07 Kalman gain and covariance.ipynb

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    "import numpy as np\n",
    "import plotly.express as ex\n",
    "import plotly.graph_objects as go\n",
    "from plotly.subplots import make_subplots\n",
    "import pandas as pd\n",
    "\n",
    "from tracking.util import to_df, colorscale"
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    "Covariace estimated by the Kalman Filter will be lower than either the initial covariance estimate, measurement noise or process noise.\n",
    "\n",
    "## No process noise\n",
    "\n",
    "Let's start with initial estimate and measurement noise being equal, no process noise. Repeated Kalman Filter updates are equivalent to an average of N normally-distributed random variables. According to the Central Limit Theorem, variance converges to $\\sigma / \\sqrt{n}$."
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    "def simulate_variance(P: float, R: float, Q: float):\n",
    "    trace = [(0, P, 1)]\n",
    "\n",
    "    for i in range(1, 100):\n",
    "        K = P / (P + R)\n",
    "        P = K**2 * R + (1-K)**2 * P + Q\n",
    "        trace.append((i, P, K))\n",
    "    \n",
    "    return np.array(trace)\n",
    "\n",
    "def simulate_variance_df(P: float, R: float, Q: float):\n",
    "    return to_df(simulate_variance(P, R, Q), columns=['t', 'P', 'K'])"
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    "def plot_variance(**kwargs):\n",
    "    col = colorscale(x=kwargs.keys())\n",
    "    fig = make_subplots(rows=1, cols=2, subplot_titles=(\"P\", \"K\"))\n",
    "    #fig.update_yaxes(type=\"log\", row=1, col=2)\n",
    "\n",
    "    for name, data in kwargs.items():\n",
    "        params = dict(x=data[:,0], mode='lines', marker=dict(color=col[name]))\n",
    "        fig.add_trace(go.Scatter(y=data[:,1], name=name, **params), row=1, col=1)\n",
    "        fig.add_trace(go.Scatter(y=data[:,2], name=name, **params, showlegend=False), row=1, col=2)\n",
    "\n",
    "    return fig"
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    "plot_variance(P_0=simulate_variance(10, 10, 2))"
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   "id": "87cce48f-d99f-402d-b134-17253ace4816",
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   "source": [
    "ex.line(simulate_variance_df(10, 10, 0).melt(id_vars=['t'], var_name='parameter'), x='t', y='value', color='parameter')"
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   "source": [
    "This is equivalent to the variance an average of N normal random variables with variance $P$. Given the [Central Limit Theorem](https://en.wikipedia.org/wiki/Central_limit_theorem):\n",
    "\n",
    "$$\n",
    "Var\\left(\\frac{1}{N}\\sum_i^N N(0, \\sigma^2)\\right) = \\frac{\\sigma^2}{N}\n",
    "$$"
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   "id": "324dfbe3-f819-4e7c-8a6d-37582682d586",
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       "8.881784197001252e-16"
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   "source": [
    "np.max(simulate_variance(10, 10, 0)[:,1] - 10/np.arange(1, 101))"
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   "source": [
    "## With process noise\n",
    "\n",
    "If we introduce process noise, variance converges to a value lower than measurement noise but higher than process noise."
