cartpole.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, theta = sympy.symbols('x, theta', cls=sympy.Function)\n",
    "m_p, m_c, r, g, t = sympy.symbols('m_p, m_c, r, g, t')\n",
    "\n",
    "x_d = sympy.Derivative(x(t), t)\n",
    "theta_d = sympy.Derivative(theta(t), t)\n",
    "\n",
    "x_dd = sympy.Derivative(x(t), (t, 2))\n",
    "theta_dd = sympy.Derivative(theta(t), (t, 2))\n",
    "\n",
    "f_x, f_theta = sympy.symbols('f_x, f_theta')\n",
    "\n",
    "sin = sympy.sin\n",
    "cos = sympy.cos\n",
    "\n",
    "q_c = sympy.Matrix([x(t), 0])\n",
    "q_p = sympy.Matrix([\n",
    "    x(t) + r * sin(theta(t)),\n",
    "    - r * cos(theta(t))\n",
    "])\n",
    "\n",
    "q_c_d = sympy.Derivative(q_c, t).doit()\n",
    "q_p_d = sympy.Derivative(q_p, t).doit()\n",
    "\n",
    "K = (1/2)*m_p*(q_p_d[0]**2 + q_p_d[1]**2) + (1/2)*m_c*(q_c_d[0]**2 + q_c_d[1]**2)\n",
    "U = -m_p*g*r*cos(theta(t))\n",
    "\n",
    "L = K - U\n",
    "\n",
    "x_eq = sympy.Derivative(sympy.Derivative(L, x_d).doit(), t).doit() - sympy.Derivative(L, x(t)).doit() - f_x\n",
    "theta_eq = sympy.Derivative(sympy.Derivative(L, theta_d).doit(), t).doit() - sympy.Derivative(L,theta(t)).doit() - f_theta\n",
    "\n",
    "q_dd = sympy.solvers.solve([x_eq, theta_eq], [x_dd, theta_dd])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{r \\left(f_{x} + m_{p} r \\sin{\\left(\\theta{\\left(t \\right)} \\right)} \\left(\\frac{d}{d t} \\theta{\\left(t \\right)}\\right)^{2}\\right) - \\left(f_{\\theta} - g m_{p} r \\sin{\\left(\\theta{\\left(t \\right)} \\right)}\\right) \\cos{\\left(\\theta{\\left(t \\right)} \\right)}}{r \\left(m_{c} + m_{p} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)}\\right)}$"
      ],
      "text/plain": [
       "(r*(f_x + m_p*r*sin(theta(t))*Derivative(theta(t), t)**2) - (f_theta - g*m_p*r*sin(theta(t)))*cos(theta(t)))/(r*(m_c + m_p*sin(theta(t))**2))"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q_dd[x_dd]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{m_{p} r \\left(f_{x} + m_{p} r \\sin{\\left(\\theta{\\left(t \\right)} \\right)} \\left(\\frac{d}{d t} \\theta{\\left(t \\right)}\\right)^{2}\\right) \\cos{\\left(\\theta{\\left(t \\right)} \\right)} - \\left(f_{\\theta} - g m_{p} r \\sin{\\left(\\theta{\\left(t \\right)} \\right)}\\right) \\left(m_{c} + m_{p}\\right)}{m_{p} r^{2} \\left(m_{c} + m_{p} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)}\\right)}$"
      ],
      "text/plain": [
       "-(m_p*r*(f_x + m_p*r*sin(theta(t))*Derivative(theta(t), t)**2)*cos(theta(t)) - (f_theta - g*m_p*r*sin(theta(t)))*(m_c + m_p))/(m_p*r**2*(m_c + m_p*sin(theta(t))**2))"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    " q_dd[theta_dd]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "z1, z2, z3, z4 = sympy.symbols('z1, z2, z3, z4')\n",
    "u1, u2 = sympy.symbols('u1, u2')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{r \\left(m_{p} r z_{4}^{2} \\sin{\\left(z_{2} \\right)} + u_{1}\\right) - \\left(- g m_{p} r \\sin{\\left(z_{2} \\right)} + u_{2}\\right) \\cos{\\left(z_{2} \\right)}}{r \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)}$"
      ],
      "text/plain": [
       "(r*(m_p*r*z4**2*sin(z2) + u1) - (-g*m_p*r*sin(z2) + u2)*cos(z2))/(r*(m_c + m_p*sin(z2)**2))"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q_dd[x_dd].subs([\n",
    "    (x_d, z3),\n",
    "    (theta_d, z4),\n",
    "    (x(t), z1),\n",
    "    (theta(t), z2),\n",
    "    (f_x, u1),\n",
    "    (f_theta, u2)\n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{m_{p} r \\left(m_{p} r z_{4}^{2} \\sin{\\left(z_{2} \\right)} + u_{1}\\right) \\cos{\\left(z_{2} \\right)} - \\left(m_{c} + m_{p}\\right) \\left(- g m_{p} r \\sin{\\left(z_{2} \\right)} + u_{2}\\right)}{m_{p} r^{2} \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)}$"
      ],
      "text/plain": [
       "-(m_p*r*(m_p*r*z4**2*sin(z2) + u1)*cos(z2) - (m_c + m_p)*(-g*m_p*r*sin(z2) + u2))/(m_p*r**2*(m_c + m_p*sin(z2)**2))"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q_dd[theta_dd].subs([\n",
    "    (x_d, z3),\n",
    "    (theta_d, z4),\n",
    "    (x(t), z1),\n",
    "    (theta(t), z2),\n",
    "    (f_x, u1),\n",
    "    (f_theta, u2)\n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "f_zu = sympy.Matrix([\n",
    "    z3,\n",
    "    z4,\n",
    "    q_dd[x_dd].subs([\n",
    "        (x_d, z3),\n",
    "        (theta_d, z4),\n",
    "        (x(t), z1),\n",
    "        (theta(t), z2),\n",
    "        (f_x, u1),\n",
    "        (f_theta, u2)\n",
    "    ]),\n",
    "    q_dd[theta_dd].subs([\n",
    "        (x_d, z3),\n",
    "        (theta_d, z4),\n",
    "        (x(t), z1),\n",
    "        (theta(t), z2),\n",
    "        (f_x, u1),\n",
    "        (f_theta, u2)\n",
    "    ])\n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "delta_t = sympy.symbols('delta_t')\n",
    "g_zu = sympy.Matrix([z1, z2, z3, z4]) + delta_t * f_zu"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "A_z = g_zu.jacobian([z1, z2, z3, z4])\n",
    "B_u = g_zu.jacobian([u1, u2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "C_2 = sympy.Matrix.hstack(B_u, A_z * B_u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-54-f6e143429ed4>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mevals\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mC_2\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mC_2\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mT\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0meigenvals\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/matrices/matrices.py\u001b[0m in \u001b[0;36meigenvals\u001b[0;34m(self, error_when_incomplete, **flags)\u001b[0m\n\u001b[1;32m   1257\u001b[0m                 \u001b[0meigs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mroots\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmat\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcharpoly\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mDummy\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'x'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msimplify\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0msimplify\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mflags\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1258\u001b[0m             \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1259\u001b[0;31m                 \u001b[0meigs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mroots\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmat\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcharpoly\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mDummy\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'x'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mflags\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1260\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1261\u001b[0m         \u001b[0;31m# make sure the algebraic multiplicty sums to the\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/polyroots.py\u001b[0m in \u001b[0;36mroots\u001b[0;34m(f, *gens, **flags)\u001b[0m\n\u001b[1;32m    972\u001b[0m         \u001b[0mzeros\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m{\u001b[0m\u001b[0mS\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0mk\u001b[0m\u001b[0;34m}\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    973\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 974\u001b[0;31m     \u001b[0mcoeff\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mpreprocess_roots\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    975\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    976\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mauto\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_domain\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Ring\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/polyroots.py\u001b[0m in \u001b[0;36mpreprocess_roots\u001b[0;34m(poly)\u001b[0m\n\u001b[1;32m    717\u001b[0m     \u001b[0mpoly_func\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mpoly\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunc\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    718\u001b[0m     \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 719\u001b[0;31m         \u001b[0m_\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mpoly\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mpoly\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mclear_denoms\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mconvert\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    720\u001b[0m     \u001b[0;32mexcept\u001b[0m \u001b[0mDomainError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    721\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mcoeff\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mpoly\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/polytools.py\u001b[0m in \u001b[0;36mclear_denoms\u001b[0;34m(self, convert)\u001b[0m\n\u001b[1;32m   2169\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2170\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrep\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'clear_denoms'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2171\u001b[0;31m             \u001b[0mcoeff\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrep\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mclear_denoms\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2172\u001b[0m         \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m  \u001b[0;31m# pragma: no cover\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2173\u001b[0m             \u001b[0;32mraise\u001b[0m \u001b[0mOperationNotSupported\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'clear_denoms'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/polyclasses.py\u001b[0m in \u001b[0;36mclear_denoms\u001b[0;34m(f)\u001b[0m\n\u001b[1;32m    587\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mclear_denoms\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    588\u001b[0m         \u001b[0;34m\"\"\"Clear denominators, but keep the ground domain. \"\"\"\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 589\u001b[0;31m         \u001b[0mcoeff\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mF\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdmp_clear_denoms\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrep\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlev\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdom\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    590\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mcoeff\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mF\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    591\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/densetools.py\u001b[0m in \u001b[0;36mdmp_clear_denoms\u001b[0;34m(f, u, K0, K1, convert)\u001b[0m\n\u001b[1;32m   1237\u001b[0m     \"\"\"\n\u001b[1;32m   1238\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1239\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mdup_clear_denoms\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mconvert\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mconvert\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1240\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1241\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mK1\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/densetools.py\u001b[0m in \u001b[0;36mdup_clear_denoms\u001b[0;34m(f, K0, K1, convert)\u001b[0m\n\u001b[1;32m   1194\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1195\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mK1\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_one\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcommon\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1196\u001b[0;31m         \u001b[0mf\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdup_mul_ground\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcommon\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1197\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1198\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mconvert\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/densearith.py\u001b[0m