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    "# Solving non-linear equations\n",
    "\n",
    "^ca27a3\n",
    "\n",
    "## Introduction\n",
    "\n",
    "## Prerequisites\n",
    "\n",
    "``` python\n",
    "import numpy\n",
    "import scipy\n",
    "import sympy\n",
    "```\n",
    "\n",
    "## fsolve from SciPy\n",
    "\n",
    "``` python\n",
    "from scipy.optimize import fsolve\n",
    "\n",
    "def equations(vars):\n",
    "    x, y = vars\n",
    "    eq1 = x**2 + y**2 - 25\n",
    "    eq2 = x**2 - y\n",
    "    return [eq1, eq2]\n",
    "\n",
    "initial_guess = [1, 1]\n",
    "solution = fsolve(equations, initial_guess)\n",
    "print(\"Solution:\", solution)\n",
    "```\n",
    "\n",
    "## root from SciPy\n",
    "\n",
    "``` python\n",
    "from scipy.optimize import root\n",
    "\n",
    "def equations(vars):\n",
    "    x, y = vars\n",
    "    eq1 = x**2 + y**2 - 25\n",
    "    eq2 = x**2 - y\n",
    "    return [eq1, eq2]\n",
    "\n",
    "initial_guess = [1, 1]\n",
    "solution = root(equations, initial_guess)\n",
    "print(\"Solution:\", solution.x)\n",
    "```\n",
    "\n",
    "## minimize from SciPy\n",
    "\n",
    "``` python\n",
    "from scipy.optimize import minimize\n",
    "\n",
    "# Define the equations\n",
    "def equation1(x, y):\n",
    "    return x**2 + y**2 - 25\n",
    "\n",
    "def equation2(x, y):\n",
    "    return x**2 - y\n",
    "\n",
    "# Define the objective function for optimization\n",
    "def objective(xy):\n",
    "    x, y = xy\n",
    "    return equation1(x, y)**2 + equation2(x, y)**2\n",
    "\n",
    "# Initial guess\n",
    "initial_guess = [1, 1]\n",
    "\n",
    "# Perform optimization\n",
    "result = minimize(objective, initial_guess)\n",
    "solution_optimization = result.x\n",
    "\n",
    "print(\"Optimization Method Solution:\", solution_optimization)\n",
    "```\n",
    "\n",
    "## nsolve from SymPy\n",
    "\n",
    "``` python\n",
    "from sympy import symbols, Eq, nsolve\n",
    "\n",
    "# Define the variables\n",
    "x, y = symbols('x y')\n",
    "\n",
    "# Define the equations\n",
    "eq1 = Eq(x**2 + y**2, 25)\n",
    "eq2 = Eq(x - y, 0)\n",
    "\n",
    "# Initial guess for the solution\n",
    "initial_guess = [1, 1]\n",
    "\n",
    "# Use nsolve to find the solution\n",
    "solution = nsolve([eq1, eq2], [x, y], initial_guess)\n",
    "print(\"Solution:\", solution)\n",
    "```\n",
    "\n",
    "## newton_method from NumPy\n",
    "\n",
    "``` python\n",
    "import numpy as np\n",
    "\n",
    "def equations(vars):\n",
    "    x, y = vars\n",
    "    eq1 = x**2 + y**2 - 25\n",
    "    eq2 = x**2 - y\n",
    "    return np.array([eq1, eq2])\n",
    "\n",
    "def newton_method(initial_guess, tolerance=1e-6, max_iter=100):\n",
    "    vars = np.array(initial_guess, dtype=float)\n",
    "    for _ in range(max_iter):\n",
    "        J = np.array([[2 * vars[0], 2 * vars[1]], [2 * vars[0], -1]])\n",
    "        F = equations(vars)\n",
    "        delta = np.linalg.solve(J, -F)\n",
    "        vars += delta\n",
    "        if np.linalg.norm(delta) < tolerance:\n",
    "            return vars\n",
    "\n",
    "initial_guess = [1, 1]\n",
    "solution = newton_method(initial_guess)\n",
    "print(\"Solution:\", solution)\n",
    "```"
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