Trefethen & Bau lecture notes in MATLAB

TB-Lecture-01.ipynb

TB-Lecture-02.ipynb

TB-Lecture-03.ipynb

TB-Lecture-04.ipynb

TB-Lecture-05.ipynb

TB-Lecture-06.ipynb

TB-Lecture-07.ipynb

TB-Lecture-08.ipynb

{
 "cells": [
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   "cell_type": "markdown",
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   "source": [
    "# Lecture 8: Gram-Schmidt orthogonalization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Modified Gram-Schmidt\n",
    "\n",
    "The classical Gram-Schmidt process is numerically unstable, as will be seen in the next lecture. A mathematically equivalent formulation, which we call MGS, turns out to be a lot better. \n",
    "\n",
    "In classical GS the key step is to project a new column of $A$ onto the orthogonal complement of the span of all the previous $q_j$s. We express this projection as $P_j=I - Q_jQ_j^*$, where $Q_j=\\bigl[ q_1 \\; q_2\\; \\cdots \\; q_j\\bigr]$. For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "\n",
      "    1.0000\n",
      "    1.0000\n",
      "    1.0000\n",
      "    0.0000\n",
      "    0.0000\n",
      "    0.0000\n"
     ]
    }
   ],
   "source": [
    "[Qj,~] = qr(rand(6,3),0);   % get a random Q with 3 ON columns\n",
    "Pj = eye(6) - Qj*Qj';\n",
    "svd(Pj)       % all 1 or 0, as required for an orthogonal projector"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's intuitively clear that we get the same result by subtracting off projections onto each column, one at a time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "\n",
      "   1.2505e-16\n"
     ]
    }
   ],
   "source": [
    "Mj = eye(6) - Qj(:,1)*Qj(:,1)' - Qj(:,2)*Qj(:,2)' - Qj(:,3)*Qj(:,3)';\n",
    "norm(Pj-Mj)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also factor this into a product of orthogonal projectors:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "\n",
      "   2.3849e-16\n"
     ]
    }
   ],
   "source": [
    "Gj = eye(6);\n",
    "for k = 1:3\n",
    "    Gj = Gj*(eye(6)-Qj(:,k)*Qj(:,k)');\n",
    "end\n",
    "norm(Gj-Pj)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This works because the \"cross-terms\" you get in the products have inner products between different $q_k$ and thus are zero. In MGS, we apply each of these factors, as soon at it is found, to \\emph{all} the columns of $A$ at once. By the time we are ready to choose a new column from $A$, it is already orthgonalized against the entire ON basis set found so far."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Operation counts\n",
    "\n",
    "It's standard in numerical analysis to compare the performance of algorithms by counting the number of floating point operations, or flops, that are performed as a function of the input size. By either form of QR factorization, the number of flops is $\\sim 2mn^2$ in asymptotic notation. Let's see if we observe the implicit proportionality with respect to $n^2$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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BGgEmIZBgZ0oajXR+WcTIP40OnO7XfoQ0AsxCIMHOoh6dp02jqNnzSCNAWgQS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\nDAoAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n_ = 50:50:500;\n",
    "time_ = [];\n",
    "for k = 1:length(n_)\n",
    "    n = n_(k);\n",
    "    A = rand(1200,n);\n",
    "    Q = zeros(1200,n);  R = zeros(600,600); \n",
    "    \n",
    "    tic\n",
    "    R(1,1) = norm(A(:,1));\n",
    "    Q(:,1) = A(:,1)/R(1,1);\n",
    "    for j = 2:n\n",
    "        R(1:j-1,j) = Q(:,1:j-1)'*A(:,j);\n",
    "        v = A(:,j) - Q(:,1:j-1)*R(1:j-1,j);\n",
    "        R(j,j) = norm(v);\n",
    "        Q(:,j) = v/R(j,j);\n",
    "    end\n",
    "    time_(k) = toc;\n",
    "end\n",
    "loglog(n_,time_,'-o',n_,(n_/500).^2,'--')\n",
    "axis tight, xlabel('n'), ylabel('elapsed time')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Flops aren't everything. On massively parallel computers, for example, the time needed for memory access and communication often competes with or dwarfs the floating point arithmetic. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Matlab",
   "language": "matlab",
   "name": "matlab"
  },
  "language_info": {
   "codemirror_mode": "octave",
   "file_extension": ".m",
   "help_links": [
    {
     "text": "MetaKernel Magics",
     "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
    }
   ],
   "mimetype": "text/x-octave",
   "name": "matlab",
   "version": "0.11.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}

TB-Lecture-09.ipynb

TB-Lecture-10.ipynb

TB-Lecture-11.ipynb

TB-Lecture-12.ipynb