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ParkSuMin committed
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@@ -1,47 +1,155 @@
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 22,
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"id": "18bd929b",
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"cell_type": "markdown",
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"id": "fda98006",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"image/png": 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truncated
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"text/plain": [
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"<Figure size 640x480 with 1 Axes>"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"\n",
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"N = 20\n",
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"x = np.linspace(0, 10, N + 1)\n",
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"with open(\"matrix_linear.txt\", 'r') as file:\n",
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" data_str = file.read()\n",
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"y = np.fromstring(data_str, sep=' ')\n",
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"plt.plot(x, y, label=\"Linear element solution\", color='red')\n",
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"plt.title(\"Linear element\")\n",
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"plt.grid(True)\n",
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"plt.legend()\n",
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"plt.show()"
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"# Solution of second-order linear ordinary differential equation"
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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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"execution_count": 135,
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"id": "7fced95c",
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"metadata": {},
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"outputs": [],
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"source": [
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"def write_to_file(name: str, array):\n",
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" with open(f\"{name}\", 'w') as file:\n",
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" for element in array:\n",
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" file.write(f\"{element} \")"
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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": 136,
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"id": "d20bde93",
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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 matplotlib.pyplot as plt"
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]
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},
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{
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"cell_type": "markdown",
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"id": "8af07a56",
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"metadata": {},
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"source": [
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"## Solution of $5u'' + 4u' + 1 = 0, u'(0) = u(0), u(10) = 5$"
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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": 137,
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"id": "d0fe0703",
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"metadata": {},
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"outputs": [],
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"source": [
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"def u(x):\n",
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" return (9 * np.exp(8) * (x - 30) + 155 * np.exp(8 - 4 * x / 5) - 5 * (x + 1)) / (20 - 36 * np.exp(8))"
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]
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},
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{
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"cell_type": "markdown",
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"id": "efac514a",
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"metadata": {},
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"source": [
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"## Linear element"
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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": 138,
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"id": "18bd929b",
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"metadata": {},
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"outputs": [],
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"source": [
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"def show_plot_linear(filename: str, elements: int):\n",
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" with open(filename, 'r') as file:\n",
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" data_str = file.read()\n",
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"\n",
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" x = np.linspace(0, 10, elements + 1)\n",
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" y = np.fromstring(data_str, sep=' ')\n",
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" y_real = u(x)\n",
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"\n",
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" plt.plot(x, y, label=\"Linear element solution\", color='red')\n",
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" plt.plot(x, y_real, label=\"Numeral solution\", color='black')\n",
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" plt.title(f\"Linear element, elements = {elements}\")\n",
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" plt.grid(True)\n",
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" plt.xlabel(\"X\")\n",
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" plt.ylabel(\"Y\")\n",
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" plt.legend()\n",
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" plt.savefig(f\"linear_{elements}.png\", dpi=300)\n",
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" plt.show()\n",
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"\n",
