diff --git "a/\320\2403208/kaparulin_408737/lab1/Comp_math_1.pdf" "b/\320\2403208/kaparulin_408737/lab1/Comp_math_1.pdf" new file mode 100644 index 0000000..49c16ec Binary files /dev/null and "b/\320\2403208/kaparulin_408737/lab1/Comp_math_1.pdf" differ diff --git "a/\320\2403208/kaparulin_408737/lab1/SysOfLinEq.py" "b/\320\2403208/kaparulin_408737/lab1/SysOfLinEq.py" new file mode 100644 index 0000000..058c160 --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab1/SysOfLinEq.py" @@ -0,0 +1,157 @@ +from itertools import permutations + +import numpy as np + +class SysOfLinEq: + def __init__(self, file_name=''): + self.answ = [] + self.variance = float('inf') + self.iter_cnt = 0 + + if not file_name: + self.__input_params() + else: + self.__read_params(file_name) + + self.__is_epsilon_correct() + + assert self.__is_solvable(), "The condition of predominance of diagonal elements is not fulfilled" + + self.__create_coeff_matrix() + self.__solution() + + def get_variance(self): + return self.variance + + def get_iteration_count(self): + return self.iter_cnt + + def get_matrix_norm(self): + return np.max(np.sum(np.abs(self.KMatrix[:, :-1]), axis=1)) + + def get_solution(self): + return self.solution + + # решение СЛАУ + def __solution(self): + """ + У нас есть расширенная матрица со свободными коэфицентами + тогда чтобы они имели вес ровно 1, добавляем единичный столбец + """ + self.solution = np.concat(( self.KMatrix[:, self.dims], [1] )) + + while np.max(self.variance) >= self.epsilon: + self.iter_cnt+=1 + old_solution = np.copy(self.solution) + + for line in range(self.dims): + self.solution[line] = np.sum(self.KMatrix[line] * self.solution) + + self.variance = np.abs(self.solution - old_solution) + + self.solution = self.solution[:-1] + self.variance = self.variance[:-1] + + # создание матрицы коэфициентов + def __create_coeff_matrix(self): + self.KMatrix = np.empty((self.dims, self.dims+1)) + + for line in range(self.dims): + self.KMatrix[line] = np.concat(( -self.matrix[line], [self.answ[line]])) + self.KMatrix[line] /= self.matrix[line][line] + self.KMatrix[line, line] = 0 + + + # выполняется ли условие сходимости + def __is_solvable(self): + # проверка, что вообще есть число большее суммы остальных в строке + if self.__is_not_diagonally_dominant(np.abs(self.matrix)): + return False + + # попытка поставить на диагональ максимальные числа + for line in range(self.dims): + sorted_indices = np.argsort(self.matrix[line:][:, line])[::-1] + self.matrix[line:] = self.matrix[line:][sorted_indices] + + if self.__is_diagonally_dominant(np.abs(self.matrix)): + return True + + # можно переставновками проверить, но это дольше + # чем две верхних проверки + for perm in permutations(range(self.dims)): + permuted_matrix = self.matrix[list(perm), :] + + if self.__is_diagonally_dominant(np.abs(permuted_matrix)): + self.matrix = permuted_matrix + return True + + return False + + # проверка условия преобладания диагональных элементов + def __is_not_diagonally_dominant(self, matrix): + return np.any( (np.sum(matrix, axis=1) - 2*np.max(matrix, axis=1)) > 0 ) + + # проверка условия преобладания диагональных элементов + def __is_diagonally_dominant(self, matrix): + one_strict = False + diagonally_dominant = np.sum(matrix, axis=1) - 2*np.diag(matrix) + + if np.any(diagonally_dominant > 0): + return False + + if np.any(diagonally_dominant < 0): + one_strict = True + + return True & one_strict + + + def __is_epsilon_correct(self): + if self.epsilon < 0: + raise ValueError("e must be positive") + + # ввод всех параметров + def __input_params(self): + + print("Input n: ", end="") + self.dims = self.__get_param("n", int, input) + print("Input e: ", end="") + self.epsilon = self.__get_param("e", float, input) + + self.matrix = np.empty((self.dims, self.dims)) + + print("\nInput matrix:") + + for line_cnt in range(self.dims): + line_input = input().split() + self.answ.append(line_input[-1]) + line = np.array(list(map(float, line_input[:-1])), dtype=np.float64) + assert line.shape == (self.dims,), "Incorrect line shape" + self.matrix[line_cnt] = line + + print() + + # чтение всех параметов из файла + def __read_params(self, file_name): + with open(file_name) as f: + + self.dims = self.