abhigyan/Jupyter/Plotting Multivariable Functions.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "var('x,y')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "r(x,y) = sqrt(x^2+y^2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "U(x,y) = (a/r(x,y))*e^(-1*b*r(x,y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ux(x,y)=U.derivative(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "Uy(x,y)=U.derivative(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( x, y \\right) \\ {\\mapsto} \\ -\\frac{a b x e^{\\left(-\\sqrt{x^{2} + y^{2}} b\\right)}}{x^{2} + y^{2}} - \\frac{a x e^{\\left(-\\sqrt{x^{2} + y^{2}} b\\right)}}{{\\left(x^{2} + y^{2}\\right)}^{\\frac{3}{2}}}</script></html>"
      ],
      "text/plain": [
       "(x, y) |--> -a*b*x*e^(-sqrt(x^2 + y^2)*b)/(x^2 + y^2) - a*x*e^(-sqrt(x^2 + y^2)*b)/(x^2 + y^2)^(3/2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( x, y \\right) \\ {\\mapsto} \\ -\\frac{a b y e^{\\left(-\\sqrt{x^{2} + y^{2}} b\\right)}}{x^{2} + y^{2}} - \\frac{a y e^{\\left(-\\sqrt{x^{2} + y^{2}} b\\right)}}{{\\left(x^{2} + y^{2}\\right)}^{\\frac{3}{2}}}</script></html>"
      ],
      "text/plain": [
       "(x, y) |--> -a*b*y*e^(-sqrt(x^2 + y^2)*b)/(x^2 + y^2) - a*y*e^(-sqrt(x^2 + y^2)*b)/(x^2 + y^2)^(3/2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Ux.show()\n",
    "Uy.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "Variable 'x' not found",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/ext/fast_callable.pyx\u001b[0m in \u001b[0;36msage.ext.fast_callable.ExpressionTreeBuilder.var (build/cythonized/sage/ext/fast_callable.c:6744)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    726\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--> 727\u001b[0;31m             \u001b[0mind\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_vars\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mindex\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvar_name\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    728\u001b[0m         \u001b[0;32mexcept\u001b[0m \u001b[0mValueError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mValueError\u001b[0m: 'x' is not in list",
      "\nDuring handling of the above exception, another exception occurred:\n",
      "\u001b[0;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-21-7a59d8ec12e6>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mplot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mU\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/misc/decorators.py\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwds)\u001b[0m\n\u001b[1;32m    491\u001b[0m                 \u001b[0moptions\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m'__original_opts'\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mkwds\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    492\u001b[0m             \u001b[0moptions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mupdate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 493\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mfunc\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[0moptions\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    494\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    495\u001b[0m         \u001b[0;31m#Add the options specified by @options to the signature of the wrapped\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/plot/plot.py\u001b[0m in \u001b[0;36mplot\u001b[0;34m(funcs, *args, **kwds)\u001b[0m\n\u001b[1;32m   1939\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1940\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfuncs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'plot'\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-> 1941\u001b[0;31m         \u001b[0mG\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfuncs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mplot\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[0moriginal_opts\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   1942\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1943\u001b[0m         \u001b[0;31m# If we have extra keywords already set, then update them\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression.pyx\u001b[0m in \u001b[0;36msage.symbolic.expression.Expression.plot (build/cythonized/sage/symbolic/expression.cpp:62908)\u001b[0;34m()\u001b[0m\n\u001b[1;32m  12062\u001b[0m                     \u001b[0mparam\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mA\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[1;32m  12063\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> 12064\u001b[0;31m                         \u001b[0mf\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_plot_fast_callable\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mparam\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  12065\u001b[0m                     \u001b[0;32mexcept\u001b[0m \u001b[0mNotImplementedError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m  12066\u001b[0m                         \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunction\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mparam\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression.pyx\u001b[0m in \u001b[0;36msage.symbolic.expression.Expression._plot_fast_callable (build/cythonized/sage/symbolic/expression.cpp:63287)\u001b[0;34m()\u001b[0m\n\u001b[1;32m  12108\u001b[0m         \"\"\"\n\u001b[1;32m  12109\u001b[0m         \u001b[0;32mfrom\u001b[0m \u001b[0msage\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mext\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfast_callable\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mfast_callable\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m> 12110\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mfast_callable\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mvars\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mvars\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexpect_one_var\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  12111\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m  12112\u001b[0m     \u001b[0;31m############\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/ext/fast_callable.pyx\u001b[0m in \u001b[0;36msage.ext.fast_callable.fast_callable (build/cythonized/sage/ext/fast_callable.c:4743)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    494\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    495\u001b[0m         \u001b[0metb\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpressionTreeBuilder\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvars\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mvars\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdomain\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mdomain\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 496\u001b[0;31m         \u001b[0met\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_fast_callable_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0metb\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    497\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    498\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdomain\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mRealField_class\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression.pyx\u001b[0m in \u001b[0;36msage.symbolic.expression.Expression._fast_callable_ (build/cythonized/sage/symbolic/expression.cpp:62141)\u001b[0;34m()\u001b[0m\n\u001b[1;32m  11945\u001b[0m         \"\"\"\n\u001b[1;32m  11946\u001b[0m         \u001b[0;32mfrom\u001b[0m \u001b[0msage\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msymbolic\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexpression_conversions\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mfast_callable\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m> 