in \u001b[0;36mdup_mul_ground\u001b[0;34m(f, c, K)\u001b[0m\n\u001b[1;32m    272\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    273\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 274\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0;34m[\u001b[0m \u001b[0mcf\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mc\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mcf\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mf\u001b[0m \u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    275\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    276\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/densearith.py\u001b[0m in \u001b[0;36m<listcomp>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m    272\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    273\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 274\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0;34m[\u001b[0m \u001b[0mcf\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mc\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mcf\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mf\u001b[0m \u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    275\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    276\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/domains/expressiondomain.py\u001b[0m in \u001b[0;36m__mul__\u001b[0;34m(f, g)\u001b[0m\n\u001b[1;32m     86\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     87\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mg\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 88\u001b[0;31m                 \u001b[0;32mreturn\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msimplify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mex\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mex\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     89\u001b[0m             \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     90\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mNotImplemented\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/domains/expressiondomain.py\u001b[0m in \u001b[0;36msimplify\u001b[0;34m(f, ex)\u001b[0m\n\u001b[1;32m     46\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     47\u001b[0m         \u001b[0;32mdef\u001b[0m \u001b[0msimplify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mex\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 48\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__class__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mex\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcancel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     49\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     50\u001b[0m         \u001b[0;32mdef\u001b[0m \u001b[0m__abs__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36mcancel\u001b[0;34m(self, *gens, **args)\u001b[0m\n\u001b[1;32m   3310\u001b[0m         \u001b[0;34m\"\"\"See the cancel function in sympy.polys\"\"\"\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3311\u001b[0m         \u001b[0;32mfrom\u001b[0m \u001b[0msympy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpolys\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mcancel\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3312\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mcancel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0mgens\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3313\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3314\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0minvert\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0mgens\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/polytools.py\u001b[0m in \u001b[0;36mcancel\u001b[0;34m(f, *gens, **args)\u001b[0m\n\u001b[1;32m   6646\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mxreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdict\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mreps\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   6647\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 6648\u001b[0;31m     \u001b[0mc\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mP\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mQ\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mF\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcancel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   6649\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   6650\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mtuple\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mTuple\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/polytools.py\u001b[0m in \u001b[0;36mcancel\u001b[0;34m(f, g, include)\u001b[0m\n\u001b[1;32m   3687\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3688\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mF\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'cancel'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3689\u001b[0;31m             \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mF\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcancel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mG\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0minclude\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0minclude\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3690\u001b[0m         \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m  \u001b[0;31m# pragma: no cover\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3691\u001b[0m             \u001b[0;32mraise\u001b[0m \u001b[0mOperationNotSupported\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'cancel'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/polyclasses.py\u001b[0m in \u001b[0;36mcancel\u001b[0;34m(f, g, include)\u001b[0m\n\u001b[1;32m    695\u001b[0m             \u001b[0mF\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mG\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdmp_cancel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mF\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mG\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlev\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdom\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0minclude\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    696\u001b[0m         \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 697\u001b[0;31m             \u001b[0mcF\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcG\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mF\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mG\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdmp_cancel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mF\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mG\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlev\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdom\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0minclude\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    698\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    699\u001b[0m         \u001b[0mF\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mG\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mF\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/euclidtools.py\u001b[0m in \u001b[0;36mdmp_cancel\u001b[0;34m(f, g, u, K, include)\u001b[0m\n\u001b[1;32m   1867\u001b[0m         \u001b[0mcp\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcq\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mone\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mone\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1868\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1869\u001b[0;31m     \u001b[0m_\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mp\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mq\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdmp_inner_gcd\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1870\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1871\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mK0\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/euclidtools.py\u001b[0m in \u001b[0;36mdmp_inner_gcd\u001b[0;34m(f, g, u, K)\u001b[0m\n\u001b[1;32m   1584\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1585\u001b[0m     \u001b[0mJ\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdmp_multi_deflate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1586\u001b[0;31m     \u001b[0mh\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcff\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcfg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_dmp_inner_gcd\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1587\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1588\u001b[0m     return (dmp_inflate(h, J, u, K),\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/euclidtools.py\u001b[0m in \u001b[0;36m_dmp_inner_gcd\u001b[0;34m(f, g, u, K)\u001b[0m\n\u001b[1;32m   1553\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_ZZ\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mquery\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'USE_HEU_GCD'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1554\u001b[0m             \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1555\u001b[0;31m                 \u001b[0;32mreturn\u001b[0m \u001b[0mdmp_zz_heu_gcd\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1556\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mHeuristicGCDFailed\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1557\u001b[0m                 \u001b[0;32mpass\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/euclidtools.py\u001b[0m in \u001b[0;36mdmp_zz_heu_gcd\u001b[0;34m(f, g, u, K)\u001b[0m\n\u001b[1;32m   1311\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mdup_zz_heu_gcd\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1312\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1313\u001b[0;31m     \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_dmp_rr_trivial_gcd\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1314\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1315\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mresult\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/euclidtools.py\u001b[0m in \u001b[0;36m_dmp_rr_trivial_gcd\u001b[0;34m(f, g, u, K)\u001b[0m\n\u001b[1;32m    911\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mdmp_one\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    912\u001b[0m     \u001b[0;32melif\u001b[0m \u001b[0mquery\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'USE_SIMPLIFY_GCD'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 913\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0m_dmp_simplify_gcd\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    914\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    915\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/euclidtools.py\u001b[0m in \u001b[0;36m_dmp_simplify_gcd\u001b[0;34m(f, g, u, K)\u001b[0m\n\u001b[1;32m    959\u001b[0m     \u001b[0mh\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdmp_gcd\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mF\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mG\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    960\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 961\u001b[0;31m     \u001b[0mcff\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m \u001b[0mdmp_quo\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mh\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mcf\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mf\u001b[0m \u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    962\u001b[0m     \u001b[0mcfg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m \u001b[0mdmp_quo\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mh\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mcg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mg\u001b[0m \u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    963\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
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      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/densearith.py\u001b[0m in \u001b[0;36mdmp_quo\u001b[0;34m(f, g, u, K)\u001b[0m\n\u001b[1;32m   1665\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1666\u001b[0m     \"\"\"\n\u001b[0;32m-> 1667\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0mdmp_div\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1668\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1669\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
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      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/densearith.py\u001b[0m in \u001b[0;36mdmp_rr_div\u001b[0;34m(f, g, u, K)\u001b[0m\n\u001b[1;32m   1389\u001b[0m     \u001b[0;32mwhile\u001b[0m \u001b[0;32mTrue\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1390\u001b[0m         \u001b[0mlc_r\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdmp_LC\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1391\u001b[0;31m         \u001b[0mc\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mR\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdmp_rr_div\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mlc_r\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlc_g\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1392\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1393\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mdmp_zero_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mR\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/densearith.py\u001b[0m in \u001b[0;36mdmp_rr_div\u001b[0;34m(f, g, u, K)\u001b[0m\n\u001b[1;32m   1398\u001b[0m         \u001b[0mq\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdmp_add_term\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mq\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mc\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mj\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1399\u001b[0m         \u001b[0mh\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdmp_mul_term\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mc\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mj\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1400\u001b[0;31m         \u001b[0mr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdmp_sub\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mh\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1401\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1402\u001b[0m         \u001b[0m_dr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdmp_degree\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/densearith.py\u001b[0m in \u001b[0;36mdmp_sub\u001b[0;34m(f, g, u, K)\u001b[0m\n\u001b[1;32m    655\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    656\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mdf\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0mdg\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 657\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mdmp_strip\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m \u001b[0mdmp_sub\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mzip\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    658\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    659\u001b[0m         \u001b[0mk\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mabs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdf\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mdg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/densearith.py\u001b[0m in \u001b[0;36m<listcomp>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m    655\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    656\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mdf\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0mdg\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 657\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mdmp_strip\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m \u001b[0mdmp_sub\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mzip\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    658\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    659\u001b[0m         \u001b[0mk\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mabs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdf\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mdg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/densearith.py\u001b[0m in \u001b[0;36mdmp_sub\u001b[0;34m(f, g, u, K)\u001b[0m\n\u001b[1;32m    655\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    656\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mdf\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0mdg\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 657\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mdmp_strip\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m \u001b[0mdmp_sub\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mzip\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    658\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    659\u001b[0m         \u001b[0mk\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mabs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdf\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mdg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/densearith.py\u001b[0m in \u001b[0;36m<listcomp>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m    655\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    656\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mdf\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0mdg\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 657\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mdmp_strip\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m \u001b[0mdmp_sub\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mzip\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    