsTstNrpxrNDY2DmbXAmgivWjkIBFjNoGrBR2nSwNum0zba0SMWUD0kUq0lIzrmhS7jrlXE7Iyc/Vk5ugJiL3sb/47ADYV/AWWYco1jERmwh1sLPgz3rvyT+xt+j53FjyD1TD7IhhUVG4lREnmQNMA2y/20twXIsWk5QtrCtiYY8BmHNk/PxJtzSKdrSILl5hITB7+er/fPyS7FkAfageiWhQI1KEVDOg1448EnAq3XcGYa4J/7U0fL/MWRf+oZyr9+P1B9jf/AyHJy8aCPyXx6pgTId26gE0Ff0Eo4uXdph8QlmZf5I6Kyq2AJyTxynknn3/tMj862gnA11dn8vNtxXx6Zd6kxD7glzl3yk+KQ0vx+xKsbuTy5csoikJJyfXFpS7UjoIm6sMXezHrU8blPooFt90KX9FakXRJ6CYo+Bpt1LXz3i4Px84fpd/awrq8r5NqnjNpWxyWEjYW/Al7G7/Pua7fsDz7M5MeK1bIskx/fz/9/f2DeQNarXbI/61WKxrNbbdWUPmA0eEJs73Gzd6GfkKSQlmWlW8sTKUs0zIlgY2WPfajyLBspQVhlE3duro6UlJShsbYhzqQDGmg0ROM9E3bhi3choIPEDFmjztS50YSk7XMv8PE6f5DGIVUcmzLp2yLwzKPktQt1Ll3UpC8Fodl3pTHHC9+v5+Wlhbcbjc+n4/Ozk76+vqQ5dEbweh0OtLS0sjIyCA9PZ2MjAySk5OnbZWiojISiqJQ0xPgtRo3x1u9aDWwsTCJRxekUJgyckDFRGiuD+PsilC63IzVNvLTgd/vp62tjRUrVgz5buhC7YhXF4p+sRe7pTgmdo2H21PwDVkYfBdBDoNm/Js0AJlzPMh1tSg9WwkWgjkG4fSl6R+hzXOCyvZ/Z8uc/w+tJn5F2yKRCI2NjdTU1HDlyhVkWUYQBFJTU0lKSqKoqGjw34IgEIlEkCRp8P+iKOJ2u+nq6uL8+fNEItH63nq9nry8PObPn09RURE63W350ZpxJDnClf5jtHtOE5EDROTQ1f+CROQgsiKRYV1MfvIaMqyL0QgTd2fMRiRZ4UiLh+01bi65gtgMGj6y2M4D81NINcfus+jzSFSfCZCWqaOgeHTtqK+vR1EU5s6dO/g7QfKilQbwG7NRFIVgpFdd4ccb0ZiNgIIu3EnElD+ha5v7D4GgoO9bxeljftZsso76SDce9FoTy7M+w3tX/omLzrdYnL5t7IsmgKIotLe3c/HiRerq6giHw1itVpYtW8bcuXNJTU0lMzNzwpl9siwPin9XVxeNjY00NDRgMBiYO3cuCxYsIDs7+wO78tdEBtAHm9EHmtAHryDI1/dZhBYtqbIEgKKxIpoLEE0FhE0FKDOQXxEQ+7jcu5fGuv34w25MuiSM2kR0GiM6jQmTLhmdxoiiSLR5TtHUfxCj1kZe0moKktZgN5d8IP9OflFi9+V+3rjYS7dPJMum54srMrh7ThImXWzdjrKscOqoH41GYOmKsd1CdXV1pKamDik9c82zEDFmIcp+JEWctpBMuE0FP2KMJn/pQh0TEnxFkWnse4906yKKS/M4Uxngcm2IkoVTf1TMtpWRl7iKaud28pJWkmiMTYJae3s7+/fvp6enB71eT3FxMQsXLiQnJ2fKfniNRoPD4cDhcLB48WJkWaa1tZWLFy9y6dIlLly4gM1mY+HChSxduhSzeXoiESaNHMbkPYs+0IAh0IQ20guAIugRTXlI+ut+WK3JSCQYAkAT6cPcfxRL30EAIno7oqkQ0TyHYEIpxOmJTVEUnP5L1Ll30TpwAgWJ/NQKChI+S1bCkhHL7UqySIf3DFf6j9DYu4969y6segeL0z9MUfL6uNgaa5x+kTcv9rKzvg+fKLMozcxTy9NZkZMQtybgteeD9Lkllq+1YLaM/t3xer20tbWxatWqIb/X3yD4ftENTF9IJtymgi/rkpE1JvShdoJjnz5Ij78Wn9jNHemPkZdkoLszwsVzQRzpOpKHybCbKOVZn6LTe44T7f/OXYXfnlJ9bK/Xy6FDh6itrcVqtbJ582bmzp07mNodDzQaDfn5+eTn5yOKIg0NDVy8eJHjx49z+vRpysrKWLZsGSZTbHypsUKQfJj7j2DpO4JG9iNrEwibCvAnrUE0F0YXCO9zfTgcDgZufCJSIuiCbRiCzeiDzRh9NZg9J7G63iGQtI5A0ioUbexueKGIh2NtP6HDexa9xsJc+72UpGymKGfxmE9qWo2e3MQKchMrEKUArZ6TXHbv4XjbT+nx1VCe9YfoNCNHnswkDe4gr9W4Odg8gAKsybOxbWEq8xzxXUw4u0Xqa0LkFxnIzhvbDXz58mWAIe4ciPrvJV0SitZKMNAATF/SFdymgo8gTGrjtrH3AHqNmdzEiqvNz830uiKcOupn4xYbuinWzjfpkijL/ASV7S/Q0Luf4tS7JjyGJElUVVVx/PhxJElixYoVVFRUxFXoh0Ov1zN//nzmz5+Py+Xi2LFjVFZWUlVVNWuEXyO6sfQdxDxwAkERCVkW4k/ZgGgqHLF89ogIOiLmAiLmgujPioI+cBlL3wES3Duw9L5LMGkl/qR1yFNsZef013G45V8JSQOUZTxBcepdo2Z4j4Zea6YoeT0FSWu50PM7qnu24wpcZl3e10g05kzJzlghKwqn2n28VuPmXJcfk07DA/NTeHh+ChkJE9uDmwzhkMzpo36sCRoWLxvfjeXSpUvY7XZSU4fm5+hCHYMehoAYfYKcjtaGg/NP20yzjIghC/PAcVBkGMdKWpQCtAwcpzB5/eDqx2DQUL7KyuF3vZw/NTQLd7IUJW+iue8wZ7peItu2bEIfhitXrrBv3z76+vooKipiw4YNJCdP34dpJOx2Ow888ABOp5Pjx49TWVnJmTNnKCsrY/ny5dN+M9KIvSS4fo/Rew7QELSV4U/ZgGTIiN0kgoBoKaHfUoIu1IGl9wDmvsOY+w4TtC3FZ78PWZc4oSEVReGSewdnOl/Cok/lnqLvkmoujIm5GkFDafqHSbPM42jr/2NXw3dZnvVZCpPXxWT8yRCWZPY1RjtKtQ6EsVt0fGZZGltKkrEapmezWVEUzpwIEAoprL/HOq5FncfjoaOjg9WrVw89IIfRij2EEu4AGCyrYJrGFb6gDFffeBbR3j6xeHkA+aWfo+tsRRTFEc8x5ykkV0DPboh4xv4jNuR5OVXay12H0rH3D33cvZS0nvqktZQ5Xyfbf3HUcfR6/ah2AXisIrvWd5LVbWbN6bEzcBXgtNbKEW0CyYrEemmAAjk85nUTtStWuAQdldoELmtN2JQImyIj2xtbuxTMBZBYCgjgbwDfZZCDE38ym4xdWrOCtQQsRaBI0H8Ggq1EjRkDUSdzotRNW1aA7E4zFWdTMURuXqjE4v0KGCWOlblw2kMUXbFSVp2MVp7afs9E7OrXGPm9ZT7vWOczoDUzJ+ziEV81awNN6IitXI1lV4t1Cefs97Ggdx9zPMfHNeYZrYWDukT+INxDsiJdnytFwXEnuI9CqEPg9OJermT5eXT3zU9SlnmLCD36yQm/HohzeeQPKmK0SB26JIh4xj6/KddHokdHav/Nj5Al/Ydxmgo4n7qFlFA7ZmlgSrbZfHoW1idxYX4/XS1BMpwjP65HgHd1SVzSmimRAtwd6We2d+K1KxHui/TRLunZp0viTX0qJVKA9REPVkbPAZgsGpNC0jIwZUKoB/pPgeSf3qgUKSAwcA58jQrJyyFlBQSyYaBKQQ6PbEufLczRchc+c4QlNUnMbbQhjOMmMVnMIS0bj6dxYW4/tSUeBhJENh5Pm7Loj0WrLpE3rIvYb5lDWNBREWzhkd5qFoe74vhqR8arS6U65W7swSaKxin2AHUaEw5ZHCL2APqrdRIjV7UnYJQwh6Y3gfGWXOHDOBoIKBJpDd/Dn7QWn+P+UccaCLXxTv23WJrxCRY4Hhj2HJ9X4sAOD7ZkLWvvShjS3mxCdl1FkkXerv8zTLpkNhd9d9gQMJ/Px5tvvklXVxerV6++KcFjIsxUgwpJkjh58iSVlZVotVrWrl1LaWnp4OuYsl2KgtF7BlvP6whKBK/9PgJJq8flxhuNqdslY+l7D6trF4rWxEDah4bt0eAONLGv6QfoNCbW5H6FNOv8YQaLoV3v40r/MY60/pjcxArW5n510oEEI9mlKArnu/1sr3FT2ebDoBW4qyiJRxakkJsU/43jERugSAoHd3sJ+GXuvM+GyTy+1z0wMMAvf/lL1q5dO6R+DoCt+3cYvWdxFv01CAK7Gr6LXmPhzsK/GLdd40Fd4Q+HoCViyBhXTZ3G3gMIaEb1Z1oTtJQut3D6mJ+66iDz75ha1IBWo2eR41FOdPw7Hd4qsm1D20J2d3fzxhtvEAqFeOCBB4bU6vggodVqWblyJfPmzePdd99l3759XLx4kXvvvXfYRjoTQZB82Lpfw+Q7j2jMYyDj8WhK+2xA0OBP2UTIMp/E7ldI7vwVQVsZHscjg9E8/cE29jf/HXqtlXuKvoNFbx9j0NiTn7SKgOimquvXVHW9xLLMJ2IybkRWONg8wOsX3Vx2h0gyavlEqYP75yWTZJp5Wbp4NshAn8SK9dZxiz1Ek62AYb+PulD71Yiv6GLGG+4mN3FFbAweJzP/zs4gEWM2Rm/1qE3NZSVCU/8hsm1lY9auzy000NMlculCiNQ03YgNU8ZLUcoGapxvcq77t2QllA2ueuvq6ti1axcmk4nHH3+ctLRZImJTIDk5mW3btlFbW8uBAwd46aWX2LRpExs2bJjUeNqwk6SOF9GK/Xjt9+FP3jDlVX08kIyZ9OZ+Gav7XSy976ILtdOX9VkGlDD7m/8OjaDlzoJvzYjYX2Oe/T58opNLrnew6u3Ms2+d9Fi+sMSO+j7erO3F5Y+Qm2jgK6sy2VSYiDHGiVKTpbNNpOFSiMISA5k5E/sO19XVkZ6efnOwhCKhC3cSSIzG5YciXsKSF1ssAwXGwe0t+IYszHIlGmkAeQQx7/CeJRjppyh507jGLF1uoc/t4dQRP5u2jv9RcDg0go7F6R/ieNtPafOcIDdxBbW1tezYsYOsrCwefPBBLJZbp1WiIAgsWLCA3NxcduzYwe7du+nu7mbNmjUYjeN/vNcFr5Dc/p+AQm/O56+HSs5WBC0++2bC5iKSOv8L/ZV/YX/EjaSI3FX4bWzG6RWFm8wTBMoy/wC/6OJ0539j0dvJTawY+8Ib6BgI8h8nu9hV308wIlOaYeHLKzMpz7aimUUZvn6vRNUxP0kpWhaVTewpvb+/n66uLtatu9kToBWdCEpkMCTTG45W7bQZMqdu9ASYHbfUGUIczLgd2a3T2HsAozaRLNuScY2p0wlUrLUiRRROHfEhy1PbIilIWovNkMW57t/S2NjArl27yMnJ4UMf+tAtJfY3kpCQwIc+9CFWr17NuXPneOmll+jq6hrXtQbvBVLafo6sNdKb+6XZL/Y3IFqK6cj8JK+HrhCK9LI542Mkm/Jm2iwgGra5OvdLpJrncLT133D668d13SVngL9/r42P/vIEb9f2sio3gR/eX8j3N+dTkZMwq8RekhROHPajoFCx1oJ2hIYmI3HNnfP+ZCu4XlLhmuZ4wtHPs82oCv60IRkzURBGFAFgxtMAACAASURBVPxgpJ92TxWFyevRCON/GLIlaSmtsODqkag9P5Fc3pvRCBruSH+MgVAbe078CrvdzkMPPXTLFyfTaDSsXLmSz372s0iSxCuvvMKpU6cYLcbA3HeYpM7/JmLIpDfnS0gfsKYyYcnH3o5f4kHhAfM85rh2YRo4OdNmDaLTGNmQ/8eY9SkcvPJDvOHhb8KSrHC0xcMzO5v5sx3NVHX4+Hh5Dj/bVswfr8umOHV2ZVpfo7oqQH+vRNlKC5aEicf5X7p0iYyMDBITb86viDYt1w3uIXnDnQgIWPXpU7Z7ItzWgq9ojEh6+4gZt+2eKhQkCpLXTnjsvEID+UUG6mtCdHVMLS7aHJ6D5LNgndPGw488NCH3xgedwsJCnnjiCQoLCzl48CBvvfUW4fD7YvYVGavzHWzONwhbF9Cb83mUcbSbnE1Issh7V37IQKiVdXl/hLnwTxHNc0jsfhWLe090n2kWYNIlsjH/T1FQOND8T0Tk0OCxUETm7Uu9fPXNBv72QBtOv8hTy9P5xYeK+cr6IhyW2Rsw3N4Spqk+zJx5RrJyJ56929fXR09Pz7Cre4jW0IkYMgZLdHjCXVj09rhWxh2OmAh+VVUV3/jGN/ja177Ga6+9dtPxtrY2/vIv/5InnniC119/PRZTxoyIMWvESB1XoB69xkKycXKP1XeUm0lM0nD6qJ+Af3Lx5X19fWzf/jpi2xw0Jj894dOTGueDjMlk4sEHH2TDhg00Njbyyiuv0N9/NZhZkbF1v4q17wD+pNX0Z35ywiWvZwOnO3+F03+JVTlfIsu2BEVjoi/7MwRs5SS4d5PQs33WiL7NmMXa3K/iCXdwtusVegMRflXVw1O/q+enlV1YDVr+fH02P3mkmEcWpGLRz+4SzF6PxJnjflLsWhYumdzTR01NDYIgMG/eMP0sFOVqhM71znieUCcJ0+y/hxgIvizLvPDCC3z729/m+eef59ChQ7S2tg45JyEhgc9+9rM8/PDDU50u5kSM2WgjvQhS4KZjLv9l7JaSSccea3UCy9dZkWWFk4cn7s/3+Xy89tpryLLMQxs/R4qpkAs9ryHJkUnZ80FGEASWLVvGo48+itfr5eWXX6a1pYWEntcxe07jTd2M1/HIrIzEGYumvoNc7t3LAvuD5CfdUF1R0OJJ/wi+5I1YBo5hdb0za0Q/I2ExGda7uOTaybd2vsurF1wszrDw3L35/MPWAtYVJMatamUsiURkTh72I2gEytdYR2xEPhqyLFNTU0NBQQEJCTc/WWqkATSyn4gh6r9XFAVvuHPaI3QgBoJfX19PZmYmGRkZ6HQ61q5dS2Vl5ZBzkpKSKCkpQaudfXf6iOFqU/PwULeOKAXoD7ViN0+tG02CTcvSFRZ6XRI1Z8bvzw+FQrz22msEAgEeeeQR7HY7d6R/GJ/YQ2PfgSnZ9EEmPz+fj370o5jNJnw1/4ll4Bi+5E34U++ZeMGzWUBfsIUT7S+SZllAacbjN58gCPjs9+FPWo217z0sve9Ov5E3oCgKpzt8fHdvC//3SAV+MYlNc97mXx/K4ZmNuSxMn1r7wOnm2HtOBvoklq2yYLFOTg6vXLmC1+tl0aJFwx7X3dC0HCAkDSDKARKmecMWYiD4brd7SIF/u92O2+2e6rDThnhDbfwbcQcaAAWHZXif3ETIyTdQWGKg4VKIhrqx6zgoisKePXtwu908+OCDZGZGPxhZCUuxm0uo7tmONMFaObcSKSkpPHVfFhvmeDnWbOGt88Yx2zLORsKSn0MtP0KvNbMm9ysjd58SBLyOhwnYlpHg3oW579D0GgqIksyey3184+0mvre3hebeIB8vzebuwi+i1Tjp9m+fdpumSvPlEJeqByhZaCQje/K+9OrqakwmE0VFRcMejzYtFwYFfzBCZwZcOlMO9RguamIqd/jdu3eze/duAJ577rkhzX8ngk6nG+e1DpS2JBJwYb3h/CZf9K48N3cFxhhsAG7crOD3tnFobzcPfjiXVMfIG6+VlZXU19dz7733Ul5ePuTYOt2TvH7223SJJ1iS88iU7brG+N+v6WVYuzr3oOnfi2xfhasvh7OHj+Dz+fn4xz8+bRvaU32/FEXh99U/wRfuZtvSvyM7eRyZ0o4volz6CTbnm1gTHZB+c7x3rP+OA0GR353t5LdnO3D5wsyxW/j2vXO5d14aBp0GWICfas61v8kdeZvJTr5j2HFm2+eruzPI+VN95ORbWXdn5oilUMbC5/PR2NjIypUrycgY3kUjuFxgSsOeHi2S1tMZ3YfLz1hAsmX49yRe79eUBd9ut+NyuQZ/drlcU0qJ37x5M5s3bx78ebL1JCZSiyJJn4FmoIneG86/4jxLojEbT18Qz4TapIzM0pUGDu0R2fVmGxvuTcBgvPkBy+l08vbbb5Ofn8+CBQtueg0mJRe7eS4nm18h07ByQuGiozFTtXTG4v12mQZOktj9KiHrIvqTH6aiQovFYmXPnj387Gc/49FHH52W/ISpvl8XnW/T4DzE0oyPY4hMoL1k6odJDnrRN/wHA/4woYTSmNp1jQ5PmNcvutlzuZ+QpFCWZeVrqzIoy4y6bAb6rj/Fz018hIaeo+yq+Ue2Fj87bPOU2fT5CgZk3tvlwWTWsOnedNxu19gXjUBVVRWSJFFUVDTi67N7mhBNuYMNczpc9QhoCPt0OP3DXxOvWjpTdukUFxfT0dFBd3c3kUiEw4cP31Q0aLYTMeagC3dFm5oTXX25AvXYzbGtT2Mya7jrviwCAZlTR/0o79vEFUWRd955B6PRyJYtW4Z9UhIEgUVpD+MXnVzpPxZT+2Y7Ru95bN2/JWwuoT/j44MhbosWLeLhhx+mt7eX3/zmN/T19c2wpaPT46vlbNfL5NoqmG8fvhjfiAg6+rI+hWjKJ7HzZQy+2pjZpSgKNd1+/vZAK196vYGd9X2sK0jkXx4o5P/cnceyLOuwn0m91sTKnM/jDXdxruuVmNkTD2QpGkARDitUrLNiNE1+X1FRFKqrq0lPTx9xNS5IPrSR3sEMW4jG4FsNaTFbrE2EKQu+VqvlySef5Nlnn+Wb3/wma9asIS8vj507d7Jz504gGlr49NNP89Zbb/G///u/PP300/j9/jFGnj5EUwECMvpgCwDecBdhyRtzwQdIzzRRWm6mpzPCxfclZe3fv5/e3l62bt066io1K2EpScZcapxvoCgfPN/1ZNAFr5DY9TIRUx59WZ+6qU9sYWEhjz32GOFwmN/85jfjzsydboKRfg63/l+shjRW5Hx+cu5PjYH+rE8TMaST2PlrtBPs3PZ+JFnhUPMAf76jmW/tusL5Lj8fXmzn59tK+MaaLApTxg5VTLcupCR1M5fcO+mJ4U0o1lyoCuB2RpOrklKmFkTS09OD0+kccbMWQB9oAoh2UbuKJ9w1IxE6EKNaOuXl5Tf5mrds2TL47+TkZH7yk5/EYqq4IJoKUBDQBxsRLcW4AtEUabslPhUoC4qN9Lkl6mtCJKVoyc4zUFtbS3V1NStWrCAvb/S4f0HQsMDxEMfafkK7t4ocW/mo53/Q0UT6Ser4FbI2kb6sPxwxzj4zM5PHH3+c7du389vf/pYHHniAwsLC6TV2FBRF4UT7i4QlH5sK/hyDdvKuJ0Vrpj/7M6S0/Jjkjv/EnfuVCSeb+UWJPZf7ef1iL90+kSybni+uyODuOUmYJlHIbEn6x+jwnOF4+8/ZWvwDdLMsH+JKQ4im+jDF843k5E/dturqarRaLfPnj1yy2hBsQhF0iKZc4HpIZppl9DLX8eKDF7QcBxStiYgxC8PVu7HLX49OY4prT887ys2k2LVUHffTesXF3r17yc7OvqnL/UjkJ63GqndQ0/PGqOUGPvDIYZI6/gtBDtGX9YcoWuuop6ekpPD444+TkpLCm2++SU1NzTQZOjbN/Ydo85ykNP0jJJvypzyerEukP+uTaCQvSZ3/Dcr48jOcfpFfnurmc7+7zC9OduOw6HhmYw4/fmgOD8xLmZTYQ9S1syL7KbzhLi65fj+pMeJFryvCuZMBHBk6FkwyuepGIpEItbW1lJSUjBoooA80IRrz4Kr7JhjpJyKHZiRCB1TBH0Q0FaIPXgFFwhmox24uRhPHJB6tVqBinRWNVubtt3+PRqNh69ataDTjm1MjaJlvfwBXoB6n/1Lc7JxRFAXh8i/RhdoZyPw40jirRlqtVh577DFycnLYtWsXZ86cibOhY+MX3Zzq+C/s5rnMs4/ecGciREx5DKR/GEOwCVvP66MmZjW4gzx/qJ0vvHaZ7RfdlGVZ+YetBfztlgJW59likiiVkbCYHFs5Nc43CF5r7TTDBAMyJw75MJoEytdYJh2RcyOXL18mFAqN6s4R5BC6UDviDX2HPVerZCbMUAVUVfCvIpoLERQRAo30B1vi4r9/PyazBn3CBYJhF3kZ67FaJvZIXpSyEaPWRo3zjThZOLNYevchuCrx2bcQti6c0LVGo5GHH36YOXPmsH//fk6cOBEnK8dGURQq23+BrERYlfPFmC8kQrYyfCl3Yh6ohK69Q47JisKJNi/f2X2Fb77TxNFWLw/MT+Enj8zhzzfkMM8xtUY9w7Ek4+NIcpgL3b+L+dgTJRJRqDzoQwwrrFhvxThMZNxkqK6uxmazkZubO+I5uuAVBOQh/vuZKos8aNOMzDoLCV/9o/