d8660091/gist:b0b7ac78eaf56dcb6d6c

## gistfile1.txt
{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy\n",
    "from IPython.display import Image\n",
    "# Enviroment initialization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###Here, we try to test the [theorem 3](http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1047025&tag=1). \n",
    "If $(E,A)$ is regular, then for each $Q > 0$ there exist $P > 0$ which satisfied $A^T P E + E^T P A + E^T Q ^E =0 $\n",
    "\n",
    "Take the following example (used by the paper)\n",
    "\n",
    "$$\n",
    "  E= \\begin{bmatrix}\n",
    "    1 & 0 & 0\\\\\n",
    "    0 & 0 & 0\\\\\n",
    "    0 & 1 & 0\n",
    "  \\end{bmatrix} ,\n",
    "  A= \\begin{bmatrix}\n",
    "    0 & 0 & 0 \\\\\n",
    "    0 & 0 & 1 \\\\\n",
    "    -1 & -1 & 0\n",
    "  \\end{bmatrix}\n",
    "  $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will find that, the equation can't be solved by the solver(also can't be solved by the solver in MATLAB), which indicates that both the solver in Python and MATLAB are wrong or the theorem. In fact, both program use the same routine, which are SLICOT routines SB03MD and SG03AD for Lyapunov equations and SB04MD (SLICOT) and ZTRSYL (LAPACK) for Sylvester equations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "The pencil A - lambda * E has a                                 degenerate pair of eigenvalues. That is,                                 lambda_i = lambda_j for some i and j, where                                 lambda_i and lambda_j are eigenvalues of                                 A - lambda * E. Hence, the equation is                                 singular;  perturbed values were                                 used to solve the equation (but the matrices                                 A and E are unchanged)",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-60-3056efdbc153>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[0;32m      2\u001b[0m \u001b[0mA\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mnumpy\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mmatrix\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'0 0 0;0 0 1;-1 -1 0'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m;\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      3\u001b[0m \u001b[0mQ\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mnumpy\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mmatrix\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'1 0 0;0 1 0;0 0 1'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m;\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 4\u001b[1;33m \u001b[0mX\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mlyap\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mA\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtranspose\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mE\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtranspose\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0mQ\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0mE\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mNone\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mE\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtranspose\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;32m/usr/lib/python2.7/site-packages/control/mateqn.pyc\u001b[0m in \u001b[0;36mlyap\u001b[1;34m(A, Q, C, E)\u001b[0m\n\u001b[0;32m    223\u001b[0m                                 A and E are unchanged)\")\n\u001b[0;32m    224\u001b[0m                 \u001b[0me\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0minfo\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mve\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0minfo\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 225\u001b[1;33m             \u001b[1;32mraise\u001b[0m \u001b[0me\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    226\u001b[0m     \u001b[1;31m# Invalid set of input parameters\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    227\u001b[0m     \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mValueError\u001b[0m: The pencil A - lambda * E has a                                 degenerate pair of eigenvalues. That is,                                 lambda_i = lambda_j for some i and j, where                                 lambda_i and lambda_j are eigenvalues of                                 A - lambda * E. Hence, the equation is                                 singular;  perturbed values were                                 used to solve the equation (but the matrices                                 A and E are unchanged)"
     ]
    }
   ],
   "source": [
    "E=numpy.matrix('1 0 0;0 0 0;0 1 0');\n",
    "A=numpy.matrix('0 0 0;0 0 1;-1 -1 0');\n",
    "Q=numpy.matrix('1 0 0;0 1 0;0 0 1');\n",
    "X = lyap(A.transpose(),E.transpose()*Q*E,None,E.transpose())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The solver is working for nonsingular E A matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.14702869 -0.20543033  0.13268443]\n",
