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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 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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}
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