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" write_to_file(f\"linear_real_y_{elements}.txt\", y_real)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "d9a38740",
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"metadata": {},
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"source": [
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"## Cubic element"
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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": 139,
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"id": "73525bc7",
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"metadata": {},
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"outputs": [],
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"source": [
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"def show_plot_cubic(filename: str, elements: int):\n",
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" with open(filename, 'r') as file:\n",
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" data_str = file.read()\n",
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"\n",
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" x = np.linspace(0, 10, 3 * elements + 1)\n",
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" y = np.fromstring(data_str, sep=' ')\n",
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" y_real = u(x)\n",
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"\n",
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" plt.plot(x, y, label=\"Cubic element solution\", color=\"green\")\n",
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" plt.plot(x, y_real, label=\"Numeral solution\", color='black')\n",
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" plt.title(f\"Cubic element, elements = {elements} \")\n",
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" plt.xlabel(\"X\")\n",
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" plt.ylabel(\"Y\")\n",
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" plt.grid(True)\n",
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" plt.legend()\n",
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" plt.savefig(f\"cubic_{elements}.png\", dpi=300)\n",
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" plt.show()\n",
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"\n",
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" write_to_file(f\"cubic_real_y_{elements}.txt\", y_real)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "3584baa4",
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"metadata": {},
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"source": [
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"## Main part"
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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": 140,
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"id": "b8a772e3",
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"metadata": {},
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"outputs": [
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{
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"data": {
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||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 640x480 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 640x480 with 1 Axes>"
|
||||
]
|
||||
@@ -52,16 +160,82 @@
|
||||
],
|
||||
"source": [
|
||||
"N = 20\n",
|
||||
"x = np.linspace(0, 10, 3 * N + 1)\n",
|
||||
"with open(\"matrix_cubic.txt\", 'r') as file:\n",
|
||||
" data_str = file.read()\n",
|
||||
"y = np.fromstring(data_str, sep=' ')\n",
|
||||
"plt.plot(x, y, label=\"Cubic element solution\")\n",
|
||||
"plt.title(\"Cubic element\")\n",
|
||||
"plt.grid(True)\n",
|
||||
"show_plot_linear(\"matrix_linear.txt\", N)\n",
|
||||
"show_plot_cubic(\"matrix_cubic.txt\", N)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "22fe3de6",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Errors"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 142,
|
||||
"id": "e0e9266d",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def read_data_from_file(filename):\n",
|
||||
" with open(filename, 'r') as file:\n",
|
||||
" content = file.read().strip()\n",
|
||||
" data = np.array([float(x) for x in content.split()])\n",
|
||||
" return data\n",
|
||||
"\n",
|
||||
"def show_error(program_file: str, analytical_file: str):\n",
|
||||
" # Считывание данных из файлов\n",
|
||||
" real = read_data_from_file(analytical_file)\n",
|
||||
" matrix = read_data_from_file(program_file)\n",
|
||||
" \n",
|
||||
" # Расчет абсолютных разностей\n",
|
||||
" absolute_errors = np.abs(real - matrix)\n",
|
||||
"\n",
|
||||
"plt.show()"
|
||||
" # Расчет относительных погрешностей (в долях)\n",
|
||||
" # Добавляем маленькое значение к знаменателю для избежания деления на ноль\n",
|
||||
" relative_errors = absolute_errors / (np.abs(real))\n",
|
||||
"\n",
|
||||
" # Расчет относительных погрешностей в процентах\n",
|
||||
" relative_errors_percent = relative_errors * 100\n",
|
||||
"\n",
|
||||
" # Нахождение максимальной относительной погрешности\n",
|
||||
" max_relative_error = np.max(relative_errors)\n",
|
||||
" max_relative_error_percent = np.max(relative_errors_percent)\n",
|
||||
" max_error_index = np.argmax(relative_errors)\n",
|
||||
"\n",
|
||||
" # Вывод результатов\n",
|
||||
" print(f\"\\nМаксимальная относительная погрешность: {max_relative_error:.6f} ({max_relative_error_percent:.2f}%)\")\n",
|
||||
" print(f\"Индекс точки с максимальной погрешностью: {max_error_index}\")\n",
|
||||
" print(f\"Значения в точке с максимальной погрешностью:\")\n",
|
||||
" print(f\" cubic_real: {real[max_error_index]}\")\n",
|
||||
" print(f\" matrix_cubic: {matrix[max_error_index]}\")\n",
|
||||
" print(f\" Абсолютная разность: {absolute_errors[max_error_index]:.6f}\")"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 143,
|
||||
"id": "89918171",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"\n",
|
||||
"Максимальная относительная погрешность: 0.107830 (10.78%)\n",
|
||||
"Индекс точки с максимальной погрешностью: 0\n",
|
||||
"Значения в точке с максимальной погрешностью:\n",
|
||||
" cubic_real: 3.1950864987612366\n",
|
||||
" matrix_cubic: 2.85056\n",
|
||||
" Абсолютная разность: 0.344526\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"show_error(\"matrix_linear.txt\", \"linear_real_y_20.txt\")"
|
||||
]
|
||||
}
|
||||
],
|
||||
|
||||
Reference in new issue
Block a user