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       "}});\n",
       "\n",
       "// Listen for the removal of the full notebook cells\n",
       "var notebookContainer = gd.closest('#notebook-container');\n",
       "if (notebookContainer) {{\n",
       "    x.observe(notebookContainer, {childList: true});\n",
       "}}\n",
       "\n",
       "// Listen for the clearing of the current output cell\n",
       "var outputEl = gd.closest('.output');\n",
       "if (outputEl) {{\n",
       "    x.observe(outputEl, {childList: true});\n",
       "}}\n",
       "\n",
       "                        })                };                });            </script>        </div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "data = pd.DataFrame(dict(P_2=simulate_variance(10, 10, 2),\n",
    "                         P_3=simulate_variance(10, 10, 3),\n",
    "                         P_4=simulate_variance(10, 10, 4),\n",
    "                         P_5=simulate_variance(10, 10, 5),\n",
    "                         time=range(101)))\n",
    "data = data.melt(id_vars=['time'])\n",
    "ex.line(data, x='time', y='value', color='variable')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bbb72901-1d9f-4e32-a736-9d2cb6d9a256",
   "metadata": {},
   "source": [
    "The limiting variance can be calculate from the following:\n",
    "\n",
    "$$\n",
    "K = \\frac{P}{P+R}\n",
    "$$\n",
    "$$\n",
    "P = K^2 R + (1-K)^2 P + Q\n",
    "$$\n",
    "\n",
    "which can be refomulated as:\n",
    "\n",
    "$$\n",
    "P^2 R + R^2 P = (P-Q)(P+R)^2\n",
    "$$\n",
    "\n",
    "which for $P=R=10$ and $Q=2$ result in $P=1+\\sqrt{21}\\approx5.58$ while for $Q=5$ results in $P=10$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "id": "dfb2ce65-02f1-4093-aa50-fe428719eb10",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7603.84"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "87.2**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 262,
   "id": "e5e2177b-ab7f-4e29-b250-73904537ff3a",
   "metadata": {},
   "outputs": [],
   "source": [
    "H = np.array([[1, 0]])\n",
    "P = np.eye(2) * 1\n",
    "R = np.eye(1) * 1\n",
    "Q = np.eye(2) \n",
    "F = np.array([[1, 1], [0, 1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 263,
   "id": "3ef2aba5-a0a0-44dd-9f3b-76d3acede936",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([[0.82184641, 0.42208244],\n",
       "        [0.42208244, 1.94712297]]),\n",
       " array([[0.82184641],\n",
       "        [0.42208244]]))"
      ]
     },
     "execution_count": 263,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "for _ in range(100):\n",
    "    P = F @ P @ F.T + Q\n",
    "    S = H @ P @ H.T + R\n",
    "    K = P @ H.T @ np.linalg.inv(S)\n",
    "    \n",
    "    P = P - K @ S @ K.T\n",
    "    \n",
    "P, K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 145,
   "id": "41d2fccc-b2ed-444f-ac82-35357c18151c",
   "metadata": {},
   "outputs": [],
   "source": [
    "class KalmanGain:\n",
    "    def __init__(self, Px, Pv, Pxv, Qx, Qv, R):\n",
    "        self.Px = Px\n",
    "        self.Pv = Pv\n",
    "        self.Pxv = Pxv\n",
    "        self.Qx = Qx\n",
    "        self.Qv = Qv\n",
    "        self.R = R\n",
    "\n",
    "    @property\n",
    "    def S(self):\n",
    "        return self.Px + 2*self.Pxv + self.Pv + self.Qx + self.R\n",
