{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Integração Numérica\n",
    "Existem diversos métodos comumente utilizados para encontrar soluções aproximadas para sistemas de EDOs. Neste documento introduziremos os mais usados assim com exemplos do seu uso no Sage.\n",
    "## Método de Euler \n",
    "\n",
    "No método de Euler aproximamos a EDO de primeira ordem por uma série de pontos no tempo a intervalos regulares de tamanho $h$ definido pelo usuário. Logo, $t_n=t_0 + n h$. Assumindo um EDO da forma $\\frac{dy}{dt}=f(t,y(t))$, a solução aproximada pelo método de Euler é $$y_{n+1} = y_n + hf(t_n,y_n)$$.\n",
    "\n",
    "Abaixo temos um exemplo interativo para um sistema relativamente simples: $$\\frac{dx}{dt} = sin(x)$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "%display typeset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "520c2fbbf1cb48d6a301429e298b846b",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Interactive function <function euler_method at 0x7f6b8c611280> with 7 widgets\n",
       "  y_exact_in: TransformText(valu…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def tab_list(y, headers = None):\n",
    "    '''\n",
    "    Converte uma lista em uma tabela html\n",
    "    '''\n",
    "    s = '<table border = 1>'\n",
    "    if headers:\n",
    "        for q in headers:\n",
    "            s = s + '<th>' + str(q) + '</th>'\n",
    "    for x in y:\n",
    "        s = s + '<tr>'\n",
    "        for q in x:\n",
    "            s = s + '<td>' + str(q) + '</td>'\n",
    "        s = s + '</tr>'\n",
    "    s = s + '</table>'\n",
    "    return s\n",
    "\n",
    "var('x y')\n",
    "@interact\n",
    "def euler_method(y_exact_in = input_box('-cos(x)+1.0', type = str, label = 'Solução Exata = '), y_prime_in = input_box('sin(x)', type = str, label = \"y' = \"), start = input_box(0.0, label = 'Val. inicial de x: '), stop = input_box(6.0, label = 'Valor de parada de x: '), startval = input_box(0.0, label = 'Valor inicial de y: '), nsteps = slider([2^m for m in range(0,10)], default = 10, label = 'Número de passos: '), show_steps = slider([2^m for m in range(0,10)], default = 8, label = 'Numero de passos mostrados na tabela: ')):\n",
    "    y_exact = lambda x: eval(y_exact_in)\n",
    "    y_prime = lambda x,y: eval(y_prime_in)\n",
    "    stepsize = float((stop-start)/nsteps)\n",
    "    steps_shown = min(nsteps,show_steps)\n",
    "    sol = [startval]\n",
    "    xvals = [start]\n",
    "    for step in range(nsteps):\n",
    "        sol.append(sol[-1] + stepsize*y_prime(xvals[-1],sol[-1]))\n",
    "        xvals.append(xvals[-1] + stepsize)\n",
    "    sol_max = max(sol + [find_local_maximum(y_exact,start,stop)[0]])\n",
    "    sol_min = min(sol + [find_local_minimum(y_exact,start,stop)[0]])\n",
    "    show(plot(y_exact(x),start,stop,rgbcolor=(1,0,0))+line([[xvals[index],sol[index]] for index in range(len(sol))]),xmin=start,xmax = stop, ymax = sol_max, ymin = sol_min)\n",
    "    if nsteps < steps_shown:\n",
    "        table_range = range(len(sol))\n",
    "    else:\n",
    "        table_range = list(range(0,floor(steps_shown/2))) + list(range(len(sol)-floor(steps_shown/2),len(sol)))\n",
    "    html(tab_list([[i,xvals[i],sol[i]] for i in table_range], headers = ['step','x','y']))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Campos Vetoriais e o Método de Euler"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "1366df4241ac47b4ba6cfa64aa064ce6",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Interactive function <function _ at 0x7f6b72f70820> with 10 widgets\n",
       "  f: EvalText(value='y', description='f', …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x,y = var('x,y')\n",
    "from sage.ext.fast_eval import fast_float\n",
    "@interact\n",
    "def _(f = input_box(default=y), g=input_box(default=-x*y+x^3-x),\n",
    "      xmin=input_box(default=-1), xmax=input_box(default=1),\n",
    "      ymin=input_box(default=-1), ymax=input_box(default=1),\n",
    "      x_ini=input_box(default=0.5), y_ini=input_box(default=0.5),\n",
    "      tam_passo=(0.01,(0.001, 0.2, .001)), passos=(600,(0, 1400)) ):\n",
    "    ff = fast_float(f, 'x', 'y')\n",
    "    gg = fast_float(g, 'x', 'y')\n",
    "    passos = int(passos)\n",
    "    \n",
    "    points = [ (x_ini, y_ini) ]\n",
    "    for i in range(passos):\n",
    "        xx, yy = points[-1]\n",
    "        try:\n",
