{
 "metadata": {
  "name": ""
 },
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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from __future__ import division\n",
      "\n",
      "import numpy as np\n",
      "import numpy.linalg as la\n",
      "\n",
      "import matplotlib.pyplot as pt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.random.seed(40)\n",
      "\n",
      "# Generate matrix with eigenvalues 1...25\n",
      "n = 25\n",
      "eigvals = np.linspace(1., n, n)\n",
      "eigvecs = np.random.randn(n, n)\n",
      "print eigvals\n",
      "\n",
      "A = la.solve(eigvecs, np.dot(np.diag(eigvals), eigvecs))\n",
      "print la.eig(A)[0]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[  1.   2.   3.   4.   5.   6.   7.   8.   9.  10.  11.  12.  13.  14.  15.\n",
        "  16.  17.  18.  19.  20.  21.  22.  23.  24.  25.]\n",
        "[ 25.  24.  23.   1.   2.   3.  22.   4.  21.  20.   5.   6.   7.  19.  18.\n",
        "   8.   9.  17.  16.  10.  11.  12.  15.  14.  13.]\n"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Initialize Arnoldi iteration\n",
      "\n",
      "# Set up Q and H\n",
      "Q = np.zeros((n, n))\n",
      "H = np.zeros((n, n))\n",
      "\n",
      "k = 0\n",
      "\n",
      "# Pick a starting vector, normalize it\n",
      "x0 = np.random.randn(n)\n",
      "x0 = x0/la.norm(x0)\n",
      "\n",
      "# Poke it into the first column of Q\n",
      "Q[:, k] = x0\n",
      "\n",
      "del x0\n",
      "\n",
      "# ritz_values = []"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 82
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Carry out one iteration of Arnoldi iteration\n",
      "# Run this cell in-place (Ctrl-Enter) until H is filled.\n",
      "\n",
      "print k\n",
      "\n",
      "u = np.dot(A, Q[:, k])\n",
      "\n",
      "# Carry out Gram-Schmidt on u against Q\n",
      "for j in range(k+1):\n",
      "    qj = Q[:, j]\n",
      "    H[j,k] = np.dot(qj, u)\n",
      "    u = u - H[j,k]*qj\n",
      "\n",
      "if k+1 < n:\n",
      "    H[k+1, k] = la.norm(u)\n",
      "    Q[:, k+1] = u/H[k+1, k]\n",
      "\n",
      "k += 1\n",
      "\n",
      "pt.spy(H)\n",
      "\n",
      "# ritz_values.append(la.eig(H)[0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "24\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPwAAAD7CAYAAABOrvnfAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nQwgeMITgAUMIHjCE4AFD/gKflH2/NE4ELQAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0xa57d810>"
       ]
      }
     ],
     "prompt_number": 107
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Check that Q^T A Q = H\n",
      "\n",
      "la.norm(\n",
      "        np.dot(np.dot(Q.T, A), Q)\n",
      "        - H) / la.norm(A)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 108,
       "text": [
        "6.9230906536862802e-06"
       ]
      }
     ],
     "prompt_number": 108
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Check that Q is orthogonal\n",
      "\n",
      "la.norm(\n",
      "        np.dot(Q.T, Q) - np.eye(n))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 109,
       "text": [
        "1.3525100273749116e-05"
       ]
      }
     ],
     "prompt_number": 109
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### Plot convergence of Ritz values"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Enable the Ritz value collection above to make this work.\n",
      "\n",
      "for i, rv in enumerate(ritz_values):\n",
      "    plot([i] * len(rv), rv, \"x\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stderr",
       "text": [
        "/usr/lib/pymodules/python2.7/numpy/core/numeric.py:460: ComplexWarning: Casting complex values to real discards the imaginary part\n",
        "  return array(a, dtype, copy=False, order=order)\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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wORByAABwORByAABwORByAABwORByAABwORByAABwORByAABwOf8f\nSUMc7mhho+8AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x95d9d50>"
       ]
      }
     ],
     "prompt_number": 110
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}