    "\n",
    "    @property\n",
    "    def Kx(self):\n",
    "        return (self.Px + 2*self.Pxv + self.Pv + self.Qx) / self.S\n",
    "\n",
    "    @property\n",
    "    def Kv(self):\n",
    "        return (self.Pxv + self.Pv) / self.S"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 264,
   "id": "551a6a6a-9c3d-4bb2-9fc9-f65e4b624b94",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.811419670437016, 0.4342590113110481)"
      ]
     },
     "execution_count": 264,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "kg = KalmanGain(0.81142, 2.1142, 0.18858, 1, 1, 1)\n",
    "kg.Kx, kg.Kv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 241,
   "id": "72006673-7aab-462c-a719-61321942a1be",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.solvers import solve, solveset, nsolve, nonlinsolve\n",
    "from sympy import Symbol, symbols, N\n",
    "from sympy.matrices import Matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 251,
   "id": "3d5dbf57-cc1c-4268-96ed-74606968a5fb",
   "metadata": {},
   "outputs": [],
   "source": [
    "Px, Pv, Pxv = symbols(\"P_x, P_v, P_xv\", positive=True)\n",
    "Qx, Qv, R, t = 1, 1, 1, 1\n",
    "\n",
    "F = Matrix([[1, t], [0, 1]])\n",
    "H = Matrix([[1, 0]])\n",
    "Q = Matrix([[Qx, 0], [0, Qv]])\n",
    "Rm = Matrix([R])\n",
    "\n",
    "P = Matrix([[Px, Pxv], [Pxv, Pv]])\n",
    "Pn = F*P*F.T + Q\n",
    "S = H*Pn*H.T + Rm\n",
    "K = Pn*H.T*S.inv()\n",
    "\n",
    "X = P - Pn + K*S*K.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 252,
   "id": "c8388e7c-e7cd-4c54-9dcf-dc13ee631101",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}P_{v} + P_{x} + 2 P_{xv} + 1 & P_{v} + P_{xv}\\\\P_{v} + P_{xv} & P_{v} + 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[P_v + P_x + 2*P_xv + 1, P_v + P_xv],\n",
       "[            P_v + P_xv,    P_v + 1]])"
      ]
     },
     "execution_count": 252,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 253,
   "id": "8f307fa0-4dc9-4662-9afd-a396fb4c362b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}P_{v} + P_{x} + 2 P_{xv} + 2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([[P_v + P_x + 2*P_xv + 2]])"
      ]
     },
     "execution_count": 253,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 254,
   "id": "e9feacdf-6370-480a-b16d-8fa239e6b332",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{P_{v} + P_{x} + 2 P_{xv} + 1}{P_{v} + P_{x} + 2 P_{xv} + 2}\\\\\\frac{P_{v} + P_{xv}}{P_{v} + P_{x} + 2 P_{xv} + 2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[(P_v + P_x + 2*P_xv + 1)/(P_v + P_x + 2*P_xv + 2)],\n",
       "[            (P_v + P_xv)/(P_v + P_x + 2*P_xv + 2)]])"
      ]
     },
     "execution_count": 254,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 255,
   "id": "84c7847c-0a25-4204-a241-57d00230e958",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}P_{v} + P_{x} + 2 P_{xv} - \\frac{\\left(P_{v} + P_{x} + 2 P_{xv} + 1\\right)^{2}}{P_{v} + P_{x} + 2 P_{xv} + 2} + 1 & P_{v} + P_{xv} - \\frac{\\left(P_{v} + P_{xv}\\right) \\left(P_{v} + P_{x} + 2 P_{xv} + 1\\right)}{P_{v} + P_{x} + 2 P_{xv} + 2}\\\\P_{v} + P_{xv} - \\frac{\\left(P_{v} + P_{xv}\\right) \\left(P_{v} + P_{x} + 2 P_{xv} + 1\\right)}{P_{v} + P_{x} + 2 P_{xv} + 2} & P_{v} - \\frac{\\left(P_{v} + P_{xv}\\right)^{2}}{P_{v} + P_{x} + 2 P_{xv} + 2} + 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[P_v + P_x + 2*P_xv - (P_v + P_x + 2*P_xv + 1)**2/(P_v + P_x + 2*P_xv + 2) + 1, P_v + P_xv - (P_v + P_xv)*(P_v + P_x + 2*P_xv + 1)/(P_v + P_x + 2*P_xv + 2)],\n",