    "            points.append( (xx + tam_passo * ff(xx,yy), yy + tam_passo * gg(xx,yy)) )\n",
    "        except (ValueError, ArithmeticError, TypeError):\n",
    "            break\n",
    "    equilibria = solve([f,g], [x,y])\n",
    "    xnull = plot(solve(f,y)[0].rhs(),(x,xmin,xmax), color='red', legend_label=\"Nuliclina de X\")\n",
    "    ynull = plot(solve(g,y)[0].rhs(),(x,xmin,xmax), color='green', legend_label=\"Nuliclina de Y\")\n",
    "    starting_point = point(points[0], pointsize=50)\n",
    "    solution = line(points)\n",
    "    vector_field = plot_vector_field( (f,g), (x,xmin,xmax), (y,ymin,ymax) )\n",
    "\n",
    "    result = vector_field + starting_point + solution +xnull+ ynull\n",
    "\n",
    "    pretty_print(html(r\"$\\displaystyle\\frac{dx}{dt} = %s$ <br>$\\displaystyle\\frac{dy}{dt} = %s$\" % (latex(f),latex(g))))\n",
    "    result.show(xmin=xmin,xmax=xmax,ymin=ymin,ymax=ymax)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Campo Vetorial com Runge-Kutta Fehlberg"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "52e94b8dd4e2466ba28717cb24fbbbf4",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Interactive function <function _ at 0x7f6b72dd8700> with 12 widgets\n",
       "  fin: EvalText(value='-x*y^3 - 1/12*(4*y^…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "var('x y')\n",
    "@interact\n",
    "def _(fin = input_box(default=y+exp(x/10)-1/3*((x-1/2)^2+y^3)*x-x*y^3), gin=input_box(default=x^3-x+1/100*exp(y*x^2+x*y^2)-0.7*x),\n",
    "      xmin=input_box(default=-1), xmax=input_box(default=1.8),\n",
    "      ymin=input_box(default=-1.3), ymax=input_box(default=1.5),\n",
    "      x_start=(-1,(-2,2)), y_start=(0,(-2,2)), error=(0.5,(0,1)),\n",
    "      t_length=(23,(0, 100)) , num_of_points = (1500,(5,2000)),\n",
    "      algorithm = selector([\n",
    "         (\"rkf45\" , \"runga-kutta-felhberg (4,5)\"),\n",
    "         (\"rk2\" , \"embedded runga-kutta (2,3)\"),\n",
    "         (\"rk4\" , \"4th order classical runga-kutta\"),\n",
    "         (\"rk8pd\" , 'runga-kutta prince-dormand (8,9)'),\n",
    "         (\"rk2imp\" , \"implicit 2nd order runga-kutta at gaussian points\"),\n",
    "         (\"rk4imp\" , \"implicit 4th order runga-kutta at gaussian points\"),\n",
    "         (\"bsimp\" , \"implicit burlisch-stoer (requires jacobian)\"),\n",
    "         (\"gear1\" , \"M=1 implicit gear\"),\n",
    "         (\"gear2\" , \"M=2 implicit gear\")\n",
    "      ])):\n",
    "    f(x,y)=fin\n",
    "    g(x,y)=gin\n",
    "\n",
    "    ff = f._fast_float_(*f.args())\n",
    "    gg = g._fast_float_(*g.args())\n",
    "\n",
    "    #solve\n",
    "    path = []\n",
    "    err = error\n",
    "    xerr = 0\n",
    "    for yerr in [-err, 0, +err]:\n",
    "      T=ode_solver()\n",
    "      T.algorithm=algorithm\n",
    "      T.function = lambda t, yp: [ff(yp[0],yp[1]), gg(yp[0],yp[1])]\n",
    "      T.jacobian = jacobian([f,g],[x,y])\n",
    "      T.ode_solve(y_0=[x_start + xerr, y_start + yerr],t_span=[0,t_length],num_points=num_of_points)\n",
    "      path.append(line([p[1] for p in T.solution]))\n",
    "\n",
    "    #plot\n",
    "    vector_field = plot_vector_field( (f,g), (x,xmin,xmax), (y,ymin,ymax) )\n",
    "    starting_point = point([x_start, y_start], pointsize=50)\n",
    "    xnull = plot(solve(f,y)[0].rhs(),(x,xmin,xmax), color='red', legend_label=\"Nuliclina de X\")\n",
    "    ynull = plot(solve(g,y)[0].rhs(),(x,xmin, xmax), color='green', legend_label=\"Nuliclina de Y\")\n",
    "    show(vector_field + starting_point + sum(path)+xnull, aspect_ratio=1, figsize=[8,9])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}y = -\\frac{4^{\\frac{2}{3}} {\\left(x {\\left(i \\, \\sqrt{3} + 1\\right)} \\left(-\\frac{4 \\, x^{3} - 4 \\, x^{2} + x - \\sqrt{\\frac{16 \\, x^{7} - 32 \\, x^{6} + 24 \\, x^{5} - 8 \\, x^{4} + x^{3} + 144 \\, x e^{\\left(\\frac{1}{5} \\, x\\right)} - 24 \\, {\\left(4 \\, x^{4} - 4 \\, x^{3} + x^{2}\\right)} e^{\\left(\\frac{1}{10} \\, x\\right)} - 16}{x}} - 12 \\, e^{\\left(\\frac{1}{10} \\, x\\right)}}{x}\\right)^{\\frac{2}{3}} - i \\cdot 4^{\\frac{2}{3}} \\sqrt{3} + 4^{\\frac{2}{3}}\\right)}}{16 \\, x \\left(-\\frac{4 \\, x^{3} - 4 \\, x^{2} + x - \\sqrt{\\frac{16 \\, x^{7} - 32 \\, x^{6} + 24 \\, x^{5} - 8 \\, x^{4} + x^{3} + 144 \\, x e^{\\left(\\frac{1}{5} \\, x\\right)} - 24 \\, {\\left(4 \\, x^{4} - 4 \\, x^{3} + x^{2}\\right)} e^{\\left(\\frac{1}{10} \\, x\\right)} - 16}{x}} - 12 \\, e^{\\left(\\frac{1}{10} \\, x\\right)}}{x}\\right)^{\\frac{1}{3}}}</script></html>"