QPnERBjpv//kZcLhc1F8+QnzcPKZTD2ZMBlq4wj3sjT6cxMs++lXPdr9IXvBITN8FsweCtJsG9E8W+Ev84exG8H51Ox/3338/u3bs5fPgwoiiyevXqac8Ebeh9l07vOcoz/zBute19qfeiC3VhaPoN+mwrPuMc9jUOsL3GTetAGLtZx6eXpbGlJJkEQ3wz3hONWRSn3s1l917m2reQaBw5TDCeKIpC1TE/fe5o56qklNjI3cDAAC0tLaxatWrUz5Ih0ISCgGi+/r30hLvQCNoZa2ijrvCvouhsRPR23P6LAFMuqTDmfIrCvn370Ov1bNm6kXmLjbQ0hrlcGxr74hsoSd2MTmOixvlmnCydfrShThK7XkY05qIUf3pKJRO0Wi333nsvixcvprKykvfee29a9zy84W6qun5NhnUxJan3xG8iQcNA5seQjBmY2/6bv37rFD8+1olBK/DNtVn8bFsxjy2yx13sr7E47UNoNQbOdL08LfMNR+35IB2tIouWmibcuWo0ru0LLVw4+lOnPthExJiNorm+Z+AJdWLVp4+cVR1nVMG/AdFcRE+ogwRDBkadLa5z1dbW0tbWxtq1a7FYLMxbbCI7T0/NmSAdreMvm2DQWilJuZuW/qN4w91xtHh6ECQ/yR3/iaIx0p8Vm8qXGo2Gu+++m6VLl1JVVcW77747LaKvKDLH234OCKzI/tyki/CNh9b+ED+u7OWTlesIRRR+MHcPP7g7nR/eX8idRUnoprmQmUmXyCLHI7R7TtHtm/6N85amMHXVIfKLDMyZH7vs62ux93l5ecPWvb9+YgR98MoQdw4wY0XTrqEK/g2EjQV0yQEccYzOgWhhtIMHD5KRkcHixYuBq23kVlpITtVy+qifPvf4K2LOs9+HIGi56Hw7XiZPD4pCYveraCID9Gd9csS2k5NBEAQ2btxIRUUF58+fZ+fOnXGvv1Pn3kWP/yLLMj+FNQ7NWBRF4VyXj+/va+Erbzayt6GfJXPm0OX4GBn6ftYIO5nJMmZz7Vux6O1Udf56WvNFXD0Rzlb6safrKF0+fhfpeGhpacHj8Qx+b0dCF2xDUCJDNmwVRcYb7pqRomnXUAX/Bnr1SfiRydTGt3nG0aNH8fv93HXXXUOicrQ6gRXrreiNApUHfeOuoW/Wp1CYvJ7GvgOzJjJiMpj7D2H01eB13E8kDvsRgiCwdu1a1qxZQ21tbVxFfyDUwdmul8lOKKMoeXKN2EciIivsb+znT37fxHd2t1DrDPKxUju/2FbMtzbPJTltIb7UzZi8ZzANzNxmtU5joDT9cXqDTVzq3jctc/q8EicO+TBbNFSstUyq3PFonD17FpPJxJw5c0Y9zxBsAq7vDQIEIr1IijhjG7agCv4QnOFo7YosRYrbHO3t7Zw9e5YlS5aQnn5zezOTWcOqDQmIYjS6IBIZn+thgf0hFCUy6+Kfx4sueIUE5zsErYsJJE28w9hEWLFiBevWrePSpUtxEX1FkTnR/gJajYGK7KditsL0hSV+V+3iC9sv88PDHQQjCl9emckvthXzxJI0ks3XNyX9KXcSNpdgc74+5UzcqVCQtIYUUyFHG39JJM4VXsWwzPH3fCgKrNxoHbZW1VTo6+ujoaGB0tLSMVuM6gONRPRpQ5LhPKGZjdABVfCH4AzUo0VDRtgdl0YTiqLw5ptvYjKZWLNmzYjnJSZrWb7GSn+fNO7GKTZjBrmJK6l37yEszZ6yFeNBkAIkdf4Psi4JT/qHp6Wu/fLlywdFf8eOHTEV/Ybe/fT4a1ma8QRmffKUx+v2irxwsounfneZX57uIctm4DubcvnXh4rYOjcZ43CJUoKG/oyPImvMJHX+D4I8sWCAWCEIGpZmfgJvqIc61464zSNJ0QWSzytTsdZCgi32m6JnzpxBo9FQWlo6+omKjD7YPMSdA9fLIieoPvzZgStQj8OQiV72oIn0xnz8Cxcu0Nrayvr168cs45uRrWfJcjPdHRHOVgbGtcm4wPEQohzgcu/eMc+dNQz67T30Z34CRRv7uPCRWL58OevXr6euri5moh8Q+zjT9RLploVTduVccgb4+/fa+OLrl3mrtpeVuQn88P5Cnt2cz4rcBDRj3BgVnY2BjI+hFZ0z2iIxw7qIQvsqqp2vx8XlqMgKp4/6cfVILFtpwZER+z6xoVCI6upq5s6dO2xXqxvRhbvQyMFhN2y1gh6LPjXm9o0XNQ7/KpIcpi/YzIKkjRCowxBoIhjDP4zf7+fQoUMUFBSwYMGCcV1TUGwkGJC5dCGEySKwoHR0MUw1F5JhXcwl1++Zl7oF7SyrZTIc5v7DGH3VeBwPEjFNrnfwVLhWA+rgwYMoisLWrVun1JntdOevkBSRiuzPTsqVI8kKlW1ette4qe4JYNFreHRBKg/OTyHNOnEhEy3F+FLvJsG9B9E8h2DizFSyXVP0JC+5nuai8y3KMp+I2biKonD+dCAafllmIqcgPp/5CxcuIIoiy5YtG/NcfaARgLB5aEMUT7gLqyE9rtFaY6EK/lV6g03IikRKQilyqBV9sJFgYuyKkh05cgRRFHn44YcnJATzFpsIBhTqqkMYTRqK5o7+ZLDQ8TD7mp+jqe8Qxal3TdXsuKILtpDgfIeQdSGBpJubeUwXN4o+MGnRb/ecpmXgGKXpH8F2Q8Pq8RCKyOxp6Of1i246PCLpVh1PLU9nc3HSlJuA+1PuxhCIll4QjXnjLlERS1Kt+eQnraXevYcFjgcxxSgCq/5itCDanPlGiudPvUbOcMiyzJkzZ8jOzh523+396INNSLokZN1Qd140JHNin4tYo7p0ruLyXwbAbpmLaCoYLGsaC/r6+qiurqa0tHRcH5gbEQSB0uVmMrJ1nD8VoL1l9I2vdOsiUkxFXHS9jTyLSydf99vbGEj/yIz3oy0vL2f9+vXU19dPaiNXlIKc7PgPEo05zLc/OO7regMRflXVw1O/q+enlV0kGLT86bpsfvJIMY8sSJ2y2APRpKyMjyFrjCR1/nqw78N0syjtESRFpNb5TkzGa2kMcfFskJx8PYuWxkfsIVo3x+PxjGt1j6JEC6aZCod8pmVFxhvuntEYfFAFfxBXoB6r3oFZn4xoLkInOhEiY/efHQ/Hjh1Dq9VOujGMRiNQvsZKij0ao+/sHjlGXxAEFjoewhvupM1zcrImxx1bz3Y0kf6rfvv4d6gaD9dEv66ujl27dk1I9M93v4pfdLMi+ym0mrEfnJv7QvzoSAefe+0yr15wsSjdwg/uzecfthawoTAxJoXMbkQe9Of3YHO+FdOxx0uiMZv8pNXU9+4mNMXvVleHyJnKaPXLspXxbZxeVVVFYmLiiD1rb0QrutBKnpvcOQHRhaxESJjBCB1QBX8Qp/96h6vw1d31a7G0U8HlclFbW8uSJUuwWkcv7TsaOp3Ayg1WLAkaKg96GegbOXQ0J7GCBEPmrC2dbPRUYfKewZe6OS7x9lOhvLx8ME5/z54943r/XIEG6tw7KUm5e9Sm94qiUNXh43t7W/j6W4281zzAvcVJ/PjhOXx7Uy6L0+MrXKKlBH/yBswDxzHMQPYrwCLHI0TkMLVTCB/udUU4eciHLUl7teJs/N6zzs5OOjo6KCsrG1clW/1VzXj/hu21Kpm2GUy6AlXwgWjp2kDEPVgwLWLMRhH0MXHrHDt2DL1ez/Lly6c8lsGoYdXGBHQ6gSP7vHgHhhd9jaBhgf0BeoONdPuqpzxvLNGIvdh6thM2FeBPmVxRtHizYsUKVq1aRU1NDXv37h1V9GUlwon2FzDpkinN+Oiw54iSwt6Gfv7o7Sa+u7eFxt4gf7DEwQvbinl6ZSY5idO3ue6z34toyCKx+7cxe4KdCEmmXPISV1Dn3kko4p3w9f29Esf2+zCaNKzaaEWvj68rsKqqCoPBMGoZ5BvRB5qQNRYkQ9qQ31+LwZ/JkExQBR+INjwBrpdEFnSIprzBu/Vk6enpob6+nrKyMszm2IQbWqwa1twZDQs7ss+L3zu86Bcmr8OkS5pdRdUUmcSuV0CRGcj4KMxgtMJYrFy5khUrVnDhwgX27ds3ouhfcu2gL3iF8qxP3dTByhOSePW8i89vv8y/HOlAVhS+tjqaKPXRUgeJphmImRB0DGR8DEEOkdj92xkJ1VyUto2IHOSSe2Jx+Z4BiaP7vWh1sOYuKyZzfD8/Ho+H+vp6Fi1ahMEwvpuyPtgUjb9/32fbE+5CKxgw61LiYOn4mb3fuGnEHWhAI+hINhUM/k40FaILdSBIwVGuHJ1jx45hMBjGt9kzARIStay5MwFJgiP7hi/BoNUYmJd6H12+87hjuAE9FSx9BzEEG/GmPYI8g7HI40EQBFavXs3y5cs5d+7csFU2fWEn57v/l2xbOTm26/szHZ4wP6vs5Knf1fNfZ3ooSDby3bty+dGDRWwuTkavndmvnWTMwGu/D6O/FtPA8WmfP9mUR66tgjrXDsKSb1zX+LwSR/d5EQRYc1cCFmv8q02ePXsWRVEoKysb1/mayAA60TWknMI1vOFObMbMaS/N/X5UwQc84WiFzBs328LmIgQU9MHmSY3Z1dVFQ0MD5eXlmEyxjyBITNayeqOVcEjmyD4voeDNol+cejd6jZmLs2CVrwu1Y3XtJGi9g+AHpAfvtdo7ZWVlVFVVcejQoUHRVxSFUx3/gSAIlGd+CkEQqOnx89yBVr70egM76vtYV5DIvzxQyP+5O4/y7IQZ/7LfSCBpDSHzXGzOt9CGe6Z9/kVp2xDlAJfGkX0b8MscedeLJMHqTQlxyaJ9P6Iocv78eebMmTN6VcwbGGxY/r4MW4iu8GfanQMxisOvqqrixRdfRJZl7rnnHrZt2zbkuKIovPjii5w+fRqj0ciXv/zlMYsPTSfDdZEXTfkoaNAHmwhbJ95w+OjRo5hMJpYuXRorM28i2a5j5cYEju33cmSfl7V3JQypH2LQWihOvYda51t4Ql1xa74xJrJIYtfLyFoLnvRtMx6COREEQWDDhg3IssypU6fQarWsWbOGRtcR2r1VLEn/BFUdBrZfbKLWGSTBoOHDi+08OD+FVPMsTnMRNHgyPkLqlX8msetlenO/BNNYoz3FXECOrZxLrh3Ms983YkP3YCAq9qKosObOBBKTp8fGmpoaQqHQhJ7O9cEmZMFA5H0NX2RFwhfuIS9xRazNnDBTXuHLsswLL7zAt7/9bZ5//nkOHTpEa2vrkHNOnz5NZ2cnP/rRj/jCF77AL37xi6lOGzOUq/GxN4VLaaJ/uMls3La3t9Pc3Mzy5cvHLKEwVexpOlast+LzyBzdH23ldiPzUrciCDpqXTNXOjnB9Xt04W48GR8Zswn5bEQQBDZt2jTYROXo8YPsr/t/CGTzz4dL+PuD7fQHJb5QkcELHyrhU2Vps1vsryLrEvGkP4Y+1IbVvWfa54+u8v3UuXcOezwUkjm6z0swKLNqYwLJqdPznkqSxMmTJ8nIyCAra/yJUvpAUzTq7H03Tl/YiYI04yGZEAPBr6+vJzMzk4yMDHQ6HWvXrqWysnLIOSdOnGDjxo0IgsC8efPw+Xz09sa+Vs1k8ItuZEUcNiFCNBehD7aALE5ozKNHj2I2m1myZEmszByVtEw9FeusDPRJHDvgHSL6Zn0yRckbaOw7gF90T4s9N2LwX8LSfxh/0lrClnnTPn+sEASBu+++m8KSeVQ7X8MXcrLj0r2kmI18a2MO//bwHB6cn4JpuEJms5hQwh0EbMux9O6LabLheEg1F5GdUMYl1+8RpcBQu0IyR/dFi6GtXG8l1TF9N9Dq6mo8Hs+YLQxvRJAC6MKdgyHdN3K9j+0t4NJxu93Y7df7M9rtdurq6m46x+FwDDnH7XaTknLzjvXu3bvZvXs3AM8999yQ6yaCTqcb17WB3hYActLm40h53/napQh97+Ew9ELy+MKyGhoaaG1t5f777x92dTBeuyaKwwEJCV727ejkxKEgWx7OxmiKrjTWJvwhjcf30+zby4aSp4e9Pi52RXwIV36HYs7CNP8PME2itk+83q+JcqnHy0un2jgZNHHvvG4CbWl8Oj+Tx7ZOPdw2lkzq/Ur+NJxrJtn5vyhL/hq0sd9zGsmutYbP8OrpP6I9dITl+dGw1oA/wnu72vF5ZDY/mEVOfvyeCt9vlyiKnDx5kry8PJYvXz7+fRf3KQQULJllWBKHvs62YDT8tCBrERbD+KJ04vW5n7LgDxeu9v43aTznXGPz5s1s3rx58Gen0zkpuxwOx7iubXPXAiAHzTefL6eRdHrmQQAAIABJREFUJugJdhzBGxlfSYSdO3ditVopKioadv7x2jUZrIlQsc7KiUM+3vztFVbfacVo1ABaCpLWcaH9bQqtm4ct2RsPuxK7XsYYHqA395NE3AOTGiOe79dYyIrCqXYf22vcnO3yY9Ip/z97bx4eR3nlbd/V+yp1S61dsiVL3oR3bLxhG2OBAWMwhBDI9mZhMkkm4QskeROSIeEbhgkT4PIME5iQCckkmWQChGA2x4Cx8Y5XGS+yLMmWtVlr7/tW9f4hS9hYS0vdWmzVfV2+LFVXPXW6unT6qfOc8zvcUf4eGqUJq/JGju7ZhlUjDS6XO4oM93qpbXdhafk1oZo/4su6c9TsUpJJrnE2R5v+SoHuemJhFfu2+wgGRK5bYURrCNLVFexjxJGx6+jRo3g8HtasWYPdbk94HHP7h2gVerrCFvjE+2xznkGl0OF3xwgIiX02ydz3+fn9N41P+vkzMzPzkgtjt9svm7lnZmZeYnxf+4wVH+fH9qFbrtAQMUxF66tKKF+5tbWV8+fPc+211w7aIGGkyMlXs+h6Iz5vnH3bP87eKc+6A1GKU20fnbJ6re8EOu9RAtbVxHQj2zIy1UTiIu/Wufj2W/U8/kEzzZ4IX5yXxU9WN6NSNLIg73N85p7PU1xczPbt26mqGl/FbcMhqp9CMH0ZBveHqAO1gx+QQmZk3U4o5qa2Yyd7t/kIBkUWrzKNiMzxQESjUQ4dOkRhYSFFRUNQbpXiaP3VhI0z+lz47hZNG/uUTEiBwy8tLaW1tZWOjg5isRh79+69TDNm4cKF7Ny5E0mSqKmpwWAwjBuH7wu3YdLk9CtZGjaWo4x7UIVbBh1rqFV5I0V2nprrVhjx+0T2bvcRCoqYNDlMtizjjGPbiLdBFGI+zJ2biGrz8Y9zxc6LcYdi/Pl4Fw9sOsNz+9tQKwW+szSPX91ZyrrpKk7b/0K2sZzJ6ctQqVTcdtttTJo0ia1bt1JdXT3W5ieNL3MtMXVWdxVufORm1Z8k2zATi6aEE21vEwpHWbLKRGbW6E+Yjh07RiAQYMmSJUM6Th08h0IMEjb23efWG24f0y5XF5O0w1cqlXzlK1/hiSee4KGHHmLp0qUUFRXx7rvv8u673avv8+fPJzs7mwcffJAXXniBBx54IGnDU0VfKZkXEzbOQEJAO4hEQU9V3qxZsxKuyhtJsnLULFlpIhgQu2dNAZFy252IUnRkm51LEubOTQjxEJ7se0c11W+4NHvCPL+/jQc2neF/j3VRlqHj8TVFbLy1mNVT0lErBSrb/khcinBt3pd6Z2oqlYp169ZRWFjIe++9R01NzRi/kyRRqPHkfBpFzItpFGs3An6RWOMaRHUXJddVj+oCbQ+RSITDhw8zefLkAUMifaH1n0QSVET60FGKiREC0c4xbVx+MSm5sgsWLOjVFO/h5ptv7v1ZEIRx5eR7EKU4/mgHBWn9L7xJSiNRfQlafxX+zJv73e+jjz4CGLXMnETIzFaxZJWJ/Tt97NnmY8mqrAua5FsvaJInVlAyFLS+o+j8J/Fl3jomuuuJIkkSJzuCbDrl4GCLD7VC4IaSNO6YmcGk9EtTaVt9x2h07+OarA2kfULnXq1Ws379el5//XXeeecdFAoFZWVlo/lWUkpMV0TAugqjczth4zVETCP7tNqTWaaIz8FQkEtj+G/MkJaNevjj6NGjhEKhIc/ukSS0/qpuZ99HUoIzdA4JiYw+qm/HgisrhyzFBKJ2RCk+aLpU2FiOKtKOMtL3IkokEuHkyZOUlpYmXJU3WmTYVN0yDDGJPe/7KNSsIy5FqbGnRpP8YhQxN+bON7qF0SzXp3z8VBATJXae8/DdLQ38eGsjp7uCfGZ2Jr/eUMq3luRd5uxjYojD53+LWZPHTNv6PsdUq9Xccccd5ObmsmXLFs6ePTsab2XE8Gfc2C2w1vkaQnzoAmeJYu+MsWdbt4Db8tVpXJOzDleogXb/iRE7Z1+Ew2EqKyspKSkhJ2dokxRV+DzKmJuwse8vxl6dLsP4mARMaIffK1k6SBeang9T009YZzhVeaOJJUPF8jUmlCqBY7vMZGsWUetIXpP8EiQJc8erCFIcb/Y9404YzR+J81qVnb9//QzP7DlPMCryjety+PWGUj47JwtLP4VSJztewx/tYmH+lwdsGanRaLjjjjvIyspi8+bNnDt3boTeySggqPDk3IsQD2LuGJleuK3NET78wIdWp2D5GjNplu5MMr3KOuqCf5WVlYTD4aHP7gGtvwoJoXvBtg+6+2xkpazDV7KMr7/KUcYXTqyLvKi2EtXk9RnHlySJjz76iJycHHJzx0ecri9MZiXXrzFhNCnwVlUQE8PUJKFJ/kl0ngNoA7X4bLcS14x93nwPHb4oLx5u56uvneG/KzvJNan58aoCnltfwi1TrWgHKJRyBhs4bd/CFMsNZBtnDnourVbLhg0byMzM5O2336ahYXg6TOOBuDYXf2YFOv8JtL6jKR274UyYQ3sDpFmULF9jwmDs/gyUCjXTMtfS4a/CHhydpyS/309lZSVlZWVkZWUNfsAn0PqriOqKkZR9Nza3B+rGzeweJrjD90ZaUSl0CX37hk3lqEONl2mI19fX43K5mD9//rhIuxoInV7BstVmbGlFKN3zqO56N2G1woFQRu2YujYT0ZcRTBv6LGkkqLUHeWp3C3//xhneOu1kUYGJZ24p5ombJnNdoRnFIJ+VKIkcPP8iWpWZubn3JXxerVbLXXfdhdVq5a233qKxsTHZtzJmBCwrieomYe58A0UKMrskSaLmZIhjh4Jk56pYutp0oU7kY0qtN6JWGEZN8G/Pnj1Eo1EWL1485GOVUTuqSBvhftY5uvtsOD+WXR8HTHCH334hJXNwRx02XoOAhNZ/afrd0aNHMZlMlJaWjpSZKUWtEVi80kiecDsiIfZUvZVcVyxJxNz+l+6+qdmfGlNhtLgo8WGTl0febeB7Wxo4ct7PHTMy+NWdpXz3+nzKMhOvIK1zvIczVM/83M+jGaL+j06n46677sJisfDmm29euU5fUODJ/jSCFMfc8dekQjuiKHH8cJDTJ0IUFnfXiqhUl98raqWesow1NHsO4Q23JmP9oLhcLj788EOmT59+iVpAomh83U/8/cfvu+sZ5Bn+OKGnICIR4ppc4iorWv/J3m2dnZ00Nzczd+5clMrxn37Yg1IpsGzxVMzx+XRI77N/XyfR6PAanutde9CEzuGzrUfso4J3NAjHRP5W4+Rbb53lZztb6PRH+cqCbF68q5QvL8gmyzi0Ah5/pIvjHa+QZ5pLUdrQZ34Aer2eu+++G6vVyptvvnnFhnfiGtsF7fwadJ6Dgx/QB9GIxNa3W2k4E6F0hpZ51xlQDNCzd1rmWhSCiuoRFPyTJImdO3eiVCpZvnz5sMbQ+quIavL67e3QFTyDUlBj0Y6fNp4T1uGLUgx/pCtxQSNBIGwsRxM8gyCGge7ZvUql4ppr+i64GM8ICoGlU+8BZYiW6Fts/mtLn41UBkIZbsfkeJewsZyQefQXrJ3BGP9ztJOvvlbHLw+2Y1Ar+e7yfF64s5Q7Z2ZgUA/9S1iSJA63/g5Jki7JuR8Oer3+kvDOler0g+lLiOhLMXW9jWKIAnx+X5zd73s53xxgzkI95XP1g15TnSqdEssKzrl2E4y6kjG9X86ePcu5c+dYvXo1JlPf8feBEGI+1KEGIv3M7qF7hm/VlyTU1H60mLAO3x/pREIcUkFE2HQNghRDE6ghEAhw+vRpZs6cOSINTkYDq34SJZaVxG07cQfOs+s9L057LLGDpThpHS8jKbR4su4a1VBOgyvMs/taeWDTGf5y0k55toF/qZjE07dMZmVxGsoBZo+D0ew5QKvvKLOz78GYgsXnq8LpCwo82feAIPS2qEwEe0eMXe/5CIck1q7PZ3Jp4lLhM2y3IUnxEUkfjkaj7NixA5vNNqzYPYA2UI2A1G/8Pi5GcYYayNSPr1DvhHX43kh3hs5QJEujukmICiNaXxXHjh1DFMWE25+NV2ZnfwqFoMQ67z2USoG92