11947\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mfast_callable\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0metb\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  11948\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m  11949\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mshow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression_conversions.py\u001b[0m in \u001b[0;36mfast_callable\u001b[0;34m(ex, etb)\u001b[0m\n\u001b[1;32m   1887\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1888\u001b[0m     \"\"\"\n\u001b[0;32m-> 1889\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0mFastCallableConverter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mex\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0metb\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   1890\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1891\u001b[0m \u001b[0;32mclass\u001b[0m \u001b[0mRingConverter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mConverter\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression_conversions.py\u001b[0m in \u001b[0;36m__call__\u001b[0;34m(self, ex)\u001b[0m\n\u001b[1;32m    215\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mgetattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'use_fake_div'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0moperator\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0m_operator\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmul\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0moperator\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mmul_vararg\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    216\u001b[0m                 \u001b[0mdiv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_fake_div\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[0;32m--> 217\u001b[0;31m                 \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marithmetic\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdiv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdiv\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0moperator\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    218\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marithmetic\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mex\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0moperator\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    219\u001b[0m         \u001b[0;32melif\u001b[0m \u001b[0moperator\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrelation_operators\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression_conversions.py\u001b[0m in \u001b[0;36marithmetic\u001b[0;34m(self, ex, operator)\u001b[0m\n\u001b[1;32m   1815\u001b[0m         \u001b[0;32melif\u001b[0m \u001b[0moperator\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0mmul_vararg\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1816\u001b[0m             \u001b[0moperator\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_operator\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmul\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1817\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mreduce\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;32mlambda\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0metb\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0moperator\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0moperands\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   1818\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1819\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0msymbol\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression_conversions.py\u001b[0m in \u001b[0;36m<lambda>\u001b[0;34m(x, y)\u001b[0m\n\u001b[1;32m   1815\u001b[0m         \u001b[0;32melif\u001b[0m \u001b[0moperator\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0mmul_vararg\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1816\u001b[0m             \u001b[0moperator\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_operator\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmul\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1817\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mreduce\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;32mlambda\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0metb\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0moperator\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0moperands\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   1818\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1819\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0msymbol\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/ext/fast_callable.pyx\u001b[0m in \u001b[0;36msage.ext.fast_callable.ExpressionTreeBuilder.call (build/cythonized/sage/ext/fast_callable.c:7199)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    780\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mbase\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mexponent\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    781\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--> 782\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mExpressionCall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfn\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\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[0m\u001b[1;32m    783\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    784\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mchoice\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcond\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0miftrue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0miffalse\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/ext/fast_callable.pyx\u001b[0m in \u001b[0;36msage.ext.fast_callable.ExpressionTreeBuilder.__call__ (build/cythonized/sage/ext/fast_callable.c:6334)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    655\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mconstant\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    656\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 657\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mfc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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;34m\u001b[0m\u001b[0m\n\u001b[1;32m    659\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m_clean_var\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression.pyx\u001b[0m in \u001b[0;36msage.symbolic.expression.Expression._fast_callable_ (build/cythonized/sage/symbolic/expression.cpp:62141)\u001b[0;34m()\u001b[0m\n\u001b[1;32m  11945\u001b[0m         \"\"\"\n\u001b[1;32m  11946\u001b[0m         \u001b[0;32mfrom\u001b[0m \u001b[0msage\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msymbolic\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexpression_conversions\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mfast_callable\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m> 11947\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mfast_callable\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0metb\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  11948\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m  11949\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mshow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression_conversions.py\u001b[0m in \u001b[0;36mfast_callable\u001b[0;34m(ex, etb)\u001b[0m\n\u001b[1;32m   1887\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1888\u001b[0m     \"\"\"\n\u001b[0;32m-> 1889\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0mFastCallableConverter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mex\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0metb\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   