658\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    659\u001b[0m         \u001b[0mk\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mabs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdf\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mdg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/densearith.py\u001b[0m in \u001b[0;36mdmp_sub\u001b[0;34m(f, g, u, K)\u001b[0m\n\u001b[1;32m    655\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    656\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mdf\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0mdg\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 657\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mdmp_strip\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m \u001b[0mdmp_sub\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mzip\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    658\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    659\u001b[0m         \u001b[0mk\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mabs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdf\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mdg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/densearith.py\u001b[0m in \u001b[0;36m<listcomp>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m    655\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    656\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mdf\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0mdg\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 657\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mdmp_strip\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m \u001b[0mdmp_sub\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mzip\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    658\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    659\u001b[0m         \u001b[0mk\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mabs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdf\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mdg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/densearith.py\u001b[0m in \u001b[0;36mdmp_sub\u001b[0;34m(f, g, u, K)\u001b[0m\n\u001b[1;32m    645\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    646\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mdf\u001b[0m \u001b[0;34m<\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 647\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mdmp_neg\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    648\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    649\u001b[0m     \u001b[0mdg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdmp_degree\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/densearith.py\u001b[0m in \u001b[0;36mdmp_neg\u001b[0;34m(f, u, K)\u001b[0m\n\u001b[1;32m    510\u001b[0m     \u001b[0mv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mu\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    511\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 512\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0;34m[\u001b[0m \u001b[0mdmp_neg\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mcf\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mf\u001b[0m \u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    513\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    514\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/sympy/polys/densearith.py\u001b[0m in \u001b[0;36m<listcomp>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m    510\u001b[0m     \u001b[0mv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mu\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    511\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 512\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0;34m[\u001b[0m \u001b[0mdmp_neg\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mK\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mcf\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mf\u001b[0m \u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    513\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    514\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
     ]
    }
   ],
   "source": [
    "evals = (C_2 * C_2.T).eigenvals()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & \\delta_{t} & 0\\\\0 & 1 & 0 & \\delta_{t}\\\\0 & - \\frac{2 \\delta_{t} m_{p} \\left(r \\left(m_{p} r z_{4}^{2} \\sin{\\left(z_{2} \\right)} + u_{1}\\right) - \\left(- g m_{p} r \\sin{\\left(z_{2} \\right)} + u_{2}\\right) \\cos{\\left(z_{2} \\right)}\\right) \\sin{\\left(z_{2} \\right)} \\cos{\\left(z_{2} \\right)}}{r \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)^{2}} + \\frac{\\delta_{t} \\left(g m_{p} r \\cos^{2}{\\left(z_{2} \\right)} + m_{p} r^{2} z_{4}^{2} \\cos{\\left(z_{2} \\right)} - \\left(g m_{p} r \\sin{\\left(z_{2} \\right)} - u_{2}\\right) \\sin{\\left(z_{2} \\right)}\\right)}{r \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)} & 1 & \\frac{2 \\delta_{t} m_{p} r z_{4} \\sin{\\left(z_{2} \\right)}}{m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}}\\\\0 & \\frac{2 \\delta_{t} \\left(m_{p} r \\left(m_{p} r z_{4}^{2} \\sin{\\left(z_{2} \\right)} + u_{1}\\right) \\cos{\\left(z_{2} \\right)} - \\left(m_{c} + m_{p}\\right) \\left(- g m_{p} r \\sin{\\left(z_{2} \\right)} + u_{2}\\right)\\right) \\sin{\\left(z_{2} \\right)} \\cos{\\left(z_{2} \\right)}}{r^{2} \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)^{2}} - \\frac{\\delta_{t} \\left(- g m_{p} r \\left(- m_{c} - m_{p}\\right) \\cos{\\left(z_{2} \\right)} + m_{p}^{2} r^{2} z_{4}^{2} \\cos^{2}{\\left(z_{2} \\right)} - m_{p} r \\left(m_{p} r z_{4}^{2} \\sin{\\left(z_{2} \\right)} + u_{1}\\right) \\sin{\\left(z_{2} \\right)}\\right)}{m_{p} r^{2} \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)} & 0 & - \\frac{2 \\delta_{t} m_{p} z_{4} \\sin{\\left(z_{2} \\right)} \\cos{\\left(z_{2} \\right)}}{m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}} + 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1,                                                                                                                                                                                                                                                                                                 0, delta_t,                                                            0],\n",