320NEYGPc7o2I46fB5v1gYk1dBnSENFkiSOtvp5bFsTD75dz64GDzeVpvPc+in8aFUh1+QYkl40D8d8HGn7A1ZdCVMHKLIbKleD0xfVFny29WhC59C79gy6f+PZMPt2+NBoBa6vMJFX2HeDk/4waXIoTLuOOue2lCQWXMyBAwfw+XzccMMNww7Fan1VxFUWYv2kdLtCjYhSlMw+qm/Hkgns8Hu6yA8hlVJQEjbOQOOv5uSJYxQXF48bTaDholdbmWFbR6NnD+XXt2GxKjmyL8DpE8F+F3NVoWYMzu2EzPMIm2aNqH3RuMi2s26+s/kcP93WRL0zxOfm2HhxQylfvy6XgrTUyVgcafs94ZiPRflfRZFiSYhPOv36+vqUjj8ahMwLCBvLMTneRXkhpfmTSKLEyaNBPjoYxJatYkXF8FsSzrTdTkwMUZfC5ix2u53KykrKy8uHLKHQgyCG0QRruxdr+5lk2IMXCq7kGf74wBduQ60woFWah3Rc2FSOQgqRq3ePaPvC0WR65m0YNFZOOv7M4lVGiko01JwMc3C3n0jkE4/vYpS09lcQlSa8tjtGzCZvOM5fTtj51G8P8e/7WhEliW8vyeW/NpRy72wbabrUxkWb3Ad65ROs+smDHzAMepx+T57+J/tGjHsEAU/WXUiClrSOl0GKX/JyOCTy4U4/Z0+HKS7TcN0KI2rN8F2MVV9MrmkONfZ3iImDP3UOhiRJfPDBB2g0mmEv1AJoArUIUqxfsTTozr/XqzIw9LOgO1ZMWIc/lJTMi4nopxIVFcwpig9NQnUco1bqWFz8RezBWloDh5m7SM+s+Xo62mLsfNeHy/FxXN/keA9VtANvzqeQlPqU29LmjfCrQ+189bU6/vBRJyUZBn66upBn15VQUWpBo0z9LRuKuTnc+t9YdSXMzLo95eNfTI/Tz8nJYcuWLZw6dWpEz5dqJJUJb/YG1OHzGB3be7c7OmPsfNeLoyvG3EV6Zl87cCZOosy03U447qHetTPpsU6fPk1LSwvLli1Drx/+vavxVyEq9EQHmBjYg3XYxlE6Zg8T1uH7Im2Yh6Fg5/WHqOlQU54THrR450piRu5NpGsLOdb+Z0QpTsk0LctXm5Ckbg2ec3VhVIE69K7dBNIWp7xdYXVnkCd3tvCNN8/yTq2TZZPM/Nttxfz73bNYkG8asaI2SZI4dP63RMUQiwv+HoUw8hkVPRW5PSqbx48fH/FzppKwaRYh8zwMzu0og42cOR1i73YfCqXA9WtMTJqSuj7OWYYZZOrLON31NuInniiGZHM4zO7du8nJyUkuq65X+35mv0qwwagLf7RrXBVc9TAhHX5cjBKI2oelUX3q1ClOtevQK0Oows2DH3CFoBCUzM25H1+kgzPO7pip1aZi5c1mMrNVnD7qwND0MjFVJj7bbSk5Z1yU2Nvo4f++08AP3m3geLufu8sz+dWdpXxnWT4l1pHPfmpw76XFe5jZ2Z8ifRQbtfSobPY0UTly5MionTsVeG13EFemoWv4MzXHPOTkq1l5k5l0a2q/MAVBYKbtdvzRLprc+4c9zr59+wgGg6xevTqpyYM6WH9B+36AdMzg+BJMu5gJ6fD90Q4kpEE1dD6JJElUVVXhU5ciCSp03ivrj3Qwck2zyTHO4mTnpt7MCK1WweIVBm6dvRWNws8HZ2/H601uQTMYFXnrtINvvnmWf911HncoxtcW5vDrDWV8YV4WmYbR6XQUiDo40vp7bIZpTMu8dVTOeTE9evplZWXs3r2b/fv3J1f1PIq4PGp21N+OXuni1rnbWbjcgFozMk9h+eb5pGkLONU1vKrw5uZmjh8/zuzZs8nOTqxVaX/ovYcRBW2f2vc92INnujPfdCOzFpQME9Lhe8M9KplDm+E3Nzfj8XgonT6HkGkuOs+RUe0MNNIIgsDc3PuIxP1Udb7Ru13vq8SmrKJTu4Y2Ty473/NytiY85D8+eyDK7yo7+OqmOv7rUAcWnYofrijg+fVTWDfdil49erejJEkcPP9rRCnGdflfQzFG6p5KpZJbbrmFmTNnsn//fnbt2jWunb4kStSdCrFrq482XyEO3UqyFUfRjaCksSAomGG7HXe4iVbfR0M6NhAI8M4775Cens6yZcuSskMR86L1HieUdm2f2vc92AO1WHSTB1RXHSvGTwnYKNIrizzEGH5VVRUajYbS0lKCMRN672F03sMEx6n2+3Cw6iZTYllBreNdyjJuJF1QYep8g4h+CkL+DazKgo8OBjhZGaT9fJR51xnQGwZ2lvXOEJtOOdjd4EGUYEmRmQ0zM5huS/2ib6KcdW6nzXecBXn/B/MYN2pRKBRUVFSg0Wg4evQogUCAioqKMeuL3B8BX5zK/QEcXXHyCtXMWahH1NxEtPkM5o7XuutURkgGeHL6Ek50/IVTXW+Sb06s9kWSJLZu3UooFOKOO+5IuhOdzrMfgTjB9KX97iNKcRzBeqZYx2drz/F1R40S3kg7GqVpSKJY4XCYuro6ysvLUalUxFQFRHTFGFz7CKYvG3f678kwO/semj0HOdjya+5SWnpFtBAU6PRw3QojjWcjnKwMsmOLl9kL9RRMuvSPSZIkjpz3s6nawbG2ADqVwC1TrayfbiXXPLYzH1+knaPtfyLHeA1l1hvH1JYeBEFg5cqVGI1G9u7dSzAY5LbbbkOrTd0C6HCRJImm+ggnKoMIAsxfbKBgsro3Fu7J+QwZTc+S1v4KrvyvjMjfgkJQMT3zVirb/odO/2myjNMHPaayspJz586xatWqYUkfX4IUQ+/eT9gwfUD5b1eoibgUwWYYX/n3PVw9XmoIDEU0rYeamhri8fglDcqDlmUoYw40gdOpNnFM0autzMv9LB2BaqqD1Xiz7rpEGE0QBCaXalm11ozRrODIvgBH9nXn7EfiIu/Vufj22/X80wfNNLsjfHFeFi9uKOPvFuaMubOPiRH2NP0HAkoW5T/Qb/P6sUAQBBYuXMhNN91Ec3Mzf/3rX/H7U1tlOlTCIZFDewJ8dDCIJUPFqrVpFBZrLln4jGtseLPWowmeQe/aPWK2TLGuQqM0JSSd3NbWxt69eyktLU1J21Gt7zjKuK97cjcAHxdcjb8FW0hyhu/z+di4cSOdnZ1kZWXx0EMP9SlE9Pzzz3PkyBHS09N55plnkjllSvCG2xNqaHExVVVVZGZmXrLoEzaWE1elY3DtJTLE8cY707STaRd07Il5MWlz6Us8wWjubmBRdyrM6ZMhms5H+FD0UBUJUmLV8p2leVw/OQ21cvykrx5p/T2uUAMrJn03JVo5I8HMmTPR6/Vs3ryZV155hQ0bNmCxjK4SqSRJNDdEqToaJBaVKJ+rY8p0bb8ZLiHzQjT+05js7xI1lBHTDq+KdSBUCh3TMtZyovNVXKEmLLq+62DC4TBbtmzBaDSyZs2alKT0Glx7iamziAySeWMP1KFTpWNQj897K6npzaZNm5g9ezbPPvsss2fPZtOmTX3ud8MNN/CjH/0omVOljJgYJhhzDGmGb7fbaW9vp7y8/NKbR1ASTFuCJliHMtJ3qfmViBAPkd7xCqt0xQgKDQdbfo3Uj2BWqy/K+34Xb4l2HNEYy8R0vpWdxxMrJ7F6Svq4cvZnnTuod+2g3HZHwnHgsaK4uJi7776bSCTCyy+/TFtb26id2+eN8+EHfo7uD2A0KVhxk5nSGbqBHacg4M2+C1FpJK3tz5CCyti+KMuoQKXQUt31dp+vS5LEtm3b8Hq93HLLLSkRNlSFGlGHm7tj94M8EdqDdWTqy8ZtM6SkHP7BgwdZtWoVAKtWreLgwb71ssvLy4clQToS+CIdAENaqKuqqkKhUDB9+uVxw2D6IiRBhd61L2U2jimSRFrHKyhibsS8z18I7Zy6RM9EkiROtAd4Ykcz//DmWbaecTOr2MCNa82Uz9MR9Ujs2OLlTHUIURwfGSfOYANHWn9HjvEarsn+1FibkxC5ubnce++9aDQaXn31VU6fHtnQoRjv7ki1Y4sXlzPG7Gv1LF9jIs2SWBqupDTiyfk0ymgXaZ2vjUgvXK3KxBTrjTS69/Vm213MyZMnqa2tZenSpeTlDdyrOlEMrr2ICi2htAUD7heOefFF2sdl/n0PSYV03G53r3iY1WrF4/GkxKiRxDdE0bR4PE51dTUlJSUYDJcr/klKIyHTPPTeI/gz146I3MBoYnDtQuuvwmtbR0w/mRLdJJo8BzjW8RLZxjl81Kbn9VMO6hwhzFoln56Vybpp1o+bgFshr1DDiSMBqj4K0dwQZc5CPdbMscsPiMT97Gl6Fo3SxJLCb4xZCuZwsFgs3HvvvWzevJl33nkHu93O0qVLUz6DtHfGOHYogM8jkl+k5pr5enT6oV+nqKEMf8YaTI6tRHTFhNKHJz88EDMyb+OM431Odm5iSeHXe7d3dHSwY8cOJk2axLXXXpuScyliHrS+4wTTlyIpBl5AtwfPAOM3fg8JOPzHH38cl+vyJgT33Zd4n8+hsHXrVrZu3QrAk08+ic02vFiYSqXq89jGQHdP2uK8cjSqwbN0qqqqCAaDLF26tH9b9LciHD9EZvwU5Awsq9ufXWONSqXCprYj2LcgZVyLsfROjBecygrtQ7xS+U3+98Qv+Nvpz1JkMfD9G0u5ZUY2ur6ajNigaJJEw1k/+3d1snurj9JpZhYsycBkHlpRVbLXS5JENp98jmDMzl1znyI3PTXZE6P9OT7wwANs3ryZQ4cO4fV6ueeee/rM4BmqXR53lMP7ujh3xo/JrKJiXQ5FxUNr6XgZmZ9GOt2GuestTDnlYCpJ8fWyMSdwB5XNr7Js6hfIME6mq6uLN954A7PZzH333ZdwRGEwu4SmPYCEruQ2dLqB7a/ztiCgYGrhQtTK5EJJI3V/DerwH3300X5fS09Px+l0YrVacTqdpKWlJW1QRUUFFRUVvb93dXUNaxybzdbnsW3Os+hU6XhcQWDwoqkPP/wQo9GIxWIZwBYDFl0JyvNbsavnDRjn68+uscaWpkKq+SWi2obTsh7JbqfTH+XNagfv1rnJT7uRxZM28+0lZ7hxyjoUgoDP7cQ3wJimdFi51kTdqRBnT3upP+NlyjQtZTN1qNWJzVCTvV6nOt/inP1D5ud+HlU0K2XXfiw+x6VLl2I0Gtm5cyf/+Z//yfr160lPvzTvPVG7ImGR2qow9XVhFAJMu0ZH6QwtKlWQrq7kiwkF6wYyfL+A6udxFH2LzJzJKb1ek403clzxFrtrfsNsy5d45ZVXkCSJO+64g1AoRCgUSmicAa+XFMPW9gFRw3TcPgX4Bra/2X4ci24SbqcPBvzLSNKuQRhI9jmpZ9uFCxeyY8cOAHbs2MGiRYuSGW5U8EXaEpZU8Pl8NDQ0MHPmTBSKgS9Vd4qmE80nmpxfEUhxhNoXUIhh3Lmfo8Yp8vTuFr72+hnePO1kUYGJry+6k1zTbNyRTQSinQkPrVYLzJyjZ/VtaeQVqqk7FWbb2x7O1YVHPL7f7jvJ8Y6XKUpbzNSM1DU0GSsEQWDu3Lls2LABv9/PSy+9RFNT05DGEOMSZ0+H2La5u1q6aLKGG9elMX2Wrs+m4sNFUhpx534ORcxLWvtLCXfJShStysy0jLU0eQ7w+ju/JxKJcOedd6Y0m0nnPYYi7iNgGbxCV5RE7MEz407//pMk5fA3bNjAsWPHePDBBzl27BgbNmwAwOFw8LOf/ax3v3/7t3/jH//xHzl//jxf//rX2bZtW3JWJ4E30p5whk5NTQ2SJDFz5uApl2HjzO4UTffeZE0cdUz2LQjeOipVt/GDnSG+t6WBQy1+7piRwa/uLOW71+czzWZgUf5XEVCyu/HfiMQDQzqHwahgwRIjK24yYUpTcPxwd9FWS0MEaQQcvz1Qx+6mf8Osze+2e5xmTQyHoqIiPvOZz6DX69m0aRMHDhxAFAd2qKIo0VQfZvsWLyePhki3Klm11szc6wzDitUnQkxXiDdrPdpALbQMnjs/VErS1kBchZh1mvXr1yetk3MJkoTevZeYOptoAjF5d6iJmBga1wu2kOSirdls5ic/+cll2zMyMnjkkUd6f//Od76TzGlSRjQeJBRzJTzDr62tJSsrK7GuVoKSYPpSTPYtKMNtxIchvTwWKNzHMLh2s8V5Df9/jYUsQ5SvLMjmprL0y5qAG9SZLCv6NjsbnmZv07OsnPy9IcsJWzJULFttov18jOpjQY58GOD0CQVTy7UUTNakREPdFWpkZ+PTaJVp3DD5B6iv8IX0vuhZzP3ggw/48MMPaWpqYu3atZfFfePx7irZuuowQb9ImkXJ4pVGsvNGR6AulHYd6lADuua30ORlEkmgQjYRYrEY7235AJ8qB9OUFnQZ4ZSM20N3KmYL3qw7E+rX3Ojei4CCXNPslNqRaq6cdIUU4OvtYzu4M/Z4PLS3tzNtWuK678G0hUiCCqNz++A7jzGuYIzNx06jb/sLJ3w2XnWt4LvL83nhzlLunJlxmbPvIdc0i0UFX6Xdf5KD538zLKEvQRDILVCz6hYzC5cbUKoEjh4Ism2zl3N1YeLx4c/4veFWdjT8HKWg4YbiH6BXX9ktKAdCq9Vy8803U1FRQXt7O3/60596UzfjMYn6mu7w2fHDQbRagetWGFl5s2nUnD3QnZ+ftQEM+aS1v4Qi6kx6SFEUeeedd2hqamJx6X1olCZOdPwlBcZeQJIwOrcjKnSEzPMHt0eKcc69h3zzPHQjpCWUKiaUlk5P43JTAjn4Pe3nysoSf0STlEYClpUYndsI+RekbDaTShrdYV4/5eBEcxvPzdiMiApPzmf55ZpZ2O32hMYosazAH+nkZOdrGNU2ZmXfPSxbBEEgr1BDboGajtYYNSdDHD8cpLYqRPFULZNKhibD4I908UHDvyJJIqtLfoTa3NbFAAAgAElEQVRJk8JH/HGKIAiUl5eTl5fH3/72N/74xz8yuWgWWmEe0bCCjCwl867TYctRjV1YS6FBmvoNOP7PWM7/Fmfh3yMNQcfqYqLRKO+++y5nzpxh5cqVzCqfh6rrdj5q/zOdgRqyUtCYR+s/gTZwGm/mrYOmYgK0+o4RirkpsaxM+twjzQSb4ffIIifm8HNyci7LghgMf8ZqYuoszJ2bEMTUPmYOF0mS+KjNzz9tb+Lbb9VzqLGL52dtJ0cXJjr5y5Tl5Q3ZGVyTdRfFlhWc7HyNemdy7ecEQSAnX831FSaWrDJiNCupPhZi65sedm5tx2mPDTpGMOrig4YnicaDrJr8f0kbgdL+8YokSYgxM1MnryPNMIOGphM0d/6NGfNCLL/RTFaueuzXMPQ5uPO+iDLmJL31DyBGhzyE1+vllVde4ezZs6xYsYJ587qrpcsyKtCp0lMyyxfiAcydbxDVFhC0JNb3tt65C60yjTzz+O9xPcFm+G3oVVZUioFzZF0uFx0dHVx//TBkjwUV3uy7sba8gNH+Hr4R7pE6ENG4xO4GD69XO6h3hrHolHxhjpUvWd5AH7bjzvsisX70SAZDEAQW5X+FYNTJwfO/Qa+2Jh2/FASBrFw1WblqvO445+rCNJz1cea0RLpVSXGZhvxJmsuyScIxHzsafk4o5mLV5B9g1RcnZceVQiwq0dIY4VxtGI9bRKWGFdffRChWwt59H7Dl3Vdo75zP4sWLUatHMYzTD1F9CZ6cz5DW9ifS2/+MO/ez/bYJ/CStra289dZbxOPx3k5hPagUWmba1lPZ9j+0+06SYxp+C0NT198Q4gG8+V9OyLZQzM15byXTMteOSnvMZBn/FqYQf6Qzocf8nnDO1Kn9d7UZiKi+mED6EvTuvYTMc4jpJg1rnOHiC8fZUufi7dNOHMEYRekavr0kl5WTTdi6/oLOdxZP9j1Jh5wUgorlRQ/yfv3j7Gl6lhtL/jFlXX7M6UpmX2tg+eoMPjrcyrm6MB8dDHKiMkhugZqCSRqyclX4om3sbXoWb6SNlZO+h22ATkRXA/G4REdrlPONUdrORxHjkGZRMGehnoLJGnJzs+jqEigtK2LPnj0cOXKE2tpabrjhBkpKSsbafMKmWfhst2PuehNz5xvd8f1Bnj6qqqrYtm0bZrOZ9evXk5GRcdk+pdbVVHe9zfGOV8k2lg/riUYdqEPvPYTfsiph8bcG114k4ldEOAcmmMMPRJ1kJqBTXVtbS25uLmazedjn8meuReurIq3jNRxF/wCj8O3f7ovwRrWTrWdchGISc3MNfHtJLvPzjAiCgLFrMzrfR/gybu7u2pMC1Eo9Kyd/j61nH2N7/RMsyn+AovTrUjI2gEajoGSqluIyDfbOOC0NEVqbo7Q0RJEyKgnn/RGlQs2KooeTmtmNZyRRoqsjRktjlNbmCLEoaLQCRcUaCos1WDOVlzk4nU7HmjVrmDlzJtu2bePNN9+krKyMlStXjrmuVdCyDEXMg9G1g7gqjUDGmj73E0WRPXv2UFlZSVFREbfeemu/YmhKhYbyrDs53PrfNHsODv0eFKOYO18jps7E3489n0SSJM66dpChLx3VfsjJMGEcviRJhGJO9KqBszacTiddXV2sWLEiufMpdHiz78TS+gcMzl0EMkauA051Z5BNpxzsb/aiEGDF5DTunJlxSRNwvWs3RtcuAulLCFhvSOn5DeoM1pT8hL3Nv2Bv839QFriJeTn3o1SkLowgCAK2bBW2bBXl8yLsb/gTLeH3UQRKUDZ+mY9qMjif6yc7V01Wrgqt7spengoFRTrbonS0xehsixGNSKhUkFvY/XRjy1EllMKan5/P/fffz5EjRzhw4AANDQ1ce+21zJ07d0ybq/gz16KIezA5tiKq0gilXVq0abfb+eCDD2hpaWHOnDmsWLECpXLgEMsU6w2ccW7nSNsfyDFdM6QGR0bH+6iiDpz5D0CC960jVI8n3MLCvC8nfJ6xZsI4/KgYIC5F0asGrsQbTnZOf0SM5YRMszE63idsuoZ4CrNG4qLE/mYvm045Od0VxKhRcNfMDNZNt17WBFznOYK5621Cxmvw2dYnlFc8VIwaGzcW/yPHOl6ixr4Fe6COZUXfSnmmjD/Sxd7m/8ARPsu0zFu4Ztq9dOZItJ+P0tkWo6WhezEwzaIkO0+FLUeFJUOVsJTDWBGNiDgdcbraY3S0RvG6uwuptDqBnHwVOflqcvLUKIdRDatUKlm0aBFTp05l165dfPjhh1RWVjJ//vyxc/yCgDf7UyjiPswdmxAVBiKmawiFQuzfv59jx46h1WqpqKi4pOnQQCgEJYvyv8LWs49xrP1lFuYn5ohV4fMYXLsIpi0kOoROVfXOHSgFDUXpSxI+ZqyZMA4/EHUAoFdfHv+7mNraWvLz85MK51yMz7YeTaAWc8druAr+LunxglGR98+6eLPaSZsvSq5Jzd8tzGbNFMvlTcAlCYNrJyb7FiL6KXhyPjOirRiVChXzcz9HlmE6B1r+i3fPPMp1BX9HYdrCpMeWJIkW72EOnu/W5l9e9CCFF2aFBZOgYJIGSZJwO+N0tsXoaItypjpM3anuTClTmgJLhhJLhgprhhKzRYlyjLT643EJjzOO0xHH5YjhcsTxe7sdvKCATJuKwjkasnLVpFkUKcuwsVgsrF+/no6ODvbv39/r+BcsWMCcOXNG3/ELSjy5n8PS8iLpbX+kJnotr+yyEw6HmTVrFkuWLEGvH1rRXIZ+ClMz11Jj38Lk9GWDt0KU4pg7XkVUGvBl3pbweWJihEb3hxSmLUKjvFxFd7wyYRx+KNat+DnQDN/hcGC323s1/lOBqDLjs60jreNV9J4DMMysHXsgytunnbxT58IXEZlu0/N/5mexuNCMsq9HeymOqfNNDJ79hExz8GTfk/CjarIUpi3EopvE3qZfsKfp35livYEy6xosuslDdl6RWIBa+3vUOrbijZzHopvMssJv99nPQBAELBndM/qp5TqiUQlnV7dDdTm6QyPN56IX9u2WezCYFBgv/DOYlBhNCrQ6AbVaQBhm1a8oSkSjEpGQhN8n4vd1O/SAX8TvFQkGxF6peK1OwJKppKhYgyVDiTVThWqEn0ays7Mvcfz79u2jsrKS8vJyZsyYMaoqoKKg4SS3kub6M9Mth1g3KwdN2WfJSkImYVbWp2j2HOLg+d+wtvSfBw4ttm5FHT6PO/ezQ5I2b/EcIioGKLFeGYu1PUwYhx+8UOE3UOVlTzintDS1Akgh87XovEcxdb6JlJYJQuKZJOecITadcrCrwYMoweJCMxtmZjAjq/+bUxDDpLX9L9rAafyWVfgzbx71JusmTTZrSh7lWPtL1Dq2ctb5AWZNPpPTlzIpfemgDWg84RZqHVtpqN5DNB4kQz+F6wr+