__get_param("n", int, f.readline) + self.epsilon = self.__get_param("e", float, f.readline) + + + self.matrix = np.empty((self.dims, self.dims)) + + for line_cnt in range(self.dims): + line_file = f.readline().split() + self.answ.append(line_file[-1]) + line = np.array(list(map(float, line_file[:-1])), dtype=np.float64) + assert line.shape == (self.dims,), "Incorrect line shape" + self.matrix[line_cnt] = line + + self.answ = np.array(self.answ, dtype=np.float64) + + # получение 1 параметра + def __get_param(self, name, type, func): + try: + return type(func()) + except: + raise ValueError(f"Incorrect {name}") \ No newline at end of file diff --git "a/\320\2403208/kaparulin_408737/lab1/main.py" "b/\320\2403208/kaparulin_408737/lab1/main.py" new file mode 100644 index 0000000..e347254 --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab1/main.py" @@ -0,0 +1,13 @@ +from SysOfLinEq import SysOfLinEq + + + + + +if __name__ == "__main__": + sle = SysOfLinEq("pars.txt") # pars.txt + + print("Решение: ", sle.get_solution()) + print("Норма матрицы: ", sle.get_matrix_norm()) + print("Кол-во итераций: ", sle.get_iteration_count()) + print("Вектор погрешностей: ", sle.get_variance()) diff --git "a/\320\2403208/kaparulin_408737/lab1/rand_gen.py" "b/\320\2403208/kaparulin_408737/lab1/rand_gen.py" new file mode 100644 index 0000000..b1ee5d2 --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab1/rand_gen.py" @@ -0,0 +1,26 @@ +import random + +import numpy as np + +n = int(input("Input n: ")) +e = float(input("Input e: ")) + +with open("pars.txt", "w") as f: + f.write(str(n) + "\n") + f.write(str(e) + "\n") + + lines = [] + diag = list(range(n)) + + for _ in range(n): + random_number = random.choice(diag) + + diag.remove(random_number) + + line = np.random.randint(0, 10, n+1) + + lines.append(" ".join(map(str, line)) + "\n") + + + f.writelines(lines) + diff --git "a/\320\2403208/kaparulin_408737/lab2/Non_lin_equ.png" "b/\320\2403208/kaparulin_408737/lab2/Non_lin_equ.png" new file mode 100644 index 0000000..72d4142 Binary files /dev/null and "b/\320\2403208/kaparulin_408737/lab2/Non_lin_equ.png" differ diff --git "a/\320\2403208/kaparulin_408737/lab2/README.md" "b/\320\2403208/kaparulin_408737/lab2/README.md" index 2999024..913e0e4 100644 --- "a/\320\2403208/kaparulin_408737/lab2/README.md" +++ "b/\320\2403208/kaparulin_408737/lab2/README.md" @@ -1 +1,3 @@ Лабораторная работа 2 + +6 вариант \ No newline at end of file diff --git "a/\320\2403208/kaparulin_408737/lab2/System_graph.png" "b/\320\2403208/kaparulin_408737/lab2/System_graph.png" new file mode 100644 index 0000000..f06fbf3 Binary files /dev/null and "b/\320\2403208/kaparulin_408737/lab2/System_graph.png" differ diff --git "a/\320\2403208/kaparulin_408737/lab2/main.py" "b/\320\2403208/kaparulin_408737/lab2/main.py" new file mode 100644 index 0000000..1beda2c --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab2/main.py" @@ -0,0 +1,3 @@ +from non_lin_eq import Lin_Eq + +Lin_Eq() \ No newline at end of file diff --git "a/\320\2403208/kaparulin_408737/lab2/non_lin_eq/Main.py" "b/\320\2403208/kaparulin_408737/lab2/non_lin_eq/Main.py" new file mode 100644 index 0000000..2cdaab0 --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab2/non_lin_eq/Main.py" @@ -0,0 +1,109 @@ +from .PolDel import PolDel +from .Secant import Secant +from .SimpleIter import SimpleIter + +from numpy import sin, cos +import numpy as np +import matplotlib.pyplot as plt + +f = [ + [ + lambda x: 2*x**3 + 3.41*x**2 - 23.74*x + 2.95, + lambda x: 6*x**2 + 6.82*x - 23.74, + lambda x: 12*x + 6.82 + ], + + [ + lambda x: sin(x) + cos(x), + lambda x: cos(x) - sin(x), + lambda x: -sin(x) - cos(x) + ], + + [ + lambda x: x**2 + 1, + lambda x: 2*x, + lambda x: 2 + ], + + [ + lambda x: x + cos(x-1) - 0.7, + lambda x: 1 - sin(x-1), + lambda x: - cos(x-1) + ], +] + +method = [ + PolDel, + Secant, + SimpleIter +] + +class Main: + def __init__(self): + eq_nmb = 0 + + while(eq_nmb not in [1, 2, 3, 4]): + print("[1] f(x) = 2*x**3 + 3.41*x**2 - 23.74*x + 2.95") + print("[2] f(x) = sin(x) + cos(x)") + print("[3] f(x) = x**2 + 1") + print("[4] f(x) = x + cos(x-1) - 0.7") + print("Введите номер уравнения: ", end="") + eq_nmb = self.