1890\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1891\u001b[0m \u001b[0;32mclass\u001b[0m \u001b[0mRingConverter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mConverter\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression_conversions.py\u001b[0m in \u001b[0;36m__call__\u001b[0;34m(self, ex)\u001b[0m\n\u001b[1;32m    216\u001b[0m                 \u001b[0mdiv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_fake_div\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[1;32m    217\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marithmetic\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdiv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdiv\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0moperator\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--> 218\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marithmetic\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mex\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0moperator\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    219\u001b[0m         \u001b[0;32melif\u001b[0m \u001b[0moperator\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrelation_operators\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    220\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrelation\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mex\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0moperator\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression_conversions.py\u001b[0m in \u001b[0;36marithmetic\u001b[0;34m(self, ex, operator)\u001b[0m\n\u001b[1;32m   1808\u001b[0m             \u001b[0;32melif\u001b[0m \u001b[0mexponent\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;34m-\u001b[0m\u001b[0;36m0.5\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1809\u001b[0m                 \u001b[0;32mfrom\u001b[0m \u001b[0msage\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunctions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mall\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0msqrt\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1810\u001b[0;31m                 \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0metb\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0m_operator\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtruediv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0metb\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msqrt\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0moperands\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[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1811\u001b[0m         \u001b[0;32melif\u001b[0m \u001b[0moperator\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0m_operator\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mneg\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1812\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0metb\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0moperator\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0moperands\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[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/ext/fast_callable.pyx\u001b[0m in \u001b[0;36msage.ext.fast_callable.ExpressionTreeBuilder.call (build/cythonized/sage/ext/fast_callable.c:7199)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    780\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mbase\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mexponent\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    781\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--> 782\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mExpressionCall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfn\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\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[0m\u001b[1;32m    783\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    784\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mchoice\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcond\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0miftrue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0miffalse\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/ext/fast_callable.pyx\u001b[0m in \u001b[0;36msage.ext.fast_callable.ExpressionTreeBuilder.__call__ (build/cythonized/sage/ext/fast_callable.c:6334)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    655\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mconstant\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    656\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 657\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mfc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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;34m\u001b[0m\u001b[0m\n\u001b[1;32m    659\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m_clean_var\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression.pyx\u001b[0m in \u001b[0;36msage.symbolic.expression.Expression._fast_callable_ (build/cythonized/sage/symbolic/expression.cpp:62141)\u001b[0;34m()\u001b[0m\n\u001b[1;32m  11945\u001b[0m         \"\"\"\n\u001b[1;32m  11946\u001b[0m         \u001b[0;32mfrom\u001b[0m \u001b[0msage\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msymbolic\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexpression_conversions\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mfast_callable\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m> 11947\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mfast_callable\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0metb\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  11948\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m  11949\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mshow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression_conversions.py\u001b[0m in \u001b[0;36mfast_callable\u001b[0;34m(ex, etb)\u001b[0m\n\u001b[1;32m   1887\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1888\u001b[0m     \"\"\"\n\u001b[0;32m-> 1889\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0mFastCallableConverter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mex\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0metb\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   1890\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1891\u001b[0m \u001b[0;32mclass\u001b[0m \u001b[0mRingConverter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mConverter\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression_conversions.py\u001b[0m in \u001b[0;36m__call__\u001b[0;34m(self, ex)\u001b[0m\n\u001b[1;32m    216\u001b[0m                 \u001b[0mdiv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_fake_div\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[1;32m    217\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marithmetic\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdiv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdiv\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0moperator\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--> 218\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marithmetic\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mex\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0moperator\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    219\u001b[0m         \u001b[0;32melif\u001b[0m \u001b[0moperator\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrelation_operators\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    220\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrelation\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mex\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0moperator\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression_conversions.py\u001b[0m