nklpixNeG1CrBbLz1L2VpZIkEQxIuBwx3M44AZ+I3yfitHcvhPZ1vFrz8b+LQ8gaTYRIpLurUzwO0YhENCIRiYj9jmUwKbBkKimYrCbd2v20oTeM3VpDj+Nvb2/n4MGDHD16lCNHjmCz2ZgxYwbTpk0bsQVet9tNdXU1p0+fxuVyYTJm8cUV+czPriYkbsUjfnrYExS1UsfCvC+zs/EpTnW92W9hoM69H6HzdcLGcsLGWUM6R71rJ0a1jWzDjGHZOFZMHIcf63b4ugFm+LW1tRQUFKT+JhcE3LmfJ73tf9Cc+Q2GzFsIWFb2G0uXJInKVj+bTjn4qC2ATiWwdqqVO6ZbB20Croh5SG/9HapwK56sDSPSgCJRlAo18/M+T3nWnTR7DtLg3seJzlc50fkqGfop5JvnIyAQE0NExTBxMUxMDBGIOrEHa1EIKqZmr6LIsCKh7KrBEAQBg1HAYNSQf1H5gSRJRCJS7xdAJCwRjYgXHPgFRx6WiFxURxeNxInFuqfpSmX3TN2UpkCjUaHWKNBoBDQ6AeOFpwiNdvwuIufk5HD77bcTCASora2lurqa3bt3s3v3boqKipgyZQo5OTnYbDZUquG7jHA43Dv++fPnASgsLGThwoVMnToVlUqF17Ubs30zipgHd94Xhl2Rm2eew6T0pZzqeoNJ6YtJ016URSNJGB1bMTq3IVlm4864Z0jrWv5IJ+3+KmZl3YVwBTXTgYnk8KNONEoTKkXfDtNut+NwOLjhhhtG5PySUocr/0tkud7AZN+CIubBZ1t3ycw7GhfZcc7D66ccNLojZOhVfGFeFreUWTBpBy8CUYbbsbT+N0I8gDvvi0SM42P2oVWZKc24kdKMGwlE7TS6P+x2/h2vAiCgRK3UoRS0qBQ61Eo9s7M/zRTrDRTmThlx3XlBENBqBbRaBdbMxI4Zr30NksFgMDB37lzmzp2Ly+Wiurqa6urqXgl0hUJBZmYmOTk5ZGdnk5GRgUqlQqlUXvK/JEm4XC4cDgdOpxOHw4HH48HpdCJJElarlaVLlzJjxozL1sqC1hWIagtp7S9jbf5P3HlfIj7MZvPzcz9Hm+84B8//hhuLf9ztnKU45s5N6D2HCJoXop3+ANiHpu9T79oFQLEluUy+sWDiOPyYa8D4fW1tLYIgpCQ7p18EFVLZAwRjGgzuPSjiXjw59+IJw5ZaF2/XOHGF4pRYtXxnaR7XT05LqAm4EA9icH6Awb0XUaHHVfA1YuM0L9igzmSGbR0zbOuIxoMoBDVKxYS5Da8YLBYLS5YsYfHixfh8Ptrb2+no6KC9vZ3a2lpOnDiR0DhKpRKLxUJeXh5lZWVMmTKFrKysAddywqbZuJRm0lt/T0bTswQsKwhYVyaka3MxOlU683Lu58D5/+KMcztllhWkt/0v2kA1fuuN+DMq0CZY6duDJInUu3aRYyzHOMwvorFkwvylBQfIwZckiZqaGgoKCvrsW5tSBAU+2zriqjTM9r/R6nTy4ImVOCNqrs03cufMDObkGBJb3JRi6N37MTq2IYhBQub5+DPXIqqS7zw2GlyNssVXG4IgYDabMZvNvZMhSZJwu9243W7i8TixWIx4PN77syRJWCwWrFYraWlpKBSKIT8RRfXFOIq+jcm+BaNzGzrPAfwZN3UXDA7BSRdbVnDOvYdj7X9mpvcYmmhnUqHONt8JAtEu5uTcO6zjx5qJ4/CjTtJNhX2+1tXVhcvlYv78waVQk0WSJE52BNlUPQVrcAX/OGU3v5+zmaB5EfqseYjqBGKWkoTWfxKjfQuqqJ2IvhSf7baEy8FlZJJBEAQsFktKu0v1hai24sm9n0BoOaauv5HW+RoG9x58mbcSMUxPKO6uinaxwlDO6/5TvO0/zo35X0M1TGfvDbeyv+WXGNU2CsypqVQfbSaEwxclkVDM3W9Ip66uDkEQUp6dczFxUWJPo5e332umusOHWavk1qmLaM0qJdf3HrbAe9DwHlFNHmHT7I8LtSQRRcyNKtKJMtqJMtKJOtyEOnyemCYbV96XiBimjUgxlYzMeCCmm4Sr4Gto/FWY7H/D0vo7otpCorpC4uosYpps4posRGX3k60y0o7OdwKt/wSqC5Lo6wzlvB2s5Z3OV7jBUDbkcEwg6uCDhp8DsGry/+13LXC8MyEcfjjmQUJE109K5tmzZ8nPzx+RcE4gGue9OjdvVjvoDMQosuj4+qIcbpySjlalALJwWqejiDrR+k6g9Z/E5HgXk+Nd4ioLirgfQfo4z09U6IhrsvFk3TXkx1sZmSsWQSBiugaHcQZ693503qPovJUoLpIgFwUNklKPMuZGQiCqK8Zru52waRY6VTqrAnXsbHiKbeeeYHXxIwlXgfeosUbiPlYX/wizNm+k3uWIMyEcfk9KpqGPGL7H48Futw9PCnkAOv1R3jrt5N06F4GoyDXZev5uUQ63zi3G0UejEVFtJWhdQdC6AkXMjdZ3EnXoHKIyrXcGE1NndaepybN5mYmKoCRoWUbQsgwkCUXch/LC068q0oEi7sWvLyViLEdUXZoBZDOUcUPxI+xo+Fe21f8zNxQ/go2BZ/oxMcyuxmfwRdpZOel7ZOjHXnE0GSaUw+9rhl9fXw+QMunYOnuI10852N3oAWD5JDN3zsxgamb3AqUiAWctqtK7b2qWpcQmGZmrEkFAVJkRVWaiTEnokAx9MauLf8QH555k+7knSLc8CfRddyNKMfY2PYsjeIalRd++KtRYk3L4Pp+PjRs30tnZSVZWFg899NBlRUtdXV0899xzuFwuBEGgoqKC225LXLMiFQSj/csq1NfX92YUDBdRkjjU4utuHdgRRK9ScMeMDG6fbiXLOPaNJ2RkZD7GoivixpIf88G5J9l09AfMyv40BlUGOlUaOpUFrcqMgMCBlv+i1XeMhXlfoegTap5XKkk5/E2bNjF79mw2bNjApk2b2LRpE5///Ocv2UepVPKFL3yBKVOmEAwG+eEPf8icOXMoLOw7Y2YkCMacCAiXNRiORCI0Nzczd+7wWpOFYyLb6928Ue2kxRMhy6DiKwuyuaksvd8m4DIyMmNPmjafG0t+zK6mpzl0/sVPvCqgVuiJigFmZ3+a0hGUNh9tknL4Bw8e5LHHHgNg1apVPPbYY5c5fKvV2jt71uv1FBQU4HA4RtfhR51oVekoPrHA2djYiCiKQw7nuEIxNtc4+VuNC084TlmGju8uz2f5pH6EzGRkZMYdJk0On130K5raagjF3IRiHkIx14X/3aRrCynLqBhrM1NKUg7f7Xb3OnOr1YrH4xlw/46ODurr6wesZt26dStbt24F4Mknnxy2cp9Kpeo9Nt7qx6zLumysXbt2odPpmD179qDNFQDq7QFeqmzhneoOonGJ5VMyuH9BAXPz0xJWgbzYrvGEbNfQkO0aGuPZruL88RebH6nrNajDf/zxx3G5XJdtv++++4Z0olAoxDPPPMOXvvSlAdMfKyoqqKj4+Ft1uHolF1f2eQIdGNSZl4wlSRLV1dVMmjQJp7N/LQ1JkjjWHuD1Uw4On/ejUQrcOCWdO2ZkUJCmAaLY+8i6ScSu8YRs19CQ7Roasl1DIxm78vP7L8Ac1OE/+uij/b6Wnp6O0+nEarXidDpJS+u7pD8Wi/HMM8+wYsUKFi8effXGYNRJpv7Sp4q2tjaCwWC/4ZxoXGJ3g4fXqx3UO8Ok65R8do6NW6daSNNNiOQmGRmZq4ykPNfChQvZsWMHGzZsYMeOHSxadPlKtiRJ/PKXv6SgoIDbbx9e849kiItRwnHvZTo69fX1CILA5MmTL9nuC8d5p87F26ed2IMxitI1fGtxLqtK0tAorywpVBkZGZmLScrhb9iwgY0bN7Jt2zZsNhsPP/ww0N056oUXXuCRRx7h9OnT7Ny5k0mTJvH9738fgPvvv58FCxYkb30ChGJuAPTqS1Myz507R15eHjpdd6Pvdl+EN6qdbD3jIhSTmJNr4B8W57Ig35iyFnMyMjIyY0lSDt9sNvOTn/zksu0ZGRk88sgjAMyYMYOXX345mdMkRTB2oZftRTN8j8dDV1cX119/Pae7gmw65eDDJi8CsKI4jTtnZDAlQzdGFsvIyMiMDFd9MLq36OqiKtuzZ7ura19t0XDiVANGjYINM7sLpTINcqGUjIzM1cnV7/AvyCroVVZCMZH3z7g5eqAKpcJAp6jngWutVJRa0Kvl+LyMjMzVzQRw+C4EVLx8IsiW2laC4QirwnZyp8zg+7dOkQulZGRkJgxX9bT2nDNE5fkWvBEjf61yMDvHyP9XrkCQRJbNmS47exkZmQnFVTfDlySJylY/m3e1cbDRRUWZkwy9lefXTyHPrGHr1lNoNJoBixNkZGRkrkauOocfjIk8tfs8eo2KL8zLQqcOY9UXkmfWIEkS9fX1TJ48OSEpBRkZGZmriavO4RvUSh5fM4lry/JxOx28esqJXjUbgPb29gGra2VkZGSuZq7KGH5Zpg61UkE0HiQmhnpTMnuqa4uLi8fWQBkZGZkx4Kp0+D2EYj2NT7od/iera2VkZGQmEle1ww/05uBbCAQCdHZ2MmnSpDG2SkZGRmZsuKodfjB6weGrM2hubgaQHb6MjMyE5ap2+B+HdCw0Njai0WjIzs4eY6tkZGRkxoar2uEHo05UCh0qhY7GxkaKiopQKK7qtywjIyPTL1e19wvGnOhVFlwuFz6fj6KiorE2SUZGRmbMuModvgu92kpjYyMgx+9lZGQmNle3w4860ausNDU1kZaWRnp6+libJCMjIzNmXLUOX5IkgjEXOqWF5uZmioqK5M5VMjIyE5qr1uGHY15EKUospCISicjxexkZmQnPVevw/WE7AB57BEB2+DIyMhOepMTTfD4fGzdupLOzk6ysLB566CFMJtMl+0QiEX76058Si8WIx+MsWbKEe++9NymjE8Ef6e5la2/3kZ2djV6vH/FzysjIyIxnkprhb9q0idmzZ/Pss88ye/ZsNm3adNk+arWan/70pzz11FP8/Oc/5+jRo9TU1CRz2oTwhbsA6Gz1yrN7GRkZGZJ0+AcPHmTVqlUArFq1ioMHD162jyAIvWJl8XiceDw+KoungQsz/FhQJadjysjIyJBkSMftdmO1ditRWq1WPB5Pn/uJosgPfvAD2traWLt2LVOnTu13zK1bt7J161YAnnzySWw227Bsq3I5UUg6VEoNs2bNQq1WD2ucVKNSqYb9nkYS2a6hIds1NGS7hsZI2TWow3/88cdxuVyXbb/vvvsSPolCoeCpp57C7/fz9NNP09jY2O+su6KigoqKit7fu7q6Ej7PxXhDncRDKvLz83G73cMaYySw2WzDfk8jiWzX0JDtGhqyXUMjGbsGat86qMN/9NFH+30tPT0dp9OJ1WrF6XSSlpY24FhGo5Hy8nKOHj064mEWb7CDiF/JVDl+LyMjIwMkGcNfuHAhO3bsAGDHjh0sWrTosn08Hg9+vx/oztg5fvw4BQUFyZw2IbzBLsSIWo7fy8jIyFwgqRj+hg0b2LhxI9u2bcNms/Hwww8D4HA4eOGFF3jkkUdwOp0899xziKKIJEksXbqUa6+9NiXG94coiUREL4p40biMz8nIyMiMBUk5fLPZzE9+8pPLtmdkZPDII48AMHnyZH7+858nc5ohE4q6QJBIN+bIcgoyMjIyF7gqK23b7OcAyM6QwzkyMjIyPVyVDr+5/QwAhTllY2yJjIyMzPjhqnT4nc4mAGzpI784LCMjI3OlcNU5/FgshifYAZKAViXr38vIyMj0kNSi7XhEoVBQMjUfjxhAIVx132cyMjIyw+aqdPiCJoRZkNMxZWRkZC7mqpwCB6MujBrZ4cvIyMhczNXp8GNOjNqMsTZDRkZGZlxx1Tl8SRLJM80lN618rE2RkZGRGVdcdTF8QVCwpPDr41YFT0ZGRmasuOpm+DIyMjIyfSM7fBkZGZkJguzwZWRkZCYIssOXkZGRmSDIDl9GRkZmgiA7fBkZGZkJguzwZWRkZCYIssOXkZGRmSAIkiRJY22EjIyMjMzIc9XO8H/4wx+OtQl9Its1NGS7hoZs19CYaHZdtQ5fRkZGRuZSZIcvIyMjM0FQPvbYY4+NtREjxZQpU8bahD6R7Roasl1DQ7ZraEwku+RFWxkZGZkJghzSkZGRkZkgyA5fRkZGZoJwRTdA2bdvH6+88gotLS38y7/8C6WlpX3ud/ToUX77298iiiJr1qxhw4YNAPh8PjZu3EhnZydZWVk89NBDmEympO1KZNzz58+zcePG3t87Ojq49957WbduHS+//DLvv/8+aWlpANx///0sWLBgVOwC+Id/+Ad0Oh0KhQKlUsmTTz45pONHwq6uri6ee+45XC4XgiBQUVHBbbfdBpDS69XfvdKDJEn89re/pbKyEq1Wyze/+c3eWOtgxybDYGPv2rWL119/HQCdTscDDzxAcXEx0P/nOVq2nTx5kp///OdkZ2cDsHjxYu65556Ejh1Ju9544w127doFgCiKNDc38+KLL2IymUbsmj3//PMcOXKE9PR0nnnmmcteH/H7S7qCaWpqklpaWqSf/vSnUl1dXZ/7xONx6Vvf+pbU1tYmRaNR6Xvf+57U1NQkSZIk/eEPf5Bee+01SZIk6bXXXpP+8Ic/pMSuoY4bj8elBx54QOro6JAkSZJeeukl6fXXX0+JLcOx65vf/KbkdruHffxI2OVwOKQzZ85IkiRJgUBAevDBB3s/x1Rdr4HulR4OHz4sPfHEE5IoitLp06elRx55JOFjR9Ku6upqyev1SpIkSUeOHOm1S5L6/zxHy7YTJ05IP/vZz4Z17EjadTEHDx6UHnvssd7fR+qanTx5Ujpz5oz08MMP9/n6SN9fV3RIp7CwkPz8/AH3qaurIzc3l5ycHFQqFcuWLePgwYMAHDx4kFWrVgGwatWq3u3JMtRxjx8/Tm5uLllZWSk5f6rsSvXxyYxrtVp7Zzp6vZ6CggIcDkdKzt/DQPdKD4cOHWLlypUIgsC0adPw+/04nc6Ejh1Ju6ZPn977VDR16lTsdntKzp0K20bi2FSPvWfPHpYvX56Scw9EeXn5gE/FI31/XdEhnURwOBxkZmb2/p6ZmUltbS0Abrcbq9UKdDsUj8eTknMOddy+brZ33nmHnTt3MmXKFL74xS+mJHQyFLueeOIJAG666SYqKiqGfPxI2QXd4a/6+nrKysp6t6Xieg10r1y8j81mu2Qfh8OR0LHDZahjb9u2jfnz51+yra/PczRtq6mp4fvf/z5Wq5UvfOELFBUVjZtrFg6HOXr0KF/96lcv2T5S12wgRvr+GvcO//HHH8flcl22/b777mPRokWDHi/1kXUqCMKI2jUUYrEYhw8f5rOf/Wzvtptvvrk3xvnSSy/x+9//nm9+85ujZtfjjz9ORkYGbrebf/7nfyY/P5/y8vKEjx8puwBCoRDPPPMMX/rSlzAYDEBy1+tiErlX+ttnpO6zRO3q4cSJE2zfvp1/+qd/6t02Ep/nUGwrKSnh+eefR6fTceTIEZ566imefYGsEAkAAALeSURBVPbZcXPNDh8+fMkTEozsNRuIkb6/xr3Df/TRR5M6PjMz85LHW7vd3jubTE9Px+l0YrVacTqdvYt+ydo1lHErKyspKSnBYrH0brv45zVr1vCv//qvo2pXRkZG7/6LFi2irq6O8vLyMb9esViMZ555hhUrVrB48eLe7clcr4sZ6F65eJ+urq7L9onFYoMeO1wSsQugoaGBF154gUceeQSz2dy7vb/Pc7Rs6/liBliwYAEvvvgiHo8n4fc1Unb1sGfPHq6//vpLto3kNRuIkb6/rugYfiKUlpbS2tpKR0cHsViMvXv3snDhQgAWLlzIjh07ANixY0dCTwyJMJRx+wrnOJ3O3p8PHDhAUVHRqNkVCoUIBoO9Px87doxJkyYlfPxI2SVJEr/85S8pKCjg9ttvv+S1VF2vge6Vi23duXMnkiRRU1ODwWDAarUmdOxwSWTsrq4unn76ab71rW9dsq410Oc5Wra5XK7eGWpdXR2iKGI2m8f8mgEEAgGqqqoueW2kr9lAjPT9dUVX2h44cIDf/OY3eDwejEYjxcXF/PjHP8bhcPTOdACOHDnC7373O0RRZPXq1dx9990AeL1eNm7cSFdXFzabjYcffjglsfL+xv2kXeFwmG984xv84he/uGQW9B//8R+cO3cOQRDIysria1/7WkpmPonY1d7eztNPPw1APB7n+uuvHxfXq7q6+v+1a8coEMJAFIYfCGm9ga03sswBrDyBZ7C08R6SytZOGy9iK7iFmGpdt1BhN//XB4ZkeJBhVJalkiTxX9l9/fLK+3rXK845SdvoaF1XNU2jcRxljFGe534d+KjPrnBWV13X6vvez3/3VcJP7/lUbW3byjmnKIpkjJG1VmmaHp59qi5J6rpOwzCoKAp/7s47q6pK0zRpnmfFcawsy7Qsi6/p7v766cAHAHzv70c6AIANgQ8AgSDwASAQBD4ABILAB4BAEPgAEAgCHwAC8QJ2lJTZkjsHngAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = np.vander(x, 6, increasing=True)\n",
    "Q, R = gram_schmidt_naive(A)\n",
    "plt.plot(x, Q);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Full vs Reduced QR\n",
    "\n",
    "Our algorithm computes the reduced QR factorization.  There also exists a full QR factorization, which includes an orthogonal basis for the null space of $A$.  This is sometimes useful, and is implicitly constructed by the Householder method (which we'll get to later).\n",
    "\n",
    "<img src=\"img/TB-ReducedQR.png\" width=\"40%\" /> <img src=\"img/TB-FullQR.png\" width=\"40%\" />"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Theorem: all full-rank $m\\times n$ matrices ($m \\ge n$) have a unique reduced $Q R$ factorization with $R_{j,j} > 0$.\n",
    "\n",
    "### Orthogonal polynomials\n",
    "\n",
    "We used `x = numpy.linspace(-1,1)` which uses $m=50$ points by default.  The number 50 is arbitrary and as we use more points, our columns become better approximations of continuous functions and the vector inner product becomes an integral (up to scaling):\n",
    "$$ \\frac 2 m \\sum_{i=1}^m p_i q_i \\approx \\int_{-1}^1 p(x) q(x) . $$\n",
    "\n",
    "When we orthogonalize the monomials using this inner product, we get the [Legendre Polynomials](https://en.wikipedia.org/wiki/Legendre_polynomials) (up to scaling).  These polynomials have important applications in physics and engineering, as well as playing an important role in approximation (which we will go into in more detail).\n",
    "\n",
    "### Solving equations using QR\n",
    "\n",
    "To solve\n",
    "$$ A x = b $$\n",
    "we can compute $A = QR$ and then\n",
    "$$ x = R^{-1} Q^T b . $$\n",
    "\n",