__get_param("equation number", int, input) + print() + + print("Введите путь до данных(\"\" - ввод в терминале): ", end="") + file = self.__get_param("file name", str, input) + + + method_nmb = 0 + + while(method_nmb not in [1, 2, 3]): + print() + print("[1] Метод половинного деления") + print("[2] Метод секущих") + print("[3] Метод простой итерации") + print("Введите номер метода решения уравнения: ", end="") + method_nmb = self.__get_param("method name", int, input) + + print() + print("Результаты в файл?(\"\" - нет, file_path - да): ", end="") + res_file = self.__get_param("result file name", str, input) + print() + + x, f_x, iter_cnt, interval = method[method_nmb-1](f[eq_nmb-1], file).get_answer() + + if res_file == "": + print("x: ", x) + print("f(x): ", f_x) + print("Number of iteration: ", iter_cnt) + else: + resf = open(res_file, 'w') + + print("x: ", x, file=resf) + print("f(x): ", f_x, file=resf) + print("Number of iteration: ", iter_cnt, file=resf) + + resf.close() + + x_plot = np.linspace(interval[0], interval[1], 500) + y_plot = f[eq_nmb-1][0](x_plot) + + plt.title('График нелинейной функции') + plt.plot(x_plot, y_plot) + plt.xlabel('x') + plt.ylabel('y') + plt.grid(1) + plt.savefig("Non_lin_equ.png") + + + # получение 1 параметра + def __get_param(self, name, type, func): + try: + return type(func()) + except: + print(f"Incorrect {name}") + exit() + + +if __name__ == "__main__": + Main() \ No newline at end of file diff --git "a/\320\2403208/kaparulin_408737/lab2/non_lin_eq/PolDel.py" "b/\320\2403208/kaparulin_408737/lab2/non_lin_eq/PolDel.py" new file mode 100644 index 0000000..5e69da6 --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab2/non_lin_eq/PolDel.py" @@ -0,0 +1,107 @@ +import numpy as np + +class PolDel: + interval=np.array([0,0], dtype=float) + e=0.01 + + def __init__(self, funcs, file_name=''): + + if not file_name: + self.__input_params() + else: + self.__read_params(file_name) + + self.__is_correct(self.interval, self.e) + + self.__root_check(funcs[0], funcs[1], self.interval) + + self.__solution(funcs[0]) + + def get_answer(self): + return (self.solution, self.f_x, self.cnt, self.interval) + + + def __solution(self, f): + interval = self.interval.copy() + diff = float('inf') + self.cnt = 0 + + while(diff > self.e): + interval = self.__iter(f, interval) + diff = interval[1] - interval[0] + self.cnt+=1 + + self.solution = (interval[1] + interval[0])/2 + self.f_x = f(self.solution) + + + def __iter(self, f, interval): + x = (interval[1] + interval[0])/2 + + if(f(x)*f(interval[1])<0): + interval[0] = x + elif(f(x)*f(interval[1])>0): + interval[1] = x + else: + interval[0] = x + interval[1] = x + + return interval + + + def __root_check(self, f, fp, interval): + if(f(interval[0]) * f(interval[1]) > 0): + print(f"Ошибка: на данном интервале нет корней, либо их несколько") + exit() + + else: + linspace = np.linspace(interval[0], interval[1], num=500) + fp_linspace = fp(linspace) + + if not (np.all(fp_linspace[:-1] <= fp_linspace[1:]) or np.all(fp_linspace[:-1] >= fp_linspace[1:])): + print(f"Предупреждение: на данном интервале может быть несколько корней") + + + def __is_correct(self, interval, e): + if interval[0] >= interval[1]: + print(f"Ошибка: некорректный интервал - {interval}") + exit() + + if e >= 1 or e < 0: + print(f"Ошибка: некорректное значение e - {e}") + exit() + + + # ввод всех параметров + def __input_params(self): + + print("Input a: ", end="") + self.interval[0] = self.__get_param("a", float, input) + + print("Input b: ", end="") + self.interval[1] = self.__get_param("b", float, input) + + print("Input e: ", end="") + self.e = self.__get_param("e", float, input) + + print() + + # чтение всех параметов из файла + def __read_params(self, file_name): + try: + with open(file_name) as file: + + self.interval[0] = self.