       "[  P_v + P_xv - (P_v + P_xv)*(P_v + P_x + 2*P_xv + 1)/(P_v + P_x + 2*P_xv + 2),                          P_v - (P_v + P_xv)**2/(P_v + P_x + 2*P_xv + 2) + 1]])"
      ]
     },
     "execution_count": 255,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pn - K*S*K.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 256,
   "id": "dc034eb5-dbeb-4dbd-a104-5d4f02a32e63",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- P_{v} - 2 P_{xv} + \\frac{\\left(P_{v} + P_{x} + 2 P_{xv} + 1\\right)^{2}}{P_{v} + P_{x} + 2 P_{xv} + 2} - 1 & - P_{v} + \\frac{\\left(P_{v} + P_{xv}\\right) \\left(P_{v} + P_{x} + 2 P_{xv} + 1\\right)}{P_{v} + P_{x} + 2 P_{xv} + 2}\\\\- P_{v} + \\frac{\\left(P_{v} + P_{xv}\\right) \\left(P_{v} + P_{x} + 2 P_{xv} + 1\\right)}{P_{v} + P_{x} + 2 P_{xv} + 2} & \\frac{\\left(P_{v} + P_{xv}\\right)^{2}}{P_{v} + P_{x} + 2 P_{xv} + 2} - 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-P_v - 2*P_xv + (P_v + P_x + 2*P_xv + 1)**2/(P_v + P_x + 2*P_xv + 2) - 1, -P_v + (P_v + P_xv)*(P_v + P_x + 2*P_xv + 1)/(P_v + P_x + 2*P_xv + 2)],\n",
       "[   -P_v + (P_v + P_xv)*(P_v + P_x + 2*P_xv + 1)/(P_v + P_x + 2*P_xv + 2),                          (P_v + P_xv)**2/(P_v + P_x + 2*P_xv + 2) - 1]])"
      ]
     },
     "execution_count": 256,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 258,
   "id": "f6b49b74-f9fe-4265-972c-234ad94dc0fc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left\\{\\left( - \\frac{3}{4} + \\frac{\\sqrt{21}}{4} - \\frac{\\sqrt{6 - 2 \\sqrt{21}}}{8} + \\frac{\\sqrt{126 - 42 \\sqrt{21}}}{8}, \\  - \\frac{\\sqrt{6 - 2 \\sqrt{21}}}{2}, \\  - \\frac{\\sqrt{21}}{4} + \\frac{1}{4} + \\frac{\\sqrt{2} \\sqrt{3 - \\sqrt{21}}}{4}\\right), \\left( - \\frac{3}{4} + \\frac{\\sqrt{21}}{4} - \\frac{\\sqrt{126 - 42 \\sqrt{21}}}{8} + \\frac{\\sqrt{6 - 2 \\sqrt{21}}}{8}, \\  \\frac{\\sqrt{6 - 2 \\sqrt{21}}}{2}, \\  - \\frac{\\sqrt{21}}{4} + \\frac{1}{4} - \\frac{\\sqrt{2} \\sqrt{3 - \\sqrt{21}}}{4}\\right), \\left( - \\frac{\\sqrt{21}}{4} - \\frac{3}{4} + \\frac{\\sqrt{6 + 2 \\sqrt{21}}}{8} + \\frac{\\sqrt{126 + 42 \\sqrt{21}}}{8}, \\  \\frac{\\sqrt{6 + 2 \\sqrt{21}}}{2}, \\  - \\frac{\\sqrt{2} \\sqrt{3 + \\sqrt{21}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{21}}{4}\\right), \\left( - \\frac{\\sqrt{126 + 42 \\sqrt{21}}}{8} - \\frac{\\sqrt{21}}{4} - \\frac{3}{4} - \\frac{\\sqrt{6 + 2 \\sqrt{21}}}{8}, \\  - \\frac{\\sqrt{6 + 2 \\sqrt{21}}}{2}, \\  \\frac{1}{4} + \\frac{\\sqrt{2} \\sqrt{3 + \\sqrt{21}}}{4} + \\frac{\\sqrt{21}}{4}\\right)\\right\\}$"
      ],
      "text/plain": [
       "{(-3/4 + sqrt(21)/4 - sqrt(6 - 2*sqrt(21))/8 + sqrt(126 - 42*sqrt(21))/8, -sqrt(6 - 2*sqrt(21))/2, -sqrt(21)/4 + 1/4 + sqrt(2)*sqrt(3 - sqrt(21))/4), (-3/4 + sqrt(21)/4 - sqrt(126 - 42*sqrt(21))/8 + sqrt(6 - 2*sqrt(21))/8, sqrt(6 - 2*sqrt(21))/2, -sqrt(21)/4 + 1/4 - sqrt(2)*sqrt(3 - sqrt(21))/4), (-sqrt(21)/4 - 3/4 + sqrt(6 + 2*sqrt(21))/8 + sqrt(126 + 42*sqrt(21))/8, sqrt(6 + 2*sqrt(21))/2, -sqrt(2)*sqrt(3 + sqrt(21))/4 + 1/4 + sqrt(21)/4), (-sqrt(126 + 42*sqrt(21))/8 - sqrt(21)/4 - 3/4 - sqrt(6 + 2*sqrt(21))/8, -sqrt(6 + 2*sqrt(21))/2, 1/4 + sqrt(2)*sqrt(3 + sqrt(21))/4 + sqrt(21)/4)}"
      ]