      ],
      "text/latex": [
       "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}y = -\\frac{4^{\\frac{2}{3}} {\\left(x {\\left(i \\, \\sqrt{3} + 1\\right)} \\left(-\\frac{4 \\, x^{3} - 4 \\, x^{2} + x - \\sqrt{\\frac{16 \\, x^{7} - 32 \\, x^{6} + 24 \\, x^{5} - 8 \\, x^{4} + x^{3} + 144 \\, x e^{\\left(\\frac{1}{5} \\, x\\right)} - 24 \\, {\\left(4 \\, x^{4} - 4 \\, x^{3} + x^{2}\\right)} e^{\\left(\\frac{1}{10} \\, x\\right)} - 16}{x}} - 12 \\, e^{\\left(\\frac{1}{10} \\, x\\right)}}{x}\\right)^{\\frac{2}{3}} - i \\cdot 4^{\\frac{2}{3}} \\sqrt{3} + 4^{\\frac{2}{3}}\\right)}}{16 \\, x \\left(-\\frac{4 \\, x^{3} - 4 \\, x^{2} + x - \\sqrt{\\frac{16 \\, x^{7} - 32 \\, x^{6} + 24 \\, x^{5} - 8 \\, x^{4} + x^{3} + 144 \\, x e^{\\left(\\frac{1}{5} \\, x\\right)} - 24 \\, {\\left(4 \\, x^{4} - 4 \\, x^{3} + x^{2}\\right)} e^{\\left(\\frac{1}{10} \\, x\\right)} - 16}{x}} - 12 \\, e^{\\left(\\frac{1}{10} \\, x\\right)}}{x}\\right)^{\\frac{1}{3}}}$$"
      ],
      "text/plain": [
       "y == -1/16*4^(2/3)*(x*(I*sqrt(3) + 1)*(-(4*x^3 - 4*x^2 + x - sqrt((16*x^7 - 32*x^6 + 24*x^5 - 8*x^4 + x^3 + 144*x*e^(1/5*x) - 24*(4*x^4 - 4*x^3 + x^2)*e^(1/10*x) - 16)/x) - 12*e^(1/10*x))/x)^(2/3) - I*4^(2/3)*sqrt(3) + 4^(2/3))/(x*(-(4*x^3 - 4*x^2 + x - sqrt((16*x^7 - 32*x^6 + 24*x^5 - 8*x^4 + x^3 + 144*x*e^(1/5*x) - 24*(4*x^4 - 4*x^3 + x^2)*e^(1/10*x) - 16)/x) - 12*e^(1/10*x))/x)^(1/3))"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('x y z')\n",
    "\n",
    "sol = solve(-x*y^3 - 1/12*(4*y^3 + (2*x - 1)^2)*x + y + e^(1/10*x), y)\n",
    "sol[0].simplify_full()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "<ipython-input-6-9dff5e30990a>:1: DeprecationWarning: Substitution using function-call syntax and unnamed arguments is deprecated and will be removed from a future release of Sage; you can use named arguments instead, like EXPR(x=..., y=...)\n",
      "See http://trac.sagemath.org/5930 for details.\n",
      "  complex_plot(sol[Integer(0)].subs(x=z), (x,-Integer(1), Integer(1)), (y,-Integer(1),Integer(1)))\n"
     ]
    },
    {
     "ename": "TypeError",
     "evalue": "Cannot evaluate symbolic expression to a numeric value.",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-6-9dff5e30990a>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mcomplex_plot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msol\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mInteger\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[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mz\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\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[0;32m~/Downloads/SageMath/local/lib/python3.9/site-packages/sage/misc/lazy_import.pyx\u001b[0m in \u001b[0;36msage.misc.lazy_import.LazyImport.__call__ (build/cythonized/sage/misc/lazy_import.c:4032)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    358\u001b[0m             \u001b[0;32mTrue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    359\u001b[0m         \"\"\"\n\u001b[0;32m--> 360\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_object\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\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[0mkwds\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    361\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    362\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m__repr__\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~/Downloads/SageMath/local/lib/python3.9/site-packages/sage/misc/decorators.py\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwds)\u001b[0m\n\u001b[1;32m    489\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    490\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--> 491\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    492\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    493\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~/Downloads/SageMath/local/lib/python3.9/site-packages/sage/plot/complex_plot.pyx\u001b[0m in \u001b[0;36msage.plot.complex_plot.complex_plot (build/cythonized/sage/plot/complex_plot.c:6071)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    402\u001b[0m     \u001b[0mg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mGraphics\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    403\u001b[0m     \u001b[0mg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_set_extra_kwds\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mGraphics\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_extract_kwds_for_show\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0moptions\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mignore\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m'xmin'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'xmax'\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--> 404\u001b[0;31m     g.add_primitive(ComplexPlot(complex_to_rgb(z_values),\n\u001b[0m\u001b[1;32m    405\u001b[0m                                 x_range[:2], y_range[:2], options))\n\u001b[1;32m    406\u001b[0m     \u001b[0;32mreturn\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/Downloads/SageMath/local/lib/python3.9/site-packages/sage/plot/complex_plot.pyx\u001b[0m in \u001b[0;36msage.plot.complex_plot.complex_to_rgb (build/cythonized/sage/plot/complex_plot.c:3994)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    113\u001b[0m                 \u001b[0mz\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m<\u001b[0m\u001b[0mComplexDoubleElement\u001b[0m\u001b[0;34m>\u001b[0m\u001b[0mzz\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    114\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--> 115\u001b[0;31m                 \u001b[0mz\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mCDF\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mzz\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    116\u001b[0m             \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mz\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_complex\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdat\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    117\u001b[0m             \u001b[0mmag\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mhypot\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[0m\n",
      "\u001b[0;32m~/Downloads/SageMath/local/lib/python3.9/site-packages/sage/rings/complex_double.pyx\u001b[0m in \u001b[0;36msage.rings.complex_double.ComplexDoubleField_class.__call__ (build/cythonized/sage/rings/complex_double.c:5419)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    334\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mim\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[1;32m    335\u001b[0m             \u001b[0mx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mim\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 336\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mParent\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__call__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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[0m\u001b[1;32m    337\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    338\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m_element_constructor_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/Downloads/SageMath/local/lib/python3.9/site-packages/sage/structure/parent.pyx\u001b[0m in \u001b[0;36msage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9335)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    896\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mmor\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[1;32m    897\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mno_extra_args\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 898\u001b[0;31m                 \u001b[0;32mreturn\u001b[0m \u001b[0mmor\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_call_\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[0m\u001b[1;32m    899\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    900\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mmor\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_call_with_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\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~/Downloads/SageMath/local/lib/python3.9/site-packages/sage/structure/coerce_maps.pyx\u001b[0m in \u001b[0;36msage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4622)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    159\u001b[0m                 \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    160\u001b[0m                 \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_element_constructor\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_element_constructor\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 