       "[0,                                                                                                                                                                                                                                                                                                 1,       0,                                                      delta_t],\n",
       "[0,                                                 -2*delta_t*m_p*(r*(m_p*r*z4**2*sin(z2) + u1) - (-g*m_p*r*sin(z2) + u2)*cos(z2))*sin(z2)*cos(z2)/(r*(m_c + m_p*sin(z2)**2)**2) + delta_t*(g*m_p*r*cos(z2)**2 + m_p*r**2*z4**2*cos(z2) - (g*m_p*r*sin(z2) - u2)*sin(z2))/(r*(m_c + m_p*sin(z2)**2)),       1,            2*delta_t*m_p*r*z4*sin(z2)/(m_c + m_p*sin(z2)**2)],\n",
       "[0, 2*delta_t*(m_p*r*(m_p*r*z4**2*sin(z2) + u1)*cos(z2) - (m_c + m_p)*(-g*m_p*r*sin(z2) + u2))*sin(z2)*cos(z2)/(r**2*(m_c + m_p*sin(z2)**2)**2) - delta_t*(-g*m_p*r*(-m_c - m_p)*cos(z2) + m_p**2*r**2*z4**2*cos(z2)**2 - m_p*r*(m_p*r*z4**2*sin(z2) + u1)*sin(z2))/(m_p*r**2*(m_c + m_p*sin(z2)**2)),       0, -2*delta_t*m_p*z4*sin(z2)*cos(z2)/(m_c + m_p*sin(z2)**2) + 1]])"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & 0\\\\\\frac{\\delta_{t}}{m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}} & - \\frac{\\delta_{t} \\cos{\\left(z_{2} \\right)}}{r \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)}\\\\- \\frac{\\delta_{t} \\cos{\\left(z_{2} \\right)}}{r \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)} & - \\frac{\\delta_{t} \\left(- m_{c} - m_{p}\\right)}{m_{p} r^{2} \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[                                          0,                                                       0],\n",
       "[                                          0,                                                       0],\n",
       "[             delta_t/(m_c + m_p*sin(z2)**2),             -delta_t*cos(z2)/(r*(m_c + m_p*sin(z2)**2))],\n",
       "[-delta_t*cos(z2)/(r*(m_c + m_p*sin(z2)**2)), -delta_t*(-m_c - m_p)/(m_p*r**2*(m_c + m_p*sin(z2)**2))]])"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B_u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 0 & \\frac{\\delta_{t}^{2}}{m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}} & - \\frac{\\delta_{t}^{2} \\cos{\\left(z_{2} \\right)}}{r \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)}\\\\0 & 0 & - \\frac{\\delta_{t}^{2} \\cos{\\left(z_{2} \\right)}}{r \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)} & - \\frac{\\delta_{t}^{2} \\left(- m_{c} - m_{p}\\right)}{m_{p} r^{2} \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)}\\\\\\frac{\\delta_{t}}{m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}} & - \\frac{\\delta_{t} \\cos{\\left(z_{2} \\right)}}{r \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)} & - \\frac{2 \\delta_{t}^{2} m_{p} z_{4} \\sin{\\left(z_{2} \\right)} \\cos{\\left(z_{2} \\right)}}{\\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)^{2}} + \\frac{\\delta_{t}}{m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}} & - \\frac{2 \\delta_{t}^{2} z_{4} \\left(- m_{c} - m_{p}\\right) \\sin{\\left(z_{2} \\right)}}{r \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)^{2}} - \\frac{\\delta_{t} \\cos{\\left(z_{2} \\right)}}{r \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)}\\\\- \\frac{\\delta_{t} \\cos{\\left(z_{2} \\right)}}{r \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)} & - \\frac{\\delta_{t} \\left(- m_{c} - m_{p}\\right)}{m_{p} r^{2} \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)} & - \\frac{\\delta_{t} \\left(- \\frac{2 \\delta_{t} m_{p} z_{4} \\sin{\\left(z_{2} \\right)} \\cos{\\left(z_{2} \\right)}}{m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}} + 1\\right) \\cos{\\left(z_{2} \\right)}}{r \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)} & - \\frac{\\delta_{t} \\left(- m_{c} - m_{p}\\right) \\left(- \\frac{2 \\delta_{t} m_{p} z_{4} \\sin{\\left(z_{2} \\right)} \\cos{\\left(z_{2} \\right)}}{m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}} + 1\\right)}{m_{p} r^{2} \\left(m_{c} + m_{p} \\sin^{2}{\\left(z_{2} \\right)}\\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[                                          0,                                                       0,                                                                          delta_t**2/(m_c + m_p*sin(z2)**2),                                                                         -delta_t**2*cos(z2)/(r*(m_c + m_p*sin(z2)**2))],\n",
       "[                                          0,                                                       0,                                                             -delta_t**2*cos(z2)/(r*(m_c + m_p*sin(z2)**2)),                                                             -delta_t**2*(-m_c - m_p)/(m_p*r**2*(m_c + m_p*sin(z2)**2))],\n",
       "[             delta_t/(m_c + m_p*sin(z2)**2),             -delta_t*cos(z2)/(r*(m_c + m_p*sin(z2)**2)),            -2*delta_t**2*m_p*z4*sin(z2)*cos(z2)/(m_c + m_p*sin(z2)**2)**2 + delta_t/(m_c + m_p*sin(z2)**2),       -2*delta_t**2*z4*(-m_c - m_p)*sin(z2)/(r*(m_c + m_p*sin(z2)**2)**2) - delta_t*cos(z2)/(r*(m_c + m_p*sin(z2)**2))],\n",
       "[-delta_t*cos(z2)/(r*(m_c + m_p*sin(z2)**2)), -delta_t*(-m_c - m_p)/(m_p*r**2*(m_c + m_p*sin(z2)**2)), -delta_t*(-2*delta_t*m_p*z4*sin(z2)*cos(z2)/(m_c + m_p*sin(z2)**2) + 1)*cos(z2)/(r*(m_c + m_p*sin(z2)**2)), -delta_t*(-m_c - m_p)*(-2*delta_t*m_p*z4*sin(z2)*cos(z2)/(m_c + m_p*sin(z2)**2) + 1)/(m_p*r**2*(m_c + m_p*sin(z2)**2))]])"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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