in \u001b[0;36marithmetic\u001b[0;34m(self, ex, operator)\u001b[0m\n\u001b[1;32m   1815\u001b[0m         \u001b[0;32melif\u001b[0m \u001b[0moperator\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0mmul_vararg\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1816\u001b[0m             \u001b[0moperator\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_operator\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmul\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1817\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mreduce\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;32mlambda\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0metb\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0moperator\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0moperands\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   1818\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1819\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0msymbol\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression_conversions.py\u001b[0m in \u001b[0;36m<lambda>\u001b[0;34m(x, y)\u001b[0m\n\u001b[1;32m   1815\u001b[0m         \u001b[0;32melif\u001b[0m \u001b[0moperator\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0mmul_vararg\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1816\u001b[0m             \u001b[0moperator\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_operator\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmul\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1817\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mreduce\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;32mlambda\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0metb\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0moperator\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0moperands\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   1818\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1819\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0msymbol\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/ext/fast_callable.pyx\u001b[0m in \u001b[0;36msage.ext.fast_callable.ExpressionTreeBuilder.call (build/cythonized/sage/ext/fast_callable.c:7199)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    780\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mbase\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mexponent\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    781\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--> 782\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mExpressionCall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfn\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\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[0m\u001b[1;32m    783\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    784\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mchoice\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcond\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0miftrue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0miffalse\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/ext/fast_callable.pyx\u001b[0m in \u001b[0;36msage.ext.fast_callable.ExpressionTreeBuilder.__call__ (build/cythonized/sage/ext/fast_callable.c:6334)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    655\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mconstant\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    656\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 657\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mfc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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;34m\u001b[0m\u001b[0m\n\u001b[1;32m    659\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m_clean_var\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression.pyx\u001b[0m in \u001b[0;36msage.symbolic.expression.Expression._fast_callable_ (build/cythonized/sage/symbolic/expression.cpp:62141)\u001b[0;34m()\u001b[0m\n\u001b[1;32m  11945\u001b[0m         \"\"\"\n\u001b[1;32m  11946\u001b[0m         \u001b[0;32mfrom\u001b[0m \u001b[0msage\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msymbolic\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexpression_conversions\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mfast_callable\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m> 11947\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mfast_callable\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0metb\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  11948\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m  11949\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mshow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression_conversions.py\u001b[0m in \u001b[0;36mfast_callable\u001b[0;34m(ex, etb)\u001b[0m\n\u001b[1;32m   1887\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1888\u001b[0m     \"\"\"\n\u001b[0;32m-> 1889\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0mFastCallableConverter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mex\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0metb\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   1890\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1891\u001b[0m \u001b[0;32mclass\u001b[0m \u001b[0mRingConverter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mConverter\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression_conversions.py\u001b[0m in \u001b[0;36m__call__\u001b[0;34m(self, ex)\u001b[0m\n\u001b[1;32m    216\u001b[0m                 \u001b[0mdiv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_fake_div\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[1;32m    217\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marithmetic\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdiv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdiv\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0moperator\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--> 218\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marithmetic\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mex\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0moperator\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    219\u001b[0m         \u001b[0;32melif\u001b[0m \u001b[0moperator\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrelation_operators\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    220\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrelation\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mex\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0moperator\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression_conversions.py\u001b[0m in \u001b[0;36marithmetic\u001b[0;34m(self, ex, operator)\u001b[0m\n\u001b[1;32m   1815\u001b[0m         \u001b[0;32melif\u001b[0m \u001b[0moperator\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0mmul_vararg\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1816\u001b[0m             \u001b[0moperator\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_operator\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmul\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1817\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mreduce\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;32mlambda\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0metb\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0moperator\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0moperands\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   1818\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1819\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0msymbol\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression_conversions.py\u001b[0m