    "This also works for non-square systems!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "B = [[ 1.    -0.9    0.81  -0.729]\n",
      " [ 1.     0.1    0.01   0.001]\n",
      " [ 1.     0.5    0.25   0.125]\n",
      " [ 1.     0.8    0.64   0.512]] \n",
      "p = [ 3.35210084 -9.47605042 -1.74789916 12.98319328]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x1 = np.array([-0.9, 0.1, 0.5, 0.8]) # points where we know values\n",
    "y = np.array([1, 2.4, -0.2, 1.3])    # values at those points\n",
    "\n",
    "k = 4\n",
    "plt.plot(x1, y, 'o')\n",
    "B = np.vander(x1, k, increasing=True)  # Vandermonde matrix at the known points\n",
    "Q, R = gram_schmidt_naive(B)\n",
    "p = np.linalg.solve(R, Q.T @ y)      # Compute the polynomial coefficients\n",
    "plt.plot(x, np.vander(x,k,increasing=True) @ p)  # Plot the polynomial evaluated at all points\n",
    "print('B =', B, '\\np =', p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gram_schmidt_naive   7.33e-16 9.11e-09\n",
      "qr                   2.53e-15 2.36e-15\n"
     ]
    }
   ],
   "source": [
    "m = 20\n",
    "V = np.vander(np.linspace(-1,1,m), increasing=True)\n",
    "Q, R = gram_schmidt_naive(V)\n",
    "\n",
    "def qr_test(qr, V):\n",
    "    Q, R = qr(V)\n",
    "    m = len(Q.T)\n",
    "    print('{:20} {:.2e} {:.2e}'.format(\n",
    "        qr.__name__,\n",
    "        np.linalg.norm(Q @ R - V),\n",
    "        np.linalg.norm(Q.T @ Q - np.eye(m))))\n",
    "    \n",
    "qr_test(gram_schmidt_naive, V)\n",
    "qr_test(np.linalg.qr, V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gram_schmidt_classical 9.27e-16 1.42e+00\n"
     ]
    }
   ],
   "source": [
    "def gram_schmidt_classical(A):\n",
    "    Q = np.zeros_like(A)\n",
    "    R = np.zeros((len(A.T),len(A.T)))\n",
    "    for i in range(len(Q.T)):\n",
    "        v = A[:,i].copy()\n",
    "        R[:i,i] = Q[:,:i].T @ v\n",
    "        v -= Q[:,:i] @ R[:i,i]\n",
    "        R[i,i] = np.linalg.norm(v)\n",
    "        Q[:,i] = v / R[i,i]\n",
    "    return Q, R\n",
    "\n",
    "qr_test(gram_schmidt_classical, V)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Classical Gram-Schmidt is highly parallel, but unstable, as evidenced by the lack of orthogonality in $Q$.\n",
    "\n",
    "### Right-looking algorithms\n",
    "\n",
    "The implementations above have been \"left-looking\"; when working on column $i$, we compare it only to columns to the left (i.e., $j < i$).  We can reorder the algorithm to look to the right by projecting $q_i$ out of all columns $j > i$.  This algorithm is stable while being just as parallel as `gram_schmidt_classical`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gram_schmidt_modified 8.19e-16 1.33e-08\n"
     ]
    }
   ],
   "source": [
    "def gram_schmidt_modified(A):\n",
    "    Q = A.copy()\n",
    "    R = np.zeros((len(A.T), len(A.T)))\n",
    "    for i in range(len(Q.T)):\n",
    "        R[i,i] = np.linalg.norm(Q[:,i])\n",
    "        Q[:,i] /= R[i,i]\n",
    "        R[i,i+1:] = Q[:,i].T @ Q[:,i+1:]\n",
    "        Q[:,i+1:] -= np.outer(Q[:,i], R[i,i+1:])\n",
    "    return Q, R\n",
    "\n",
    "qr_test(gram_schmidt_modified, V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([4.47213595e+00, 2.71448357e+00, 1.46813359e+00, 7.74796385e-01,\n",
       "       4.02707125e-01, 2.06254652e-01, 1.03916396e-01, 5.13647014e-02,\n",
       "       2.48256542e-02, 1.16864792e-02, 5.33340139e-03, 2.34677736e-03,\n",
       "       9.88975643e-04, 3.95848907e-04, 1.48880592e-04, 5.18578998e-05,\n",
       "       1.63841809e-05, 4.54454652e-06, 1.04299294e-06, 1.71466983e-07])"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gram_schmidt_modified(V)[1].diagonal()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Stability\n",
    "\n",
    "Since QR factorization is unique (with positive diagonal of $R$), if we were to work in exact arithmetic, classical and modified Gram-Schmidt would produce the same result.  Note that modified Gram-Schmidt sequentially applies the projections into the orthogonal complement of each column $q_j$ of $Q$.  That is, given a vector $x$, we sequentially project $(I - q_j q_j^T) x$ for each column $j < i$.  This is equivalent to projecting all those columns at once due to\n",
    "\\begin{align}\n",
    "  (I - q_1 q_1^T) (I - q_0 q_0^T) x &= \\big(I - q_0 q_0^T - q_1 q_1^T + q_1 \\underbrace{q_1^T q_0}_{=0} q_0^T \\big) x \\\\\n",
    "  &= (I - q_0 q_0^T - q_1 q_1^T) x \\\\\n",
    "  &= (I - Q Q^T) x\n",
    "\\end{align}\n",
    "where $Q = [q_0 | q_1 ]$.  This identity can be applied recursively to convert modified Gram-Schmidt to classical, but the identity is not exact in finite precision arithmetic."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "norm(v) 1.4245900685395503\n",
      "r 1.7146698313477946e-07\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "v = V[:,-1]\n",
    "print('norm(v)', np.linalg.norm(v))\n",
    "print('r', R[-1,-1])\n",
    "plt.semilogy(np.abs(Q.T @ v), 'o')\n",
    "plt.title('Inner products of v with Q');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, -0.6666666666666666, -0.22222222222222224, -0.07407407407407408, -0.024691358024691364, -0.008230452674897124, -0.0027434842249657075, -0.0009144947416552361, -0.0003048315805517454, -0.00010161052685058181, -3.3870175616860605e-05, -1.1290058538953536e-05, -3.763352846317846e-06, -1.2544509487726156e-06, -4.181503162575385e-07, -1.3938343875251286e-07, -4.6461146250837626e-08, -1.548704875027921e-08, -5.162349583426404e-09, -1.7207831944754682e-09, -5.735943981584894e-10]\n",
      "seq   abs 5.9562e-22  rel 2.0768e-12\n",
      "block abs 5.3534e-17  rel 1.8666e-07\n",
      "numpy abs 7.4670e-17  rel 2.6036e-07\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXwAAAD4CAYAAADvsV2wAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjMsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy+AADFEAAAXO0lEQVR4nO3df2wbB93H8Y/TqO26lsw5dwkZqWizFpEJbbjuWgJd88PrSplEgqZI5VEn1I0pyaCUAWozKoQEeQiMKBEi0dBDFm1hlUqlhfIPbAqom1iYmjULjIbRpAJE1axu7K7N1h9bcvf8UWJiYre2c/5579df8/nO983F+9T5nr93LsuyLAEA8l5BpgsAAKQHgQ8ADkHgA4BDEPgA4BAEPgA4BIEPAA5RmOkCbubs2bNJbefxeDQ1NWVzNYtHXYmhrsRQV2Lyta6ysrKoy/mEDwAOQeADgEMQ+ADgEAQ+ADgEgQ8ADpG2b+lcvXpVP//5z1VYWKi77rpLW7duTcl+zNeOyRro17kLU5LbI1fDbhVsqU7JvgAglywq8Ht6ejQyMqKioiJ1dHSEl4+Ojqqvr0+maaqurk719fU6fvy4tmzZIp/Pp87OzpQEvvnaMVn93dL7164vCJ2X1d8tUyL0ATjeolo61dXVevLJJyOWmaap3t5ePfnkk+rs7NSrr76qM2fOKBgMyuPxXN9pQWo6SdZA/3/Cfs77164vBwCHW9Qn/MrKSgUCgYhlExMTKi0tVUlJiSSpqqpKw8PDMgxDwWBQH/3oR3WjS/APDg5qcHBQktTe3h7+RyIe5y7EGFS4MJXQ66RSYWFh1tQyH3UlhroSQ12JSVVdtvfwQ6GQDMMIPzYMQ+Pj4/rsZz+rZ555RiMjI9q4cWPM7f1+v/x+f/hxQtNmbo8UOh91eTyvM9f/V2hKKk5N/z9fJ/tShboSQ12Jyde6Yk3a2h740T69u1wuLV++XC0tLXbvLnI/Dbsje/iStHSZXA27b7ot/X8A+c72Zvpc62ZOMBiU2+22ezdRFWyplmv341LxasnlkopXy7X78bgCm/4/gHxn+yf8iooKTU5OKhAIqLi4WENDQ9q7d6/du4mpYEu1tKU68T+JQjHWjbUcAHLMogK/q6tLY2Njmp6eVlNTkxobG1VbW6s9e/aora1NpmmqpqZG5eXldtWbOsUx+v/F2XdCBwCSsajA37dvX9TlXq9XXq93MS+ddovp/wNALsj66+GnS8GWaplSyr+lAwCZQuDPM9f/B4B8xMXTAMAhCHwAcAhaOjZKx6QuACSLwLcJk7oAsh0tHZswqQsg2xH4dmFSF0CWI/DtEmsil0ldAFmCwLeJq2G3tHRZ5EImdQFkEU7a2oRJXQDZjsC3EZO6ALIZLR0AcAgCHwAcgpZOlmBKF0CqEfhZgCldAOlASycLMKULIB0I/GzAlC6ANCDwswFTugDSgMDPAkzpAkgHTtpmAaZ0AaQDgZ8lmNIFkGq0dADAIQh8AHAIAh8AHIIefh7gsgwA4kHg5zguywAgXrR0chyXZQAQLwI/13FZBgBxIvBzHZdlABAnAj/HcVkGAPHipG2O47IMAOJF4OcBLssAIB5pDfzjx49rZGREly5d0gMPPKC77747nbsHAEeLO/B7eno0MjKioqIidXR0hJePjo6qr69Ppmmqrq5O9fX1MV/j3nvv1b333qt3331X/f39BD4ApFHcgV9dXa0dO3aou7s7vMw0TfX29urgwYMyDEOtra3y+XwyTVOHDh2K2L65uVlFRUWSpBdeeEEPPPCATT8CFmNuSvfchSnJTf8fyGdxB35lZaUCgUDEsomJCZWWlqqkpESSVFVVpeHhYTU0NOjAgQMLXsOyLD3//PO65557tG7duqj7GRwc1ODgoCSpvb1dHk9yXy8sLCxMettUyqa6rrz8oi79olu6Nm9K9xfdunXVKt2yLTv+Qc6m4zUfdSWGuhKTqroW1cMPhUIyDCP82DAMjY+Px1z/N7/5jd58801dvnxZb7/9trZv375gHb/fL7/fH348NZXcAJHH40l621TKprpmn+v5T9jPuXZNl57r0Xt3bcxMUf8lm47XfNSVGOpKzGLrKisri7p8UYFvWdaCZS6XK+b6O3fu1M6dOxezS9iJKV3AURY1eGUYhoLBYPhxMBiU2+1edFFIE6Z0AUdZVOBXVFRocnJSgUBAMzMzGhoaks/ns6s2pBhTuoCzxN3S6erq0tjYmKanp9XU1KTGxkbV1tZqz549amtrk2maqqmpUXl5eSrrhY0ipnT5lg6Q9+IO/H379kVd7vV65fV6bSsI6TU3pZutJ68A2IeLpwGAQxD4AOAQXDwNSeNeukBuIfCRFO6lC+QeWjpICvfSBXIPgY/kMKUL5BwCH8lhShfIOQQ+ksKULpB7OGmLpHAvXSD3EPhIGvfSBXILLR0AcAgCHwAcgpYOMoZJXSC9CHxkBJO6QPrR0kFGMKkLpB+Bj8xgUhdIOwIfmcGkLpB2BD4ygkldIP04aYuMYFIXSD8CHxnDpC6QXrR0AMAhCHwAcAhaOshJTOkCiSPwkXOY0gWSQ0sHOYcpXSA5BD5yD1O6QFIIfOQepnSBpBD4yDlM6QLJ4aQtcg5TukByCHzkJKZ0gcTR0gEAhyDwAcAhCHwAcIi0Bv7Vq1e1f/9+nThxIp27BSKYrx3T7P5HNPvlz2t2/yMyXzuW6ZKAtIjrpG1PT49GRkZUVFSkjo6O8PLR0VH19fXJNE3V1dWpvr7+hq9z9OhRfepTn1pcxcAicFkGOFlcgV9dXa0dO3aou7s7vMw0TfX29urgwYMyDEOtra3y+XwyTVOHDh2K2L65uVn//Oc/9ZGPfEQffPCBvT8BkIAbXpaBwEeeiyvwKysrFQgEIpZNTEyotLRUJSUlkqSqqioNDw+roaFBBw4cWPAaJ0+e1LVr13TmzBktXbpUn/zkJ1VQsLCjNDg4qMHBQUlSe3u7PJ7kpicLCwuT3jaVqCsxdtd17kKMyy9cmEpoP045XnahrsSkqq6kv4cfCoVkGEb4sWEYGh8fj7n+rl27JEnHjh3TqlWrooa9JPn9fvn9/vDjqankro/i8XiS3jaVqCsxttfl9kih81GXJ7Ifxxwvm1BXYhZbV1lZWdTlSZ+0tSxrwTKXy3XT7aqrq7Vx48ZkdwssCpdlgJMl/QnfMAwFg8Hw42AwKLfbbUtRQKpwWQY4WdKBX1FRocnJSQUCARUXF2toaEh79+61szYgJbgsA5wqrsDv6urS2NiYpqen1dTUpMbGRtXW1mrPnj1qa2uTaZqqqalReXl5qusFACQprsDft29f1OVer1der9fWggAAqcHVMoEEzN08/dyFKclN/x+5hcAH4sSULnIdF08D4sTN05HrCHwgXtw8HTmOwAfixc3TkeMIfCBOTOki13HSFohTxJQu39JBDiLwgQTMTelm60W3gBuhpQMADkHgA4BD0NIB0mhuUpcrdSITCHwgTZjURabR0gHShEldZBqBD6QLk7rIMAIfSBcmdZFhBD6QJkzqItM4aQukCffTRaYR+EAacT9dZBItHQBwCAIfAByClg6QI5jSxWIR+EAOYEoXdqClA+QApnRhBwIfyAVM6cIGBD6QC5jShQ0IfCAHMKULO3DSFsgBTOnCDgQ+kCOY0sVi0dIBAIcg8AHAIWjpAA7AlC4kAh/Ie0zpYk7aAt80TR0+fFhXrlzRunXrVF1dna5dA452wyldAt9R4gr8np4ejYyMqKioSB0dHeHlo6Oj6uvrk2maqqurU319fczXeP311xUKhbRy5UoZhrH4ygHEhyld/FtcgV9dXa0dO3aou7s7vMw0TfX29urgwYMyDEOtra3y+XwyTVOHDh2K2L65uVlnz57Vhg0bdP/996ujo0Of+MQn7P1JAERX7JFC56Mvh6PEFfiVlZUKBAIRyyYmJlRaWqqSkhJJUlVVlYaHh9XQ0KADBw4seI3i4mIVFl7fXUEBXw4C0sXVsDuyhy8xpetQSffwQ6FQRGvGMAyNj4/HXH/z5s165pln9NZbb+njH/94zPUGBwc1ODgoSWpvb5fHk9ynkMLCwqS3TSXqSgx1JSZqXQ8+pCurVund55+WORVQged2rfyfJt2y7YHM1pUFnFZX0oFvWdaCZS6XK+b6y5YtU3Nz801f1+/3y+/3hx9PTSXXZ/R4PElvm0rUlRjqSkzMuu7aKNf//p+W/Pvhe5LeS2P9OXe8MmyxdZWVlUVdnnRvxTAMBYPB8ONgMCi3253sywEAUizpwK+oqNDk5KQCgYBmZmY0NDQkn89nZ20AABvF1dLp6urS2NiYpqen1dTUpMbGRtXW1mrPnj1qa2uTaZqqqalReXl5qusFACQprsDft29f1OVer1der9fWggBkFy7LkD+4tAKAmLgsQ37hC/EAYuLm6fmFwAcQG5dlyCsEPoDYuHl6XiHwAcTEzdPzCydtAcTEzdPzC4EP4Ia4eXr+oKUDAA5B4AOAQ9DSAZAyc1O65y5MSW76/5lG4ANICaZ0sw8tHQApwZRu9iHwAaQGU7pZh8AHkBpM6WYdAh9ASjClm304aQsgJSKmdPmWTlYg8AGkzNyUbrbeLNxpaOkAgEMQ+ADgELR0AGQt7qdrLwIfQFZiUtd+tHQAZCUmde1H4APITkzq2o7AB5CdmNS1HYEPICsxqWs/TtoCyErcT9d+BD6ArMX9dO1FSwcAHILABwCHoKUDIC8xpbsQgQ8g7zClGx0tHQB5hynd6Ah8APmHKd2o0tbSmZqaUm9vr1atWqWysjLV19ena9cAnKbYI4XOR1/uYHF9wu/p6dGjjz6qb3zjGxHLR0dH9bWvfU1f/epX9atf/eqGr3H27Fl5vV61tLTozJkzyVcMADfBlG50cX3Cr66u1o4dO9Td3R1eZpqment7dfDgQRmGodbWVvl8PpmmqUOHDkVs39zcrLVr12pgYEBDQ0O677777P0pAGAepnSjc1mWZcWzYiAQ0A9/+EN1dHRIkk6dOqUjR47o29/+tiRpYGBAktTQ0BB1+1//+te68847VVlZqY6OjgV/LcwZHBzU4OCgJKm9vV3vv/9+Yj/RvxUWFmpmZiapbVOJuhJDXYmhrsTka11Lly6N/rrJvmAoFJJhGOHHhmFofHw85vr33HOPjhw5oj/84Q9avXp1zPX8fr/8fn/4cbI3Ps7WmyZTV2KoKzHUlZh8rausrCzq8qQDP9ofBi6XK+b6a9asifmpHgCQekl/LdMwDAWDwfDjYDAot9ttS1EAAPslHfgVFRWanJxUIBDQzMyMhoaG5PP57KwNADLCfO2YZvc/otkvf16z+x+R+dqxTJdki7haOl1dXRobG9P09LSamprU2Nio2tpa7dmzR21tbTJNUzU1NSovL091vQCQUvl8WYa4An/fvn1Rl3u9Xnm9XlsLAoBMuuFlGXI88Lm0AgDMl8eXZSDwAWC+PL55OoEPAPPk82UZuB4+AMyTz5dlIPAB4L/k683TaekAgEMQ+ADgELR0AMBG2XzzdAIfAGyS7VO6tHQAwCbZfvN0Ah8A7JLlU7oEPgDYJcundAl8ALBJtk/pctIWAGyS7VO6BD4A2Cibp3Rp6QCAQxD4AOAQtHQAIEvMTemeuzAlue3v/xP4AJAF0jGlS0sHALJAOqZ0CXwAyAZpmNIl8AEgG6RhSpfAB4AskI4pXU7aAkAWiJjS5Vs6AJDf5qZ0PR6Ppqbsv8ImLR0AcAgCHwAcgsAHAIcg8AHAIQh8AHAIl2VZVqaLAACkXt5+wj9w4ECmS4iKuhJDXYmhrsQ4ra68DXwAQCQCHwAcYsl3v/vd72a6iFRZt25dpkuIiroSQ12Joa7EOKkuTtoCgEPQ0gEAhyDwAcAhcv5qmaOjo+rr65Npmqqrq1N9fX3E85Zlqa+vT2+88YaWLVumlpaWlPfspqam1N3drXfeeUcul0t+v187d+6MWOfkyZP60Y9+pNtvv12StHnzZj300EMprUuSHn/8cS1fvlwFBQVasmSJ2tvbI57PxPE6e/asOjs7w48DgYAaGxv1uc99LrwsXcerp6dHIyMjKioqUkdHhyTp3XffVWdnp86fP6/Vq1fr61//ulauXLlg25u9F+2uq7+/XydOnFBhYaFKSkrU0tKiW2+9dcG2N/ud213XL3/5S/3ud7/Thz70IUnSrl275PV6F2yb7uPV2dmps2fPSpIuX76sFStW6KmnnlqwbSqPV6xsSNt7zMphs7Oz1le+8hXr7bfftj744APrm9/8pvWvf/0rYp0TJ05YbW1tlmma1t/+9jertbU15XWFQiHr9OnTlmVZ1uXLl629e/cuqOsvf/mL9YMf/CDltfy3lpYW6+LFizGfz8Txmm92dtZ69NFHrUAgELE8Xcfr5MmT1unTp60nnngivKy/v98aGBiwLMuyBgYGrP7+/gXbxfNetLuu0dFRa2ZmJlxjtLos6+a/c7vrOnz4sHX06NEbbpeJ4zXfs88+ax05ciTqc6k8XrGyIV3vsZxu6UxMTKi0tFQlJSUqLCxUVVWVhoeHI9Z5/fXXdd9998nlcmnDhg167733dOHChZTW5Xa7w5+Kb7nlFt1xxx0KhUIp3addMnG85nvzzTdVWlqq1atXp22f81VWVi74ZDU8PKxt27ZJkrZt27bgPSbF9160u667775bS5YskSRt2LAhI++xaHXFIxPHa45lWfrjH/+oT3/607btL16xsiFd77GcbumEQiEZhhF+bBiGxsfHF6zj8Xgi1gmFQnK73WmpMRAI6O9//7vuvPPOBc+dOnVK3/rWt+R2u7V7926Vl5enpaa2tjZJ0v333y+/3x/xXKaP16uvvhrzf8RMHa+LFy+Gf363261Lly4tWCee92Iq/f73v1dVVVXM52/0O0+FF198Ua+88orWrVunhx9+eEH4ZvJ4/fWvf1VRUZE+/OEPx1wnHcdrfjak6z2W04FvRflGqcvlSnidVLl69ao6Ojr0pS99SStWrIh4bu3aterp6dHy5cs1MjKip556Sj/5yU9SXtP3vvc9FRcX6+LFi/r+97+vsrIyVVZWhp/P5PGamZnRiRMn9MUvfnHBc5k6XvHK5HF74YUXtGTJEm3dujXq8zf7ndtt+/bt4fMrhw8f1nPPPaeWlpaIdTJ5vG70oUJKz/G6UTbEYscxy+mWjmEYCgaD4cfBYHDBJ1HDMCJuFRZtnVSYmZlRR0eHtm7dqs2bNy94fsWKFVq+fLkkyev1anZ2Nuq/6nYrLi6WJBUVFWnTpk2amJiIeD5Tx0uS3njjDa1du1a33Xbbgucydbyk68dqrq114cKF8MnI+eJ5L6bCsWPHdOLECe3duzfm//w3+53b7bbbblNBQYEKCgpUV1en06dPL1gnU8drdnZWx48fv+FfQ6k+XtGyIV3vsZwO/IqKCk1OTioQCGhmZkZDQ0Py+XwR6/h8Pr3yyiuyLEunTp3SihUrUv7GsixLTz/9tO644w49+OCDUdd55513wv9iT0xMyDRNrVq1KqV1Xb16VVeuXAn/95///GetWbMmYp1MHK85N/rklYnjNcfn8+nll1+WJL388svatGnTgnXieS/abXR0VEePHtX+/fu1bNmyqOvE8zu32/xzPsePH4/aesvE8ZKunyMqKyuLaI3Ml+rjFSsb0vUey/lJ25GRET377LMyTVM1NTX6whe+oJdeeknS9T8tLctSb2+v/vSnP2np0qVqaWlRRUVFSmt666239J3vfEdr1qwJf+ratWtX+JPz9u3b9dvf/lYvvfSSlixZoqVLl+rhhx/Wxz72sZTWde7cOf34xz+WdP2Tzmc+85msOF6SdO3aNTU3N+unP/1p+E/c+XWl63h1dXVpbGxM09PTKioqUmNjozZt2qTOzk5NTU3J4/HoiSee0MqVKxUKhfSzn/1Mra2tkqK/F1NZ18DAgGZmZsL98fXr1+uxxx6LqCvW7zyVdZ08eVL/+Mc/5HK5tHr1aj322GNyu90ZP161tbXq7u7W+vXrtX379vC66TxesbJh/fr1aXmP5XzgAwDik9MtHQBA/Ah8AHAIAh8AHILABwCHIPABwCEIfABwCAIfABzi/wE8Z11RaNgVLAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def test_sum(n):\n",
    "    def gen(first, n, factor=2/3):\n",
    "        l = [first]\n",
    "        ifactor = 1-factor\n",
    "        for i in range(n):\n",
    "            l.append(-first * factor * ifactor**i)\n",
    "        return l, first * ifactor**n\n",
    "    def sum_seq(numbers):\n",
    "        s = 0\n",
    "        for a in numbers:\n",
    "            s += a\n",
    "        return s\n",
    "    def sum_block(numbers):\n",
    "        s = 0\n",
    "        for a in numbers[1:]:\n",
    "            s += a\n",
    "        return numbers[0] + s\n",
    "    numbers, exact = gen(1, n)\n",
    "    print(numbers)\n",
    "    plt.semilogy(np.abs(numbers), 'o')\n",
    "    seq_err = sum_seq(numbers) - exact\n",
    "    block_err = sum_block(numbers) - exact\n",
    "    numpy_err = np.sum(numbers) - exact\n",
    "    print('seq   abs {:.4e}  rel {:.4e}'.format(seq_err, seq_err/exact))\n",
    "    print('block abs {:.4e}  rel {:.4e}'.format(block_err, block_err/exact))\n",
    "    print('numpy abs {:.4e}  rel {:.4e}'.format(numpy_err, numpy_err/exact))\n",
    "    \n",
    "test_sum(20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Householder triangularization\n",
    "\n",
    "Gram-Schmidt methods perform triangular transformations to build an orthogonal matrix.  As we have seen, $X = QR$ is satisfied accurately, but $Q$ may not be orthogonal when $X$ is ill-conditioned.  Householder triangularization instead applies a sequence of orthogonal transformations to build a triangular matrix.\n",
    "\n",
    "$$ \\underbrace{Q_{n-1} \\dotsb Q_0}_{Q^T} A = R $$\n",
    "\n",
    "The structure of the algorithm is\n",
    "\n",
    "$$ \\underbrace{\\begin{bmatrix} * & * & * \\\\ * & * & * \\\\ * & * & * \\\\ * & * & * \\\\ * & * & * \\\\ \\end{bmatrix}}_{A} \\to\n",
    "\\underbrace{\\begin{bmatrix} * & * & * \\\\ 0 & * & * \\\\ 0 & * & * \\\\ 0 & * & * \\\\ 0 & * & * \\\\ \\end{bmatrix}}_{Q_0 A} \\to\n",
    "\\underbrace{\\begin{bmatrix} * & * & * \\\\ 0 & * & * \\\\ 0 & 0 & * \\\\ 0 & 0 & * \\\\ 0 & 0 & * \\\\ \\end{bmatrix}}_{Q_1 Q_0 A} \\to\n",
    "\\underbrace{\\begin{bmatrix} * & * & * \\\\ 0 & * & * \\\\ 0 & 0 & * \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ \\end{bmatrix}}_{Q_2 Q_1 Q_0 A}\n",
    "$$\n",
    "\n",
    "where the elementary orthogonal matrices $Q_i$ chosen to introduce zeros below the diagonal in the $i$th column of $R$.\n",
    "Each of these transformations will have the form\n",
    "$$Q_i = \\begin{bmatrix} I_i & 0 \\\\ 0 & F \\end{bmatrix}$$\n",
    "where $F$ is a \"reflection\" that achieves\n",
    "$$ F x = \\begin{bmatrix} \\lVert x \\rVert \\\\ 0 \\\\ 0 \\\\ \\vdots \\end{bmatrix} $$\n",
    "where $x$ is the column of $R$ from the diagonal down.\n",
    "This transformation is a reflection across a plane with normal $v = Fx - x = \\lVert x \\rVert e_1 - x$.\n",
    "\n",
    "![Householder Reflector (Trefethen and Bau, 1999)](img/TB-Householder.png)\n",
    "\n",
    "The reflection, as depected above by Trefethen and Bau (1999) can be written $F = I - 2 \\frac{v v^T}{v^T v}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1.43896426, 0.43227761, 0.99110883, 0.88292239],\n",
       "       [0.43227761, 1.64752629, 0.70840329, 0.96992974],\n",
       "       [0.99110883, 0.70840329, 0.98398427, 0.79118118],\n",
       "       [0.88292239, 0.96992974, 0.79118118, 1.76253247]])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.random.rand(4, 4)\n",
    "A = A + A.T # Random symmetric matrix\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0., 0., 0., 0.],\n",
       "       [0., 0., 0., 0.],\n",
       "       [0., 0., 0., 0.],\n",
       "       [0., 0., 0., 0.]])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.T - A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.    ,  0.    ,  0.    ,  0.    ],\n",
       "       [ 0.    , -0.3097, -0.71  , -0.6325],\n",
       "       [ 0.    , -0.71  ,  0.6151, -0.3429],\n",
       "       [ 0.    , -0.6325, -0.3429,  0.6946]])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.linalg import block_diag\n",
    "np.set_printoptions(precision=4)\n",
    "\n",
    "def reflector(v):\n",
    "    return np.eye(len(v)) - 2*np.outer(v, v)\n",
    "\n",
    "v = A[1:,0].copy()\n",
    "v[0] += np.linalg.norm(v)\n",
    "v = v / np.linalg.norm(v)\n",
    "F = reflector(v)\n",
    "Q_0 = block_diag(np.eye(1), F)\n",
    "Q_0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.     0.4323 0.9911 0.8829]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([[ 1.    ,  0.    ,  0.    ,  0.    ],\n",
       "       [ 0.    , -0.3097, -0.71  , -0.6325],\n",
       "       [ 0.    , -0.71  ,  0.6151, -0.3429],\n",
       "       [ 0.    , -0.6325, -0.3429,  0.6946]])"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = A[:,0].copy()\n",
    "v[0] = 0\n",
    "print(v)\n",
    "v[1] += np.linalg.norm(v[1:])\n",
    "v /= np.linalg.norm(v)\n",
    "reflector(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.4390e+00,  4.3228e-01,  9.9111e-01,  8.8292e-01],\n",
       "       [-1.3960e+00, -1.6266e+00, -1.4184e+00, -1.9768e+00],\n",
       "       [ 9.4328e-17, -1.0665e+00, -1.6897e-01, -8.0630e-01],\n",
       "       [ 2.0001e-16, -6.1126e-01, -2.3592e-01,  3.3943e-01]])"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q_0 @ A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.4390e+00, -1.3960e+00,  9.4328e-17,  2.0001e-16],\n",
       "       [ 4.3228e-01, -1.6266e+00, -1.0665e+00, -6.1126e-01],\n",
       "       [ 9.9111e-01, -1.4184e+00, -1.6897e-01, -2.3592e-01],\n",
       "       [ 8.8292e-01, -1.9768e+00, -8.0630e-01,  3.3943e-01]])"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A @ Q_0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "qr_householder1      1.88e-15 3.17e-16\n"
     ]
    }
   ],
   "source": [
    "def householder_Q_times(V, x):\n",
    "    \"\"\"Apply orthogonal matrix represented as list of Householder reflectors\"\"\"\n",
    "    y = x.copy()\n",
    "    for i in reversed(range(len(V))):\n",
    "        y[i:] -= 2 * V[i] * V[i].dot(y[i:])\n",
    "    return y\n",
    "\n",
    "def qr_householder1(A):\n",
    "    \"Compute QR factorization using naive Householder reflection\"\n",
    "    m, n = A.shape\n",
    "    R = A.copy()\n",
    "    V = []\n",
    "    for i in range(n):\n",
    "        v = R[i:,i].copy()\n",
    "        v[0] -= np.linalg.norm(v)\n",
    "        v = v/np.linalg.norm(v)     # Normalized reflector plane\n",
    "        R[i:,i:] -= 2 * np.outer(v, v @ R[i:,i:])\n",
    "        V.append(v)                    # Storing reflectors is equivalent to storing orthogonal matrix\n",
