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254 lines
6.9 KiB
Plaintext
254 lines
6.9 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 20,
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"id": "c8de76b1-c455-47ff-91c1-11abed30fcff",
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import itertools"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 21,
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"id": "b6479073-0cff-478f-868d-26fedd310af5",
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"metadata": {
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"editable": true,
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"slideshow": {
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"slide_type": ""
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},
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"tags": []
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},
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"outputs": [],
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"source": [
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"def get_a_ij(i: int, j: int, n: int):\n",
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" return 10.0 / (i + j + 1)\n",
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"\n",
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"def get_matrix_a(n: int) -> np.array:\n",
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" a = np.zeros((n, n))\n",
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" from_0_to_n = [i for i in range(n)]\n",
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" for i, j in itertools.product(from_0_to_n, from_0_to_n):\n",
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" a[i, j] = get_a_ij(i, j, n)\n",
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" return a\n",
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"\n",
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"\n",
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"def get_f(n: int) -> np.array:\n",
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" return np.sum(get_matrix_a(n), axis=1)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 22,
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"id": "1fb8aa25-9a4f-408a-8705-3dd9d91c7089",
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"metadata": {},
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"outputs": [],
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"source": [
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"def lower_solve(l: np.array, f: np.array):\n",
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" assert l.ndim == 2 and l.shape[0] == l.shape[1]\n",
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" dim = l.shape[0]\n",
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"\n",
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" x = np.zeros(dim)\n",
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" for i in range(dim):\n",
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" tmp = f[i]\n",
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" for j in range(i):\n",
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" tmp -= l[i, j] * x[j]\n",
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" x[i] = tmp / l[i, i]\n",
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"\n",
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" return x\n",
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"\n",
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"\n",
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"def upper_solve(u: np.array, f: np.array):\n",
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" assert u.ndim == 2 and u.shape[0] == u.shape[1]\n",
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" dim = u.shape[0]\n",
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"\n",
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" x = np.zeros(dim)\n",
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" for i in range(dim - 1, -1, -1):\n",
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" tmp = f[i]\n",
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" for j in range(i + 1, dim):\n",
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" tmp -= u[i, j] * x[j]\n",
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" x[i] = tmp / u[i, i]\n",
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"\n",
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" return x\n",
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"\n",
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"\n",
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"def lu_solve(l, u, f):\n",
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" # Ax = f, где A = LU\n",
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" # L(Ux) = f, Ux = y <=> y = Ux\n",
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" y = lower_solve(l, f)\n",
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" x = upper_solve(u, y)\n",
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" return x"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 23,
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"id": "81937d82-febb-46b0-954d-e2d631f124b8",
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"metadata": {},
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"outputs": [],
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"source": [
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"def cholesky(A):\n",
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" n = A.shape[0]\n",
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" L = np.zeros_like(A)\n",
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" for i in range(n):\n",
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" for j in range(i+1):\n",
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" s = 0\n",
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" for k in range(j):\n",
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" s += L[i][k] * L[j][k]\n",
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"\n",
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" if (i == j):\n",
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" L[i][j] = (A[i][i] - s) ** 0.5\n",
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" else:\n",
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" L[i][j] = (1.0 / L[j][j] * (A[i][j] - s))\n",
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" return L"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 24,
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"id": "06916637-b82c-4e0b-9792-6558bced33b8",
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"metadata": {},
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"outputs": [],
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"source": [
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"# Векторные и подчиненные им матричные нормы\n",
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"\n",
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"def vec_norm_max(vector: np.array):\n",
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" # |v|_1 = max(abs(v_i))\n",
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" return np.abs(vector).max()\n",
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"\n",
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"\n",
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"def vec_norm_sum(vector: np.array):\n",
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" # |v|_2 = sum(abs(v_i))\n",
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" return np.abs(vector).sum()\n",
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"\n",
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"def vec_norm_euclid(vector: np.array):\n",
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" return np.sqrt((vector ** 2.0).sum())\n",
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"\n",
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"\n",
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"def row_sum_norm(matrix: np.array):\n",
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" # |A|_1 = max( sum(a_ij) for j ) over 1 <= i <= n\n",
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" return np.sum(np.abs(matrix), axis=1).max()\n",
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"\n",
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"\n",
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"def column_sum_norm(matrix: np.array):\n",
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" # |A|_2 = max( sum(a_ij) for i ) over 1 <= j <= n\n",
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" return np.sum(np.abs(matrix), axis=0).max()\n",
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"\n",
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"\n",
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"def spectral_norm(matrix: np.array):\n",
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" matrix_h = matrix.conj().T\n",
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" eigenvalues = np.linalg.eigvals(np.matmul(matrix_h, matrix))\n",
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" return np.sqrt(eigenvalues.max())\n",
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"\n",
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"\n",
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"def mu(matrix_a: int, norm):\n",
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" inv_matrix_a = np.linalg.inv(matrix_a)\n",
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" norm_a = norm(matrix_a)\n",
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" norm_inv_a = norm(inv_matrix_a)\n",
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" return norm_a * norm_inv_a\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 25,
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"id": "6e399987-73ed-452c-927b-cc9908786520",
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"metadata": {},
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"outputs": [],
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"source": [
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"def task1(n: int) -> None:\n",
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" a = get_matrix_a(n)\n",
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" f = get_f(n)\n",
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" l = cholesky(a)\n",
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" x = lu_solve(l, l.T, f)\n",
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"\n",
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" print(\"Исходные данные:\")\n",
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" print(f\"a = \\n{a}\")\n",
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" print(f\"f = \\n{f}\")\n",
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" print(\"Решение:\")\n",
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" print(f\"x = \\n{x}\")\n",
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"\n",
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" exact_x = np.linalg.solve(a, f)\n",
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" delta = exact_x - x\n",
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"\n",
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" print(f\"|delta|_1 = {vec_norm_max(delta):#e}\")\n",
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" print(f\"|delta|_2 = {vec_norm_sum(delta):#e}\")\n",
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" print(f\"|delta|_3 = {vec_norm_euclid(delta):#e}\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 26,
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"id": "47569073-4acd-42c0-a2dd-6c8c2deee9b4",
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"metadata": {},
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"outputs": [],
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"source": [
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"n = 6"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 27,
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"id": "300cd1ef-cbee-48fb-89ef-dd3779da1b24",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Исходные данные:\n",
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"a = \n",
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"[[10. 5. 3.33333333 2.5 2. 1.66666667]\n",
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" [ 5. 3.33333333 2.5 2. 1.66666667 1.42857143]\n",
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" [ 3.33333333 2.5 2. 1.66666667 1.42857143 1.25 ]\n",
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" [ 2.5 2. 1.66666667 1.42857143 1.25 1.11111111]\n",
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" [ 2. 1.66666667 1.42857143 1.25 1.11111111 1. ]\n",
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" [ 1.66666667 1.42857143 1.25 1.11111111 1. 0.90909091]]\n",
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"f = \n",
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"[24.5 15.92857143 12.17857143 9.95634921 8.45634921 7.36544012]\n",
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"Решение:\n",
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"x = \n",
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"[1. 1. 1. 1. 1. 1.]\n",
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"|delta|_1 = 2.136471e-10\n",
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"|delta|_2 = 5.829489e-10\n",
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"|delta|_3 = 3.120671e-10\n"
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]
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}
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],
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"source": [
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"task1(n)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "0dab679e-f64b-482d-864d-2943f0e793a4",
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.11.5"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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