     },
     "execution_count": 258,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res = nonlinsolve([X[0,0], X[0, 1], X[1, 1]], [Px, Pv, Pxv])\n",
    "res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 259,
   "id": "6490814b-3084-4af3-8a4e-89a22bdb7b7c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0.821846413518260, 1.94712296670701, 0.422082440385453),\n",
       " (-4.61313426099618, -1.94712296670701, 2.36920540709247),\n",
       " (0.39564392373896 + 0.796714335936263*I,\n",
       "  -0.889543617524132*I,\n",
       "  -0.89564392373896 + 0.444771808762066*I),\n",
       " (0.39564392373896 - 0.796714335936263*I,\n",
       "  0.889543617524132*I,\n",
       "  -0.89564392373896 - 0.444771808762066*I)]"
      ]
     },
     "execution_count": 259,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "num = [tuple([x.evalf() for x in solution]) for solution in res]\n",
    "num"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 261,
   "id": "7583d014-be94-4bab-9ba0-41f312e44efb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.821846413518260, 0.422082440385453)"
      ]
     },
     "execution_count": 261,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "kg = KalmanGain(*num[0], Qx, Qv, R)\n",
    "kg.Kx, kg.Kv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 202,
   "id": "7fe159cb-dfe9-45dd-9397-56f4789ca0a7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\u001b[0;31mSignature:\u001b[0m \u001b[0mnonlinsolve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msystem\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0msymbols\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
       "\u001b[0;31mDocstring:\u001b[0m\n",
       "Solve system of $N$ nonlinear equations with $M$ variables, which means both\n",
       "under and overdetermined systems are supported. Positive dimensional\n",
       "system is also supported (A system with infinitely many solutions is said\n",
       "to be positive-dimensional). In a positive dimensional system the solution will\n",
       "be dependent on at least one symbol. Returns both real solution\n",
       "and complex solution (if they exist).\n",
       "\n",
       "Parameters\n",
       "==========\n",
       "\n",
       "system : list of equations\n",
       "    The target system of equations\n",
       "symbols : list of Symbols\n",
       "    symbols should be given as a sequence eg. list\n",
       "\n",
       "Returns\n",
       "=======\n",
       "\n",
       "A :class:`~.FiniteSet` of ordered tuple of values of `symbols` for which the `system`\n",
       "has solution. Order of values in the tuple is same as symbols present in\n",
       "the parameter `symbols`.\n",
       "\n",
       "Please note that general :class:`~.FiniteSet` is unordered, the solution\n",
       "returned here is not simply a :class:`~.FiniteSet` of solutions, rather it\n",
       "is a :class:`~.FiniteSet` of ordered tuple, i.e. the first and only\n",
       "argument to :class:`~.FiniteSet` is a tuple of solutions, which is\n",
       "ordered, and, hence ,the returned solution is ordered.\n",
       "\n",
       "Also note that solution could also have been returned as an ordered tuple,\n",
       "FiniteSet is just a wrapper ``{}`` around the tuple. It has no other\n",
       "significance except for the fact it is just used to maintain a consistent\n",