    "    Q = np.eye(m, n)\n",
    "    for i in range(n):\n",
    "        Q[:,i] = householder_Q_times(V, Q[:,i])\n",
    "    return Q, np.triu(R[:n,:])\n",
    "\n",
    "qr_test(qr_householder1, np.array([[1.,2],[3,4],[5,6]]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "qr_householder1      3.11e-15 2.70e-15\n",
      "qr                   2.53e-15 2.36e-15\n",
      "gram_schmidt_modified 8.19e-16 1.33e-08\n"
     ]
    }
   ],
   "source": [
    "qr_test(qr_householder1, V)\n",
    "qr_test(np.linalg.qr, V)\n",
    "qr_test(gram_schmidt_modified, V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "272240823.70513624"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.cond(V)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Choice of two projections\n",
    "\n",
    "It turns out our implementation has a nasty deficiency."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "qr_householder1      nan nan\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "<ipython-input-36-7c16fa44a4d4>:16: RuntimeWarning: invalid value encountered in true_divide\n",
      "  v = v/np.linalg.norm(v)     # Normalized reflector plane\n"
     ]
    }
   ],
   "source": [
    "qr_test(qr_householder1, np.eye(1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "<ipython-input-36-7c16fa44a4d4>:16: RuntimeWarning: invalid value encountered in true_divide\n",
      "  v = v/np.linalg.norm(v)     # Normalized reflector plane\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "qr_householder1      nan nan\n"
     ]
    }
   ],
   "source": [
    "qr_test(qr_householder1, np.eye(3,2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Inside `qr_householder1`, we have the lines\n",
    "```\n",
    "        x = R[i:,i]\n",
    "        v = -x\n",
    "        v[0] += numpy.linalg.norm(x)\n",
    "        v = v/numpy.linalg.norm(v)     # Normalized reflector plane\n",
    "```\n",
    "What happens when $$x = \\begin{bmatrix}1 \\\\ 0 \\end{bmatrix}$$\n",
    "(i.e., the column of $R$ is already upper triangular)?\n",
    "\n",
    "We are trying to define a reflector plane (via its normal vector) from the zero vector,\n",
    "$$v = \\lVert x \\rVert e_0 - x .$$\n",
    "When we try to normalize this vector, we divide zero by zero and the algorithm breaks down (`nan`).  Maybe we just need to test for this special case and \"skip ahead\" when no reflection is needed?  And if so, how would we define $Q$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "qr_householder1      2.20e-09 4.44e-16\n",
      "(array([[ 1.0000e+00, -2.2204e-08],\n",
      "       [ 2.2204e-08,  1.0000e+00]]), array([[1., 1.],\n",
      "       [0., 1.]]))\n"
     ]
    }
   ],
   "source": [
    "qr_test(qr_householder1, np.array([[1.,1], [2e-8,1]]))\n",
    "print(qr_householder1(np.array([[1.,1], [2e-8,1]])))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The error $QR - A$ is still $10^{-8}$ for this very well-conditioned matrix so something else must be at play here.\n",
    "\n",
    "![Choosing the better of two Householder reflectors (Trefethen and Bau, 1999).](img/TB-Householder2.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "qr_householder2      0.00e+00 0.00e+00\n",
      "qr_householder2      0.00e+00 0.00e+00\n",
      "(array([[-1.e+00,  1.e-08],\n",
      "       [-1.e-08, -1.e+00]]), array([[-1., -1.],\n",
      "       [ 0., -1.]]))\n",
      "qr_householder2      5.20e-15 3.49e-15\n"
     ]
    }
   ],
   "source": [
    "def qr_householder2(A):\n",
    "    \"Compute QR factorization using Householder reflection\"\n",
    "    m, n = A.shape\n",
    "    R = A.copy()\n",
    "    V = []\n",
    "    for i in range(n):\n",
    "        v = R[i:,i].copy()\n",
    "        v[0] += np.sign(v[0]) * np.linalg.norm(v) # Choose the further of the two reflections\n",
    "        v = v/np.linalg.norm(v)     # Normalized reflector plane\n",
    "        R[i:,i:] -= np.outer(v, 2 * (v.T @ R[i:,i:]))\n",
    "        V.append(v)                    # Storing reflectors is equivalent to storing orthogonal matrix\n",
    "    Q = np.eye(m, n)\n",
    "    for i in range(n):\n",
    "        Q[:,i] = householder_Q_times(V, Q[:,i])\n",
    "    return Q, np.triu(R[:n,:])\n",
    "\n",
    "qr_test(qr_householder2, np.eye(3,2))\n",
    "qr_test(qr_householder2, np.array([[1.,1], [1e-8,1]]))\n",
    "print(qr_householder2(np.array([[1.,1], [1e-8,1]])))\n",
    "\n",
    "qr_test(qr_householder2, V)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now have a usable implementation of Householder QR.  There are some further concerns for factoring rank-deficient matrices.  We will visit the concept of pivoting later, in the context of LU and Cholesky factorization.\n",
    "\n",
    "## Condition number of a matrix\n",
    "\n",
    "We may have informally referred to a matrix as \"ill-conditioned\" when the columns are nearly linearly dependent, but let's make this concept for precise.  Recall the definition of (relative) condition number from the Rootfinding notes,\n",
    "\n",
    "$$ \\kappa = \\max_{\\delta x} \\frac{|\\delta f|/|f|}{|\\delta x|/|x|} . $$\n",
    "\n",
    "We understood this definition for scalar problems, but it also makes sense when the inputs and/or outputs are vectors (or matrices, etc.) and absolute value is replaced by vector (or matrix) norms.  Let's consider the case of matrix-vector multiplication, for which $f(x) = A x$.\n",
    "\n",
    "$$ \\kappa(A) = \\max_{\\delta x} \\frac{\\lVert A (x+\\delta x) - A x \\rVert/\\lVert A x \\rVert}{\\lVert \\delta x\\rVert/\\lVert x \\rVert}\n",
    "= \\max_{\\delta x} \\frac{\\lVert A \\delta x \\rVert}{\\lVert \\delta x \\rVert} \\, \\frac{\\lVert x \\rVert}{\\lVert A x \\rVert} = \\lVert A \\rVert \\frac{\\lVert x \\rVert}{\\lVert A x \\rVert} . $$\n",
    "\n",
    "There are two problems here:\n",
    "\n",
    "* I wrote $\\kappa(A)$ but my formula depends on $x$.\n",
    "* What is that $\\lVert A \\rVert$ beastie?\n",
    "\n",
    "### Stack push: Matrix norms\n",
    "\n",
    "Vector norms are built into the linear space (and defined in term of the inner product).  Matrix norms are *induced* by vector norms, according to\n",
    "\n",
    "$$ \\lVert A \\rVert = \\max_{x \\ne 0} \\frac{\\lVert A x \\rVert}{\\lVert x \\rVert} . $$\n",
    "\n",
    "* This equation makes sense for non-square matrices -- the vector norms of the input and output spaces may differ.\n",
    "* Due to linearity, all that matters is direction of $x$, so it could equivalently be written\n",
    "\n",
    "$$ \\lVert A \\rVert = \\max_{\\lVert x \\rVert = 1} \\lVert A x \\rVert . $$\n",
    "\n",
    "### Stack pop\n",
    "\n",
    "Now we understand the formula for condition number, but it depends on $x$.  Consider the matrix\n",
    "\n",
    "$$ A = \\begin{bmatrix} 1 & 0 \\\\ 0 & 0 \\end{bmatrix} . $$\n",
    "\n",
    "* What is the norm of this matrix?\n",
    "* What is the condition number when $x = [1,0]^T$?\n",
    "* What is the condition number when $x = [0,1]^T$?\n",
    "\n",
    "The condition number of matrix-vector multiplication depends on the vector.  The condition number of the matrix is the worst case (maximum) of the condition number for any vector, i.e.,\n",
    "\n",
    "$$ \\kappa(A) = \\max_{x \\ne 0} \\lVert A \\rVert \\frac{\\lVert x \\rVert}{\\lVert A x \\rVert} .$$\n",
    "\n",
    "If $A$ is invertible, then we can rephrase as\n",
    "\n",
    "$$ \\kappa(A) = \\max_{x \\ne 0} \\lVert A \\rVert \\frac{\\lVert A^{-1} (A x) \\rVert}{\\lVert A x \\rVert} =\n",
    "\\max_{A x \\ne 0} \\lVert A \\rVert \\frac{\\lVert A^{-1} (A x) \\rVert}{\\lVert A x \\rVert} = \\lVert A \\rVert \\lVert A^{-1} \\rVert . $$\n",
    "\n",
    "Evidently multiplying by a matrix is just as ill-conditioned of an operation as solving a linear system using that matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(4, 4)"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.array([1,2,3])\n",
    "A.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6.280369834735101e-16\n",
      "9.402219163119398e-15\n",
      "1.0021569437765227e-14\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([1., 2., 3., 4.])"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def R_solve(R, b):\n",
    "    \"\"\"Solve Rx = b using back substitution.\"\"\"\n",
    "    x = b.copy()\n",
    "    m = len(b)\n",
    "    for i in reversed(range(m)):\n",
    "        x[i] -= R[i,i+1:].dot(x[i+1:])\n",
    "        x[i] /= R[i,i]\n",
    "    return x\n",
    "\n",
    "Q, R = np.linalg.qr(A)\n",
    "x = np.array([1,2,3,4])\n",
    "bfull = A @ x\n",
    "breduced = Q.T @ bfull\n",
    "print(np.linalg.norm(R_solve(R, breduced)\n",
    "                     - np.linalg.solve(R, breduced)))\n",
    "print(np.linalg.norm(R_solve(R, breduced)\n",
    "                     - x))\n",
    "print(np.linalg.norm(np.linalg.solve(R, breduced)\n",
    "                     - x))\n",
    "R_solve(R, breduced)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Cost of Householder factorization\n",
    "\n",
    "The dominant cost comes from the line\n",
    "```Python\n",
    "    R[i:,i:] -= 2 * numpy.outer(v, v.dot(R[i:,i:]))\n",
    "```\n",
    "were `R[i:,i:]` is an $(m-i)\\times(n-i)$ matrix.\n",
    "This line performs $2(m-i)(n-i)$ operations in `v.dot(R[i:,i:])`, another $(m-i)(n-i)$ in the \"outer\" product and again in subtraction.  As written, multiplication by 2 would be another $(m-i)(n-i)$ operations, but is only $m-i$ operations if we rewrite as\n",
    "```Python\n",
    "    w = 2*v\n",
    "    R[i:,i:] -= numpy.outer(w, v.dot(R[i:,i:]))\n",
    "```\n",
    "in which case the leading order cost is $4(m-i)(n-i)$.  To compute the total cost, we need to sum over all columns $i$,\n",
    "$$\\begin{split} \\sum_{i=1}^n 4(m-i)(n-i) &= 4 \\Big[ \\sum_{i=1}^n (m-n)(n-i) + \\sum_{i=1}^n (n-i)^2 \\Big] \\\\\n",
    "&= 4 (m-n) \\sum_{i=1}^n i + 4 \\sum_{i=1}^n i^2 \\\\\n",
    "&\\approx 2 (m-n) n^2 + 4 n^3/3 \\\\\n",
    "&= 2 m n^2 - \\frac 2 3 n^3 .\n",
    "\\end{split}$$\n",
    "Recall that Gram-Schmidt QR cost $2 m n^2$, so Householder costs about the same when $m \\gg n$ and is markedly less expensive when $m \\approx n$.\n",
    "\n",