      " [-0.20543033  0.76280738 -0.21465164]\n",
      " [ 0.13268443 -0.21465164  0.1767418 ]]\n"
     ]
    }
   ],
   "source": [
    "E=numpy.matrix('1 0 1;0 3 1;5 1 0');\n",
    "A=numpy.matrix('1 2 3;0 0 1;-1 -1 0');\n",
    "Q=numpy.matrix('1 0 0;0 1 0;0 0 1')\n",
    "X = lyap(A.transpose(),Q,None,E.transpose())\n",
    "print(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import matlab.engine\n",
    "eng = matlab.engine.start_matlab()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The vimception extension is already loaded. To reload it, use:\n",
      "  %reload_ext vimception\n"
     ]
    },
    {
     "ename": "MatlabExecutionError",
     "evalue": "\n  File /usr/local/MATLAB/R2014b/toolbox/control/control/lyap.m, line 35, in lyap\nThe input arguments of the \"lyap\" command must be numeric arrays.\n",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mMatlabExecutionError\u001b[0m                      Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-62-5a01fb6dafa4>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[0;32m      7\u001b[0m \u001b[0mQ\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mmatlab\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mdouble\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mQ\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtolist\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m;\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      8\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 9\u001b[1;33m \u001b[0meng\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mlyap\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mA\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mQ\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mE\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m     10\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/usr/lib/python2.7/site-packages/matlab/engine/matlabengine.pyc\u001b[0m in \u001b[0;36m__call__\u001b[1;34m(self, *args, **kwargs)\u001b[0m\n\u001b[0;32m     78\u001b[0m         \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     79\u001b[0m             return FutureResult(self._engine(), future, nargs, _stdout,\n\u001b[1;32m---> 80\u001b[1;33m                                 _stderr).result()\n\u001b[0m\u001b[0;32m     81\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     82\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/usr/lib/python2.7/site-packages/matlab/engine/futureresult.pyc\u001b[0m in \u001b[0;36mresult\u001b[1;34m(self, timeout)\u001b[0m\n\u001b[0;32m    107\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    108\u001b[0m             self._result = pythonengine.getFEvalResult(\n\u001b[1;32m--> 109\u001b[1;33m                 self._future,self._nargout, None, out=self._out, err=self._err)\n\u001b[0m\u001b[0;32m    110\u001b[0m             \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_retrieved\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mTrue\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    111\u001b[0m             \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_result\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mMatlabExecutionError\u001b[0m: \n  File /usr/local/MATLAB/R2014b/toolbox/control/control/lyap.m, line 35, in lyap\nThe input arguments of the \"lyap\" command must be numeric arrays.\n"
     ]
    }
   ],
   "source": [
    "%load_ext vimception\n",
    "E=numpy.matrix('1 0 0;0 0 0;0 1 0');\n",
    "E=matlab.double(E.tolist())\n",
    "A=numpy.matrix('0 0 0;0 0 1;-1 -1 0');\n",
    "A=matlab.double(A.tolist());\n",
    "Q=numpy.matrix('1 0 0;0 1 0;0 0 1');\n",
    "Q=matlab.double(Q.tolist());\n",
    "\n",
    "eng.lyap(A,Q,[],E)\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      ">> OS.pole\r\n",
      "\r\n",
      "ans =\r\n",
      "\r\n",
      "    10.2179e+027 +             i\r\n",
      "    33.8744e+018 +  8.4829e+018i\r\n",
      "    33.8744e+018 -  8.4829e+018i\r\n",
      "    -2.0000e+003 +             i\r\n",
      "    -2.0000e+003 +             i\r\n",
      "  -500.5010e+006 +  1.9364e+009i\r\n",
      "  -500.5010e+006 -  1.9364e+009i\r\n",
      "    -1.0010e+009 +             i\r\n",
      "   -28.3428e+018 +             i\r\n",
      "\r\n"
     ]
    }
   ],
   "source": [
    "%%matlab\n",
    "format shortEng;\n",
    "OS.pole"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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      ],
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image('/home/xudeng/pictures/earth.jpg')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
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    "version": 3
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   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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