{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Решение нелинейных уравнений\n",
    "\n",
    "Подмодуль [scipy.optimize](https://docs.scipy.org/doc/scipy/reference/optimize.html#module-scipy.optimize) также содержит в себе методы для поиска корней нелинейных уравнений и их систем."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Поиск корня скалярной функции одного аргумента\n",
    "\n",
    "Функция [scipy.optimize.root_scalar](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.root_scalar.html#scipy.optimize.root_scalar) позволяет искать корни функции $f: \\mathbb{R} \\to \\mathbb{R}$:\n",
    "\n",
    "$$\n",
    "f(x) = 0.\n",
    "$$\n",
    "\n",
    "Функция `root_scalar` предоставляет доступ к разным методам поиска корней, таким как `newton`, `bisect`, `secant` и многим другим. Какие-то из этих методов ищут корень внутри отрезка `bracket`, а другие ищут корень, начиная с какого-то начального приближения `x0`. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "      converged: True\n",
      "           flag: 'converged'\n",
      " function_calls: 46\n",
      "     iterations: 44\n",
      "           root: 1.0000000000002274\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy import optimize\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "\n",
    "def f(x):\n",
    "    return x**3 - 1\n",
    "\n",
    "def fprime(x):\n",
    "    return 3 * x**2\n",
    "\n",
    "\n",
    "sol = optimize.root_scalar(f, bracket=[-10, 10], method=\"bisect\")\n",
    "print(sol)\n",
    "\n",
    "\n",
    "x = np.linspace(-3, 3, 100)\n",
    "plt.plot(x, f(x))\n",
    "plt.scatter(sol.root, f(sol.root), color=\"r\")\n",
    "plt.xticks([1])\n",
    "plt.yticks([0])\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"y\")\n",
    "plt.legend([\"f(x)\", \"Корень\"])\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Метод бисекции сошелся за 44 итерации. Проверим метод Ньютона."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "      converged: True\n",
      "           flag: 'converged'\n",
      " function_calls: 22\n",
      "     iterations: 11\n",
      "           root: 1.0\n"
     ]
    }
   ],
   "source": [
    "sol = optimize.root_scalar(f, fprime=fprime, x0=10, method=\"newton\")\n",
    "print(sol)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Метод Ньютона сошелся за 11 итераций."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Решение системы нелинейных уравнений.\n",
    "\n",
    "Функция $F$ предназначена для поиска корней уравнений вида\n",
    "\n",
    "$$\n",
    "F(x) = 0,\n",
    "$$\n",
    "\n",
    "где $x$ и $F(x)$ многомерны, т.е. таких систем \n",
    "\n",
    "$$\n",
    "\\begin{cases}\n",
    "f_1(x_1, ..., x_n) = 0 \\\\\n",
    "\\cdots \\\\\n",
    "f_n(x_1, ..., x_n) = 0\n",
    "\\end{cases}.\n",
    "$$\n",
    "\n",
    "Рассмотрим поиск корня на примере функции \n",
    "\n",
    "$$\n",
    "\\begin{cases}\n",
    "x_1 + \\frac{(x_1 - x_2)^3}{2} = 1 \\\\\n",
    "\\frac{(x_2 - x_1)^3}{2} + x_2 = 1\n",
    "\\end{cases}.\n",
    "$$\n",
    "\n",
    "Матрица Якоби этого уравнения имеет вид\n",
    "\n",
    "$$\n",
    "\\begin{pmatrix}\n",
    "1 + \\frac{3}{2}(x_2 - x_1)^2 & \\frac{3}{2}(x_2 - x_1)^2 \\\\\n",
    "\\frac{3}{2}(x_2 - x_1)^2 & 1 + \\frac{3}{2}(x_2 - x_1)^2\n",
    "\\end{pmatrix},\n",
    "$$\n",
    "\n",
    "а единственный действительный корень которой\n",
    "\n",
    "$$\n",
    "\\begin{cases}\n",
    "x_1 = 1 \\\\\n",
    "x_2 = 1\n",
    "\\end{cases}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    fjac: array([[-0.89914291,  0.43765515],\n",
      "       [-0.43765515, -0.89914291]])\n",
      "     fun: array([-1.11022302e-16,  0.00000000e+00])\n",
      " message: 'The solution converged.'\n",
      "    nfev: 12\n",
      "     qtf: array([ 1.19565972e-11, -4.12770392e-12])\n",
      "       r: array([-2.16690469,  1.03701789, -1.10605417])\n",
      "  status: 1\n",
      " success: True\n",
      "       x: array([0.8411639, 0.1588361])\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy import optimize\n",
    "from matplotlib import pyplot as plt\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "\n",
    "def f(x):\n",
    "    x = np.array(x)\n",
    "    return np.array([\n",
    "        x[0]  + 0.5 * (x[0] - x[1])**3 - 1.0,\n",
    "        0.5 * (x[1] - x[0])**3 + x[1] - 1.0\n",
    "        ])\n",
    "\n",
    "\n",
    "def jac(x):\n",
    "    return np.array([[1 + 1.5 * (x[0] - x[1])**2,\n",
    "                      -1.5 * (x[0] - x[1])**2],\n",
    "                     [-1.5 * (x[1] - x[0])**2,\n",
    "                      1 + 1.5 * (x[1] - x[0])**2]])\n",
    "\n",
    "\n",
    "root = optimize.root(f, x0 = [0, 0])  \n",
    "print(root)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    fjac: array([[ 0.89914291, -0.43765515],\n",
      "       [ 0.43765515,  0.89914291]])\n",
      "     fun: array([-1.11022302e-16,  0.00000000e+00])\n",
      " message: 'The solution converged.'\n",
      "    nfev: 10\n",
      "    njev: 1\n",
      "     qtf: array([-1.19565972e-11,  4.12770392e-12])\n",
      "       r: array([ 2.16690469, -1.03701789,  1.10605417])\n",
      "  status: 1\n",
      " success: True\n",
      "       x: array([0.8411639, 0.1588361])\n"
     ]
    }
   ],
   "source": [
    "root = optimize.root(f, jac=jac, x0 = [0, 0], method=\"hybr\")  \n",
    "print(root)  "
   ]
  }
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