in \u001b[0;36m<lambda>\u001b[0;34m(x, y)\u001b[0m\n\u001b[1;32m   1815\u001b[0m         \u001b[0;32melif\u001b[0m \u001b[0moperator\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0mmul_vararg\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1816\u001b[0m             \u001b[0moperator\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_operator\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmul\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1817\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mreduce\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;32mlambda\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0metb\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0moperator\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0moperands\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   1818\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1819\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0msymbol\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/ext/fast_callable.pyx\u001b[0m in \u001b[0;36msage.ext.fast_callable.ExpressionTreeBuilder.call (build/cythonized/sage/ext/fast_callable.c:7145)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    778\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mfn\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0moperator\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpow\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    779\u001b[0m             \u001b[0mbase\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexponent\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 780\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mbase\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mexponent\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    781\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    782\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mExpressionCall\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfn\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/ext/fast_callable.pyx\u001b[0m in \u001b[0;36msage.ext.fast_callable.ExpressionTreeBuilder.__call__ (build/cythonized/sage/ext/fast_callable.c:6334)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    655\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mconstant\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    656\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 657\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mfc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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;34m\u001b[0m\u001b[0m\n\u001b[1;32m    659\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m_clean_var\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression.pyx\u001b[0m in \u001b[0;36msage.symbolic.expression.Expression._fast_callable_ (build/cythonized/sage/symbolic/expression.cpp:62141)\u001b[0;34m()\u001b[0m\n\u001b[1;32m  11945\u001b[0m         \"\"\"\n\u001b[1;32m  11946\u001b[0m         \u001b[0;32mfrom\u001b[0m \u001b[0msage\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msymbolic\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexpression_conversions\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mfast_callable\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m> 11947\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mfast_callable\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0metb\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  11948\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m  11949\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mshow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression_conversions.py\u001b[0m in \u001b[0;36mfast_callable\u001b[0;34m(ex, etb)\u001b[0m\n\u001b[1;32m   1887\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1888\u001b[0m     \"\"\"\n\u001b[0;32m-> 1889\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0mFastCallableConverter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mex\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0metb\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   1890\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1891\u001b[0m \u001b[0;32mclass\u001b[0m \u001b[0mRingConverter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mConverter\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression_conversions.py\u001b[0m in \u001b[0;36m__call__\u001b[0;34m(self, ex)\u001b[0m\n\u001b[1;32m    210\u001b[0m         \u001b[0moperator\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mex\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0moperator\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    211\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0moperator\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--> 212\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msymbol\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    213\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    214\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0moperator\u001b[0m \u001b[0;32min\u001b[0m \u001b[0marithmetic_operators\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/symbolic/expression_conversions.py\u001b[0m in \u001b[0;36msymbol\u001b[0;34m(self, ex)\u001b[0m\n\u001b[1;32m   1836\u001b[0m             \u001b[0mValueError\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0mVariable\u001b[0m \u001b[0;34m'z'\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mfound\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1837\u001b[0m         \"\"\"\n\u001b[0;32m-> 1838\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0metb\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mvar\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mSR\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[0m\u001b[1;32m   1839\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1840\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mcomposition\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mex\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfunction\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/home/abhigyan/anaconda3/envs/sage/lib/python3.7/site-packages/sage/ext/fast_callable.pyx\u001b[0m in \u001b[0;36msage.ext.fast_callable.ExpressionTreeBuilder.var (build/cythonized/sage/ext/fast_callable.c:6796)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    727\u001b[0m             \u001b[0mind\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_vars\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mindex\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvar_name\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    728\u001b[0m         \u001b[0;32mexcept\u001b[0m \u001b[0mValueError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 729\u001b[0;31m             \u001b[0;32mraise\u001b[0m \u001b[0mValueError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"Variable '%s' not found\"\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mvar_name\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    730\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mExpressionVariable\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mind\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    731\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mValueError\u001b[0m: Variable 'x' not found"
     ]
    }
   ],
   "source": [
    "plot(U(x,y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I was able to figure out that the H(s) function should be of the form $\\frac{A}{s-B}$, where $B \\in \\mathbb{R}^{-}$ and the ROC is $\\Re \\{s\\} > B$.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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