{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "5f35bc8a-ca77-4d10-8d37-6ef1ea61da36",
   "metadata": {},
   "source": [
    "# Import modules:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "9b0779af-b27a-4d61-b425-100d71ef8f55",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "422b2cc5-5567-4670-8c80-a067810e0241",
   "metadata": {},
   "source": [
    "# Define Newton Rafson function (it checked if the derivative is zero):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "ffd82ddf-658c-4dcc-bc15-34684b584edf",
   "metadata": {},
   "outputs": [],
   "source": [
    "def newton_method(f, df, x0, tolerance=1e-6, max_iterations=100):\n",
    "    \"\"\"\n",
    "    Newton's method for solving f(x) = 0.\n",
    "\n",
    "    Parameters:\n",
    "        f              : function f(x)\n",
    "        df             : derivative f'(x)\n",
    "        x0             : initial guess\n",
    "        tolerance      : stopping tolerance\n",
    "        max_iterations : maximum number of iterations\n",
    "\n",
    "    Returns:\n",
    "        Approximate root\n",
    "    \"\"\"\n",
    "\n",
    "    xn = x0\n",
    "\n",
    "    for it in np.arange(max_iterations):\n",
    "        # Avoid division by zero\n",
    "        if df(xn) == 0:\n",
    "            print(\"Derivative is zero. Newton's method cannot continue.\")\n",
    "            return None\n",
    "\n",
    "        # Newton's method formula\n",
    "        xn1 = xn - f(xn) / df(xn)\n",
    "\n",
    "        # Check convergence\n",
    "        if abs(xn1 - xn) < tolerance:\n",
    "            print(f\"Converged after {it + 1} iterations.\")\n",
    "            return xn1\n",
    "\n",
    "        xn = xn1\n",
    "\n",
    "    print(\"Maximum number of iterations reached.\")\n",
    "    return xn"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5cdf4a00-2967-4f2b-9bd8-cbdf64596a68",
   "metadata": {},
   "source": [
    "# Define a test function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "06443c7f-1250-4b17-a5f6-b281469455e9",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Converged after 13 iterations.\n",
      "Root: 2.0\n"
     ]
    }
   ],
   "source": [
    "def f1(x):\n",
    "    return x**2 - 4\n",
    "\n",
    "def df1(x):\n",
    "    return 2*x\n",
    "\n",
    "root = newton_method(f1, df1, x0=600)\n",
    "\n",
    "print(\"Root:\", root)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b25e5389-4b86-438c-92de-bc743cec009b",
   "metadata": {},
   "source": [
    "# Define a trigo function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "61f23c0e-dc76-4901-b3c3-54d147a26948",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Converged after 99 iterations.\n",
      "Root: 0.7390851332151611\n"
     ]
    }
   ],
   "source": [
    "def f2(x):\n",
    "    return np.cos(x) - x\n",
    "\n",
    "def df2(x):\n",
    "    return -np.sin(x) - 1\n",
    "\n",
    "root = newton_method(f2, df2, x0=200)\n",
    "\n",
    "print(\"Root:\", root)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "94631020-96a9-4d64-923d-4e6c976a8133",
   "metadata": {},
   "source": [
    "# Oscillating function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "ac6832df-fb35-4cd1-92e5-0d1cf84f856c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Maximum number of iterations reached.\n",
      "Root: 0.0\n"
     ]
    }
   ],
   "source": [
    "def f3(x):\n",
    "    return x**3 - 2*x + 2\n",
    "\n",
    "def df3(x):\n",
    "    return 3*x**2 - 2\n",
    "\n",
    "root = newton_method(f3, df3, x0=0)\n",
    "\n",
    "print(\"Root:\", root)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "98a83f74-192d-4bfa-aa0f-c4c0299c13c1",
   "metadata": {},
   "outputs": [],
   "source": [
    "x0 = 0\n",
    "\n",
    "nt = 100\n",
    "xn = np.empty(nt+1)\n",
    "xn[0]=0\n",
    "\n",
    "for it in np.arange(nt):\n",
    "    #print(f\"x_{i} = {x}\")\n",
    "\n",
    "    xn[it+1] = xn[it] - f(xn[it]) / df(xn[it])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "1f8e46fb-f5a7-4ad3-bcec-f2c31a7d396a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f52d8fdd8e0>]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(xn,'+')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c8a2aa1e-2282-4012-8de9-b3e7344221fc",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Numeric_Env",
   "language": "python",
   "name": "numeric_env"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.12.14"
  }
 },
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