       "output format throughout the solveset.\n",
       "\n",
       "For the given set of equations, the respective input types\n",
       "are given below:\n",
       "\n",
       ".. math:: xy - 1 = 0\n",
       ".. math:: 4x^2 + y^2 - 5 = 0\n",
       "\n",
       "::\n",
       "\n",
       "   system  = [x*y - 1, 4*x**2 + y**2 - 5]\n",
       "   symbols = [x, y]\n",
       "\n",
       "Raises\n",
       "======\n",
       "\n",
       "ValueError\n",
       "    The input is not valid.\n",
       "    The symbols are not given.\n",
       "AttributeError\n",
       "    The input symbols are not `Symbol` type.\n",
       "\n",
       "Examples\n",
       "========\n",
       "\n",
       ">>> from sympy import symbols, nonlinsolve\n",
       ">>> x, y, z = symbols('x, y, z', real=True)\n",
       ">>> nonlinsolve([x*y - 1, 4*x**2 + y**2 - 5], [x, y])\n",
       "{(-1, -1), (-1/2, -2), (1/2, 2), (1, 1)}\n",
       "\n",
       "1. Positive dimensional system and complements:\n",
       "\n",
       ">>> from sympy import pprint\n",
       ">>> from sympy.polys.polytools import is_zero_dimensional\n",
       ">>> a, b, c, d = symbols('a, b, c, d', extended_real=True)\n",
       ">>> eq1 =  a + b + c + d\n",
       ">>> eq2 = a*b + b*c + c*d + d*a\n",
       ">>> eq3 = a*b*c + b*c*d + c*d*a + d*a*b\n",
       ">>> eq4 = a*b*c*d - 1\n",
       ">>> system = [eq1, eq2, eq3, eq4]\n",
       ">>> is_zero_dimensional(system)\n",
       "False\n",
       ">>> pprint(nonlinsolve(system, [a, b, c, d]), use_unicode=False)\n",
       "  -1       1               1      -1\n",
       "{(---, -d, -, {d} \\ {0}), (-, -d, ---, {d} \\ {0})}\n",
       "   d       d               d       d\n",
       ">>> nonlinsolve([(x+y)**2 - 4, x + y - 2], [x, y])\n",
       "{(2 - y, y)}\n",
       "\n",
       "2. If some of the equations are non-polynomial then `nonlinsolve`\n",
       "will call the ``substitution`` function and return real and complex solutions,\n",
       "if present.\n",
       "\n",
       ">>> from sympy import exp, sin\n",
       ">>> nonlinsolve([exp(x) - sin(y), y**2 - 4], [x, y])\n",
       "{(ImageSet(Lambda(_n, I*(2*_n*pi + pi) + log(sin(2))), Integers), -2),\n",
       " (ImageSet(Lambda(_n, 2*_n*I*pi + log(sin(2))), Integers), 2)}\n",
       "\n",
       "3. If system is non-linear polynomial and zero-dimensional then it\n",
       "returns both solution (real and complex solutions, if present) using\n",
       ":func:`~.solve_poly_system`:\n",
       "\n",
       ">>> from sympy import sqrt\n",
       ">>> nonlinsolve([x**2 - 2*y**2 -2, x*y - 2], [x, y])\n",
       "{(-2, -1), (2, 1), (-sqrt(2)*I, sqrt(2)*I), (sqrt(2)*I, -sqrt(2)*I)}\n",
       "\n",
       "4. ``nonlinsolve`` can solve some linear (zero or positive dimensional)\n",
       "system (because it uses the :func:`sympy.polys.polytools.groebner` function to get the\n",
       "groebner basis and then uses the ``substitution`` function basis as the\n",
       "new `system`). But it is not recommended to solve linear system using\n",
       "``nonlinsolve``, because :func:`~.linsolve` is better for general linear systems.\n",
       "\n",
       ">>> nonlinsolve([x + 2*y -z - 3, x - y - 4*z + 9, y + z - 4], [x, y, z])\n",
       "{(3*z - 5, 4 - z, z)}\n",
       "\n",
       "5. System having polynomial equations and only real solution is\n",