    "### Backward Stability of Housholder"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "# Forward error\n",
      "Q error 5.656854249492381\n",
      "R error 1.0007584740439666\n",
      "# Backward error\n",
      "Q2.T @ Q2 - I 1.3765843992557918e-12\n",
      "Q2*R2 - A 1.807714603200187e-16\n",
      "Q3*R3 - A 4.548215523185648e-05\n"
     ]
    }
   ],
   "source": [
    "def qr_test_backward(qr, n):\n",
    "    from numpy.linalg import norm\n",
    "    from numpy.random import randn\n",
    "    R = np.triu(randn(n,n))\n",
    "    Q, _ = np.linalg.qr(randn(n,n))\n",
    "    A = Q @ R\n",
    "    Q2, R2 = qr(A)\n",
    "    print('# Forward error')\n",
    "    print('Q error', norm(Q2 - Q))\n",
    "    print('R error', norm(R2 - R) / norm(R))\n",
    "    print('# Backward error')\n",
    "    A2 = Q2 @ R2\n",
    "    print('Q2.T @ Q2 - I', norm(Q2.T @ Q2 - np.eye(n)))\n",
    "    print('Q2*R2 - A', norm(A2 - A) / norm(A))\n",
    "    Q3, R3 = Q + 1e-5*randn(n,n), R + 1e-5*np.triu(randn(n,n))\n",
    "    A3 = Q3 @ R3\n",
    "    print('Q3*R3 - A', norm(A3 - A) / norm(A))\n",
    "    \n",
    "qr_test_backward(gram_schmidt_modified, 20)\n",
    "#qr_test_backward(np.linalg.qr, 50)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Least squares and the normal equations\n",
    "\n",
    "A **least squares problem** takes the form: given an $m\\times n$ matrix $A$ ($m \\ge n$), find $x$ such that\n",
    "$$ \\lVert Ax - b \\rVert $$\n",
    "is minimized.  If $A$ is square and full rank, then this minimizer will satisfy $A x - b = 0$, but that is not the case in general because $b$ is not in the range of $A$.\n",
    "The residual $A x - b$ must be orthogonal to the range of $A$.\n",
    "\n",
    "* Is this the same as saying $A^T (A x - b) = 0$?\n",
    "* If $QR = A$, is it the same as $Q^T (A x - b) = 0$?\n",
    "\n",
    "In the quiz, we showed that $QQ^T$ is an orthogonal projector onto the range of $Q$.  If $QR = A$,\n",
    "$$ QQ^T (A x - b) = QQ^T(Q R x - b) = Q (Q^T Q) R x - QQ^T b = QR x - QQ^T b = A x - QQ^T b . $$\n",
    "So if $b$ is in the range of $A$, we can solve $A x = b$.  If not, we need only *orthogonally* project $b$ into the range of $A$.\n",
    "\n",
    "### Solution by QR (Householder)\n",
    "\n",
    "Solve $R x = Q^T b$.\n",
    "\n",
    "* QR factorization costs $2 m n^2 - \\frac 2 3 n^3$ operations and is done once per matrix $A$.\n",
    "* Computing $Q^T b$ costs $4 (m-n)n + 2 n^2 = 4 mn - 2n^2$ (using the elementary reflectors, which are stable and lower storage than naive storage of $Q$).\n",
    "* Solving with $R$ costs $n^2$ operations.  Total cost per right hand side is thus $4 m n - n^2$.\n",
    "\n",
    "This method is stable and accurate.\n",
    "\n",
    "### Solution by Cholesky\n",
    "\n",
    "The mathematically equivalent form $(A^T A) x = A^T b$ are called the **normal equations**.  The solution process involves factoring the symmetric and positive definite $n\\times n$ matrix $A^T A$.\n",
    "\n",
    "* Computing $A^T A$ costs $m n^2$ flops, exploiting symmetry.\n",
    "* Factoring $A^T A = R^T R$ costs $\\frac 1 3 n^3$ flops.  The total factorization cost is thus $m n^2 + \\frac 1 3 n^3$.\n",
    "* Computing $A^T b$ costs $2 m n$.\n",
    "* Solving with $R^T$ costs $n^2$.\n",
    "* Solving with $R$ costs $n^2$.  Total cost per right hand side is thus $2 m n + 2 n^2$.\n",
    "\n",
    "The product $A^T A$ is ill-conditioned: $\\kappa(A^T A) = \\kappa(A)^2$ and can reduce the accuracy of a least squares solution.\n",
    "\n",
    "### Solution by Singular Value Decomposition\n",
    "\n",
    "Next, we will discuss a factorization\n",
    "$$ U \\Sigma V^T = A $$\n",
    "where $U$ and $V$ have orthonormal columns and $\\Sigma$ is diagonal with nonnegative entries.\n",
    "The entries of $\\Sigma$ are called **singular values** and this decomposition is the **singular value decomposition** (SVD).\n",
    "It may remind you of an eigenvalue decomposition $X \\Lambda X^{-1} = A$, but\n",
    "* the SVD exists for all matrices (including non-square and deficient matrices)\n",
    "* $U,V$ have orthogonal columns (while $X$ can be arbitrarily ill-conditioned).\n",
    "Indeed, if a matrix is symmetric and positive definite (all positive eigenvalues), then $U=V$ and $\\Sigma = \\Lambda$.\n",
    "Computing an SVD requires a somewhat complicated iterative algorithm, but a crude estimate of the cost is $2 m n^2 + 11 n^3$.  Note that this is similar to the cost of $QR$ when $m \\gg n$, but much more expensive for square matrices.\n",
    "Solving with the SVD involves\n",
    "* Compute $U^T b$ at a cost of $2 m n$.\n",
    "* Solve with the diagonal $n\\times n$ matrix $\\Sigma$ at a cost of $n$.\n",
    "* Apply $V$ at a cost of $2 n^2$.  The total cost per right hand side is thus $2 m n + 2n^2$.\n",
    "\n",
    "#### Example: rank-deficient problem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0,  1],\n",
       "       [ 1,  3],\n",
       "       [ 2,  5],\n",
       "       [-1, -1]])"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = np.array([[0, 1], [1, 3], [2, 5], [-1, -1]])\n",
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1,  0,  1],\n",
       "       [ 1,  1,  3],\n",
       "       [ 1,  2,  5],\n",
       "       [ 1, -1, -1]])"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([0*X[:,0]+1, X[:,0], X[:,1]]).T\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.5641983688596716e+16\n",
      "0.8452609621100825 [ 0.2071  0.4142 -0.7071]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([ 0.0000e+00,  2.2204e-16,  4.4409e-16, -1.6653e-16])"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = X[:,0] - X[:,1] + .5\n",
    "Q, R = np.linalg.qr(A)\n",
    "print(np.linalg.cond(R))\n",
    "c_qr = R_solve(R, Q.T @ b)\n",
    "print(np.linalg.norm(c_qr), c_qr)\n",
    "A @ c_qr - b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.4564354645876387 [-0.0833 -0.1667 -0.4167]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([-1.1102e-16, -4.4409e-16, -1.3323e-15,  4.4409e-16])"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U, S, V = np.linalg.svd(A, full_matrices=False)\n",
    "c_svd = V.T @ ((U.T @ b) / S)\n",
    "print(np.linalg.norm(c_svd), c_svd)\n",
    "A @ c_svd - b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Pseudoinverse\n",
    "\n",
    "An alternative is to explicitly form the $n\\times m$ pseudoinverse $A^\\dagger = R^{-1} Q^T$ (at a cost of $mn^2$) at which point each right hand side costs $2 mn$.  Why might we do this?\n",
    "\n",
    "* Lots of right hand sides\n",
    "* Real-time solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "cond(A) =  90847309.61919233\n",
      "cond(R^{-1} Q^T A) = 1.0000000105693043\n",
      "cond(L^{-T} L^{-1} A^T A) = 1.3904370775408743\n"
     ]
    }
   ],
   "source": [
    "# Test accuracy of solver for an ill-conditioned square matrix\n",
    "\n",
    "x = np.linspace(-1,1,19)\n",
    "A = np.vander(x)\n",
    "print('cond(A) = ', np.linalg.cond(A))\n",
    "Q, R = np.linalg.qr(A)\n",
    "print('cond(R^{-1} Q^T A) =', np.linalg.cond(np.linalg.solve(R, Q.T.dot(A))))\n",
    "L = np.linalg.cholesky(A.T @ A)\n",
    "print('cond(L^{-T} L^{-1} A^T A) =', \n",
    "      np.linalg.cond(np.linalg.solve(L.T, np.linalg.solve(L, A.T @ A))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The [Singular Value Decomposition](https://en.wikipedia.org/wiki/Singular_value_decomposition)\n",
    "\n",
    "The SVD is the decomposition\n",
    "$$ U \\Sigma V^T = A $$\n",
    "where $U$ and $V$ have orthonormal columns and $\\Sigma$ is diagonal and nonnegative.\n",
    "Evidenly an arbitrary matrix consists of\n",
    "1. Orthogonal matrix -- rotation and reflection\n",
    "2. Diagonal scaling (transforms a sphere into an ellipsoid aligned to the coordinate axes)\n",
    "3. Orthogonal matrix -- rotation and reflection\n",
    "\n",
    "It is typical to order the singular values in descending order.\n",
    "\n",
    "The inverse has the same behavior:\n",
    "\n",
    "$$ A^{-1} = V \\Sigma^{-1} U^T $$\n",
    "\n",
    "The matrix norm is the largest singular value\n",
    "\n",
    "$$ \\lVert A \\rVert = \\sigma_{\\max} . $$\n",
    "\n",
    "The largest value in $\\Sigma^{-1}$ is $\\sigma_\\min^{-1}$, so\n",
    "\n",
    "$$ \\lVert A^{-1} \\rVert = \\sigma_\\min^{-1} . $$\n",
    "\n",
    "We showed previously that\n",
    "\n",
    "$$\\kappa(A) = \\lVert A \\rVert \\, \\lVert A^{-1} \\rVert $$\n",
    "\n",
    "so now we can also write\n",
    "\n",
    "$$ \\kappa(A) = \\frac{\\sigma_\\max}{\\sigma_\\min} . $$\n",
    "\n",
    "The SVD is a crucial tool in statistics and dimensionality reduction, often under names like\n",
    "\n",
    "* [Principal Component Analysis (PCA)](https://en.wikipedia.org/wiki/Principal_component_analysis)\n",
    "* Proper Orthogonal Decomposition (POD)\n",
    "* Karhunen-Loeve Expansion\n",
    "\n",
    "In this context, one computes an SVD of a data matrix and finds that the spectrum $\\Sigma$ decays rapidly and only the $k \\ll n$ components larger than some threshold are \"important\".  The dense matrix can thus be approximated as\n",
    "\n",
    "$$ \\hat U \\hat \\Sigma \\hat V^T $$\n",
    "\n",
    "by keeping only the first (largest) $k$ columns.  This reduces the storage from $mn$ entries to $mk + kn$ entries."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tinkering in class"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "63.82728259639295\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "m = 6\n",
    "x = np.linspace(-1, 1, m)\n",
    "#x = np.cos(np.linspace(0, np.pi, m))\n",
    "def f(x):\n",
    "    return 1.0*(x > 0) + (x < 0.5)\n",
    "A = np.vander(x)\n",
    "Q, R = np.linalg.qr(A)\n",
    "p = np.linalg.solve(R, Q.T @ f(x))\n",
    "y = np.linspace(-1,1,50)\n",
    "g = np.vander(y, m) @ p\n",
    "\n",
    "plt.plot(x, f(x), 'o')\n",
    "plt.plot(y, g)\n",
    "plt.plot(y, f(y))\n",
    "print(np.linalg.cond(A))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7206778.417326136"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.cond(np.vander(np.linspace(-1,1,100),20))"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}