161\u001b[0;31m             \u001b[0;32mraise\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    162\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    163\u001b[0m     \u001b[0mcpdef\u001b[0m \u001b[0mElement\u001b[0m \u001b[0m_call_with_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\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[0mkwds\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[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/Downloads/SageMath/local/lib/python3.9/site-packages/sage/structure/coerce_maps.pyx\u001b[0m in \u001b[0;36msage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4514)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    154\u001b[0m         \u001b[0mcdef\u001b[0m \u001b[0mParent\u001b[0m \u001b[0mC\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_codomain\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    155\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--> 156\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_element_constructor\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[0m\u001b[1;32m    157\u001b[0m         \u001b[0;32mexcept\u001b[0m \u001b[0mException\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    158\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mprint_warnings\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/Downloads/SageMath/local/lib/python3.9/site-packages/sage/rings/complex_double.pyx\u001b[0m in \u001b[0;36msage.rings.complex_double.ComplexDoubleField_class._element_constructor_ (build/cythonized/sage/rings/complex_double.c:5977)\u001b[0;34m()\u001b[0m\n\u001b[1;32m    366\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mt\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    367\u001b[0m         \u001b[0;32melif\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'_complex_double_'\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--> 368\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_complex_double_\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    369\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    370\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mComplexDoubleElement\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\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~/Downloads/SageMath/local/lib/python3.9/site-packages/sage/symbolic/expression.pyx\u001b[0m in \u001b[0;36msage.symbolic.expression.Expression._complex_double_ (build/cythonized/sage/symbolic/expression.cpp:11835)\u001b[0;34m()\u001b[0m\n\u001b[1;32m   1663\u001b[0m             \u001b[0;36m0.5000000000000001\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;36m0.8660254037844386\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mI\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1664\u001b[0m         \"\"\"\n\u001b[0;32m-> 1665\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_eval_self\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mR\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   1666\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1667\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m__float__\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~/Downloads/SageMath/local/lib/python3.9/site-packages/sage/symbolic/expression.pyx\u001b[0m in \u001b[0;36msage.symbolic.expression.Expression._eval_self (build/cythonized/sage/symbolic/expression.cpp:10179)\u001b[0;34m()\u001b[0m\n\u001b[1;32m   1430\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mR\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mans\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1431\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-> 1432\u001b[0;31m             \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"Cannot evaluate symbolic expression to a numeric value.\"\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   1433\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1434\u001b[0m     \u001b[0mcpdef\u001b[0m \u001b[0m_convert\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\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[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mTypeError\u001b[0m: Cannot evaluate symbolic expression to a numeric value."
     ]
    }
   ],
   "source": [
    "complex_plot(sol[0].subs(x=z), (x,-1, 1), (y,-1,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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