{ "cells": [ { "cell_type": "markdown", "id": "organized-governor", "metadata": {}, "source": [ "# Прикладные дифференциальные уравнения\n", "## Занятие 4\n", "*Илья Щуров*\n", "\n", "Факультет компьютерных наук, Прикладная математики и информатика, 2021-22 учебный год\n", "\n", "[Страница курса](http://math-info.hse.ru/2021-22/Прикладные_дифференциальные_уравнения)" ] }, { "cell_type": "code", "execution_count": 1, "id": "bright-football", "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import plotly.graph_objects as go\n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "code", "execution_count": 2, "id": "violent-height", "metadata": {}, "outputs": [], "source": [ "def matrix_exp(A, degree=100):\n", " n = A.shape[0]\n", " assert A.shape == (n, n)\n", " X = np.eye(n)\n", " terms = [X]\n", " for i in range(1, degree):\n", " X = (X @ A) / i\n", " terms.append(X)\n", " return np.array(terms).sum(axis=0)" ] }, { "cell_type": "code", "execution_count": 3, "id": "boolean-gross", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[2.71828183, 0. ],\n", " [0. , 2.71828183]])" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "matrix_exp(np.array([[1, 0], \n", " [0, 1]]), degree=100)" ] }, { "cell_type": "code", "execution_count": 4, "id": "civic-horror", "metadata": {}, "outputs": [], "source": [ "A = np.array([[0, 1, 0],\n", " [0, 0, 1],\n", " [-1, 0, 0]])" ] }, { "cell_type": "code", "execution_count": 5, "id": "atomic-czech", "metadata": {}, "outputs": [], "source": [ "def trajectory(A, x0, delta_t, steps):\n", " exp_Adt = matrix_exp(A * delta_t)\n", " traj = [x0]\n", " for i in range(steps - 1):\n", " traj.append(exp_Adt @ traj[-1])\n", " return np.array(traj)" ] }, { "cell_type": "code", "execution_count": 6, "id": "embedded-investor", "metadata": {}, "outputs": [ { "data": { "text/html": [ " \n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { 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