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     "source": [
      "Crit\u00e9rio de Routh-Hurwitz\n",
      "====\n",
      "\n",
      "O teorema de Routh pode ser colocado numa forma mais c\u00f4moda usando a *matriz de Hurwitz*\n",
      "\n",
      "Operador de Hurwitz\n",
      "----\n",
      "S\u00e3o dadas duas sequ\u00eancias $\\alpha: \\mathbb{N} \\to \\mathbb{R}$ e $\\beta: \\mathbb{N} \\to \\mathbb{R}$. Definimos o operador de Hurwitz como $\\alpha * \\beta = \\gamma:\\mathbb{N} \\to \\mathbb{R}$ tal que\n",
      ":$$ \\gamma(i) = \\frac{-1}{\\beta(1)}\\begin{vmatrix} \\alpha(1) & \\alpha(i+1)\\\\ \\beta(1) & \\beta(i+1) \\end{vmatrix}$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#operador de Hurwitz para duas sequencias finitas\n",
      "def hurwitz(a,b):\n",
      "    if len(a) <= len(b):\n",
      "        for i in range(len(a),len(b)):\n",
      "            a.append(0)\n",
      "        c=[0*u for u in b]\n",
      "    else:\n",
      "        for i in range(len(b),len(a)):\n",
      "                b.append(0)\n",
      "        c=[0*u for u in a]\n",
      "    if b[0] == 0 :\n",
      "        return c\n",
      "    for i in range(len(c) -1):\n",
      "        c[i] = (-1.0/b[0])*(a[0]*b[i+1] - a[i+1]*b[0])\n",
      "    return c    \n",
      "    \n",
      "    "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print (hurwitz([1,1],[1,0]))\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[1.0, 0]\n"
       ]
      }
     ],
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    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "A sequ\u00eancia de Hurwitz\n",
      "----\n",
      "Se $\\gamma_1$ e $\\gamma_2$ s\u00e3o duas sequ\u00eancias de n\u00fameros reais, tal que $\\gamma_2(1) \\neq 0$. Definimos por recorr\u00eancia\n",
      "$$ \\gamma_{k+1} = \\gamma_{k-1}*\\gamma_{k}$$\n",
      "at\u00e9 que $\\gamma_k(1)=0$.  A matriz de Hurwitz \u00e9 a matriz em que cada linha \u00e9 a sequ\u00eancia $\\gamma_k$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Matriz de Hurwitz a partir de duas sequencias finitas\n",
      "def MatrizHurwitz(a,b):\n",
      "    H=[a,b]\n",
      "    while H[len(H)-1][0] != 0 :\n",
      "        b2=hurwitz(H[len(H)-2],H[len(H)-1])\n",
      "        H.append(b2)\n",
      "    return H    \n",
      "    "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#teste\n",
      "a=[1,2,6]\n",
      "b=[4,1]\n",
      "H = MatrizHurwitz(a,b)\n",
      "print (H)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[1, 2, 6], [4, 1, 0], [1.75, 6.0, 0], [-12.714285714285714, -0.0, 0], [5.999999999999999, 0.0, 0], [-0.0, 0.0, 0]]\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Matriz de Hurwitz de um polin\u00f4mio $p(\\lambda)$\n",
      "Seja $p(\\lambda)= \\lambda^n +a_1\\lambda^{n-1} + \\cdots + a_n$ um polin\u00f4mio, construimos duas sequ\u00eancias\n",
      "$$ \\begin{gather}\n",
      "\\gamma_1 = 1 , a_2, a_4,\\dots, a_{2k} \\text{ com } a_{2k}=0 \\text{ se } 2k>n \\\\\n",
      "\\gamma_2 = a_1, a_3, \\dots, a_{2k+1} \\text{ com } a_{2k+1}=0 \\text{ se } 2k+1>n\n",
      "\\end{gather}$$\n",
      "e depois $\\gamma_{i+2} = \\gamma_i*\\gamma_{i+1}$ at\u00e9 $k$ tal que $\\gamma_{k+1}(0)=0$.\n",
      "A matriz $H = (h_{ij})$ com $h_{ij}=\\gamma_i(j)$ \u00e9 chamada matriz de Hurwitz.\n",
      "\n",
      "## Teorema de Routh-Hurwitz \n",
      "O polin\u00f4mio $p(\\lambda)$ \u00e9 est\u00e1vel se, e somente se, $\\gamma_{k}(0) > 0$ para todo $k \\in \\{1, \\dots, n+1\\}$ "
     ]
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     "input": [],
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