       "solved using :func:`~.solve_poly_system`:\n",
       "\n",
       ">>> e1 = sqrt(x**2 + y**2) - 10\n",
       ">>> e2 = sqrt(y**2 + (-x + 10)**2) - 3\n",
       ">>> nonlinsolve((e1, e2), (x, y))\n",
       "{(191/20, -3*sqrt(391)/20), (191/20, 3*sqrt(391)/20)}\n",
       ">>> nonlinsolve([x**2 + 2/y - 2, x + y - 3], [x, y])\n",
       "{(1, 2), (1 - sqrt(5), 2 + sqrt(5)), (1 + sqrt(5), 2 - sqrt(5))}\n",
       ">>> nonlinsolve([x**2 + 2/y - 2, x + y - 3], [y, x])\n",
       "{(2, 1), (2 - sqrt(5), 1 + sqrt(5)), (2 + sqrt(5), 1 - sqrt(5))}\n",
       "\n",
       "6. It is better to use symbols instead of trigonometric functions or\n",
       ":class:`~.Function`. For example, replace $\\sin(x)$ with a symbol, replace\n",
       "$f(x)$ with a symbol and so on. Get a solution from ``nonlinsolve`` and then\n",
       "use :func:`~.solveset` to get the value of $x$.\n",
       "\n",
       "How nonlinsolve is better than old solver ``_solve_system`` :\n",
       "=============================================================\n",
       "\n",
       "1. A positive dimensional system solver: nonlinsolve can return\n",
       "solution for positive dimensional system. It finds the\n",
       "Groebner Basis of the positive dimensional system(calling it as\n",
       "basis) then we can start solving equation(having least number of\n",
       "variable first in the basis) using solveset and substituting that\n",
       "solved solutions into other equation(of basis) to get solution in\n",
       "terms of minimum variables. Here the important thing is how we\n",
       "are substituting the known values and in which equations.\n",
       "\n",
       "2. Real and complex solutions: nonlinsolve returns both real\n",
       "and complex solution. If all the equations in the system are polynomial\n",
       "then using :func:`~.solve_poly_system` both real and complex solution is returned.\n",
       "If all the equations in the system are not polynomial equation then goes to\n",
       "``substitution`` method with this polynomial and non polynomial equation(s),\n",
       "to solve for unsolved variables. Here to solve for particular variable\n",
       "solveset_real and solveset_complex is used. For both real and complex\n",
       "solution ``_solve_using_known_values`` is used inside ``substitution``\n",
       "(``substitution`` will be called when any non-polynomial equation is present).\n",
       "If a solution is valid its general solution is added to the final result.\n",
       "\n",
       "3. :class:`~.Complement` and :class:`~.Intersection` will be added:\n",
       "nonlinsolve maintains dict for complements and intersections. If solveset\n",
       "find complements or/and intersections with any interval or set during the\n",
       "execution of ``substitution`` function, then complement or/and\n",
       "intersection for that variable is added before returning final solution.\n",
       "\u001b[0;31mFile:\u001b[0m      ~/anaconda3/envs/fusion/lib/python3.11/site-packages/sympy/solvers/solveset.py\n",
       "\u001b[0;31mType:\u001b[0m      function"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "?nonlinsolve"
   ]
  },
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   "metadata": {},
   "outputs": [],
   "source": []
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