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nihonium 2023-11-15 15:27:14 +03:00
parent 2a37c0ab08
commit 8de5db8883
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{
"cells": [
{
"cell_type": "code",
"execution_count": 15,
"id": "c8de76b1-c455-47ff-91c1-11abed30fcff",
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import itertools"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "b6479073-0cff-478f-868d-26fedd310af5",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": []
},
"outputs": [],
"source": [
"def get_a_ij(i: int, j: int, n: int):\n",
" return 10.0 / (i + j + 1)\n",
"\n",
"\n",
"# Construct the required matrix\n",
"\n",
"\n",
"def get_matrix_a(n: int) -> np.array:\n",
" a = np.zeros((n, n))\n",
" from_0_to_n = [i for i in range(n)]\n",
" for i, j in itertools.product(from_0_to_n, from_0_to_n):\n",
" a[i, j] = get_a_ij(i, j, n)\n",
" return a\n",
"\n",
"\n",
"def get_f(n: int) -> np.array:\n",
" return np.sum(get_matrix_a(n), axis=1)"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "1fb8aa25-9a4f-408a-8705-3dd9d91c7089",
"metadata": {},
"outputs": [],
"source": [
"# Solve Lx = f, where L is lower triangular\n",
"def lower_solve(l: np.array, f: np.array):\n",
" assert l.ndim == 2 and l.shape[0] == l.shape[1]\n",
" dim = l.shape[0]\n",
"\n",
" x = np.zeros(dim)\n",
" for i in range(dim):\n",
" tmp = f[i]\n",
" for j in range(i):\n",
" tmp -= l[i, j] * x[j]\n",
" x[i] = tmp / l[i, i]\n",
"\n",
" return x\n",
"\n",
"\n",
"def upper_solve(u: np.array, f: np.array):\n",
" # Solve Ux = f, where U is upper triangular\n",
" assert u.ndim == 2 and u.shape[0] == u.shape[1]\n",
" dim = u.shape[0]\n",
"\n",
" x = np.zeros(dim)\n",
" for i in range(dim - 1, -1, -1):\n",
" tmp = f[i]\n",
" for j in range(i + 1, dim):\n",
" tmp -= u[i, j] * x[j]\n",
" x[i] = tmp / u[i, i]\n",
"\n",
" return x\n",
"\n",
"\n",
"def lu_solve(l, u, f):\n",
" # Ax = f, where A = LU\n",
" # L(Ux) = f, Ux = y <=> y = Ux\n",
" y = lower_solve(l, f)\n",
" x = upper_solve(u, y)\n",
" return x"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "6e399987-73ed-452c-927b-cc9908786520",
"metadata": {},
"outputs": [],
"source": [
"def do_first_task(n: int) -> None:\n",
" a = get_matrix_a(n)\n",
" f = get_f(n)\n",
" # HERE\n",
" l = crudepy.linalg.cholesky(a)\n",
" x = lu_solve(l, l.T, f)\n",
"\n",
" print(f\"a = \\n{a}\")\n",
" print(f\"f = \\n{f}\")\n",
" print(\"Solution: -----\")\n",
" print(f\"L = \\n{l}\")\n",
" print(f\"x = \\n{x}\")\n",
"\n",
" # Find the exact solution with numpy and compare results\n",
" exact_x = np.linalg.solve(a, f)\n",
" delta = exact_x - x\n",
"\n",
" print(f\"|delta|_1 = {crudepy.linalg.vec_norm_max(delta):#e}\")\n",
" print(f\"|delta|_2 = {crudepy.linalg.vec_norm_sum(delta):#e}\")\n",
" print(f\"|delta|_3 = {crudepy.linalg.vec_norm_euclid(delta):#e}\")"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "47569073-4acd-42c0-a2dd-6c8c2deee9b4",
"metadata": {},
"outputs": [],
"source": [
"n = 6"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "300cd1ef-cbee-48fb-89ef-dd3779da1b24",
"metadata": {},
"outputs": [
{
"ename": "NameError",
"evalue": "name 'crudepy' is not defined",
"output_type": "error",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)",
"Cell \u001b[0;32mIn[20], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m \u001b[43mdo_first_task\u001b[49m\u001b[43m(\u001b[49m\u001b[43mn\u001b[49m\u001b[43m)\u001b[49m\n",
"Cell \u001b[0;32mIn[18], line 4\u001b[0m, in \u001b[0;36mdo_first_task\u001b[0;34m(n)\u001b[0m\n\u001b[1;32m 2\u001b[0m a \u001b[38;5;241m=\u001b[39m get_matrix_a(n)\n\u001b[1;32m 3\u001b[0m f \u001b[38;5;241m=\u001b[39m get_f(n)\n\u001b[0;32m----> 4\u001b[0m l \u001b[38;5;241m=\u001b[39m \u001b[43mcrudepy\u001b[49m\u001b[38;5;241m.\u001b[39mlinalg\u001b[38;5;241m.\u001b[39mcholesky(a)\n\u001b[1;32m 5\u001b[0m x \u001b[38;5;241m=\u001b[39m lu_solve(l, l\u001b[38;5;241m.\u001b[39mT, f)\n\u001b[1;32m 7\u001b[0m \u001b[38;5;28mprint\u001b[39m(\u001b[38;5;124mf\u001b[39m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124ma = \u001b[39m\u001b[38;5;130;01m\\n\u001b[39;00m\u001b[38;5;132;01m{\u001b[39;00ma\u001b[38;5;132;01m}\u001b[39;00m\u001b[38;5;124m\"\u001b[39m)\n",
"\u001b[0;31mNameError\u001b[0m: name 'crudepy' is not defined"
]
}
],
"source": [
"do_first_task(n)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "9f33c8b8-c89f-43a8-9dfb-da5313ad0045",
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.11.5"
}
},
"nbformat": 4,
"nbformat_minor": 5
}

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

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