__get_param("a", float, file.readline) + self.interval[1] = self.__get_param("b", float, file.readline) + self.e = self.__get_param("e", float, file.readline) + except: + print(f"Incorrect file name: {file_name}") + exit() + + # получение 1 параметра + def __get_param(self, name, type, func): + try: + return type(func()) + except: + print(f"Incorrect {name}") + exit() \ No newline at end of file diff --git "a/\320\2403208/kaparulin_408737/lab2/non_lin_eq/Secant.py" "b/\320\2403208/kaparulin_408737/lab2/non_lin_eq/Secant.py" new file mode 100644 index 0000000..139a833 --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab2/non_lin_eq/Secant.py" @@ -0,0 +1,112 @@ +import numpy as np + +class Secant: + interval = np.array([0,0], dtype=float) + x0 = 0 + e = 0.01 + + def __init__(self, funcs, file_name=''): + + if not file_name: + self.__input_params() + else: + self.__read_params(file_name) + + self.__is_correct(self.interval, self.e) + + self.__root_check(funcs[0], funcs[1], self.interval) + + self.__find_x0(funcs[0], funcs[2], self.interval) + + self.__solution(funcs[0]) + + + def get_answer(self): + return (self.solution, self.f_x, self.cnt, self.interval) + + + def __solution(self, f): + diff = float('inf') + x_p = self.x0 + x = x_p + self.e + self.cnt = 0 + + while(diff > self.e): + x_n = self.__iter(f, x_p, x) + self.cnt+=1 + x_p = x + x = x_n + diff = abs(x - x_p) + + self.solution = x + self.f_x = f(self.solution) + + + def __iter(self, f, x_p, x): + return x - (x - x_p)/(f(x) - f(x_p))*f(x) + + + def __find_x0(self, f, fpp, interval): + if f(interval[0])*fpp(interval[0]) > 0: + self.x0 = interval[0] + else: + self.x0 = interval[1] + + + def __root_check(self, f, fp, interval): + if(f(interval[0]) * f(interval[1]) > 0): + print(f"Ошибка: на данном интервале нет корней, либо их несколько") + exit() + + else: + linspace = np.linspace(interval[0], interval[1], num=500) + fp_linspace = fp(linspace) + + if not (np.all(fp_linspace[:-1] <= fp_linspace[1:]) or np.all(fp_linspace[:-1] >= fp_linspace[1:])): + print(f"Предупреждение: на данном интервале может быть несколько корней") + + + + def __is_correct(self, interval, e): + if interval[0] >= interval[1]: + print(f"Ошибка: некорректный интервал - {interval}") + exit() + + if e >= 1 or e < 0: + print(f"Ошибка: некорректное значение e - {e}") + exit() + + + # ввод всех параметров + def __input_params(self): + + print("Input a: ", end="") + self.interval[0] = self.__get_param("a", float, input) + + print("Input b: ", end="") + self.interval[1] = self.__get_param("b", float, input) + + print("Input e: ", end="") + self.e = self.__get_param("e", float, input) + + print() + + # чтение всех параметов из файла + def __read_params(self, file_name): + try: + with open(file_name) as file: + + self.interval[0] = self.__get_param("a", float, file.readline) + self.interval[1] = self.__get_param("b", float, file.readline) + self.e = self.__get_param("e", float, file.readline) + except: + print(f"Incorrect file name: {file_name}") + exit() + + # получение 1 параметра + def __get_param(self, name, type, func): + try: + return type(func()) + except: + print(f"Incorrect {name}") + exit() \ No newline at end of file diff --git "a/\320\2403208/kaparulin_408737/lab2/non_lin_eq/SimpleIter.py" "b/\320\2403208/kaparulin_408737/lab2/non_lin_eq/SimpleIter.py" new file mode 100644 index 0000000..020127c --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab2/non_lin_eq/SimpleIter.py" @@ -0,0 +1,118 @@ +import numpy as np + +class SimpleIter: + interval = np.array([0,0], dtype=float) + x0 = 0 + e = 0.01 + + def __init__(self, funcs, file_name=''): + + if not file_name: + self.__input_params() + else: + self.__read_params(file_name) + + self.__is_correct(self.interval, self.e) + + self.__root_check(funcs[0], funcs[1], self.interval) + + self.__find_x0_and_lambda(funcs[1], self.interval) + + self.__is_algorithm_convergence(funcs[1]) + + self.__solution(funcs[0]) + + + def get_answer(self): + return (self.solution, self.f_x, self.cnt, self.interval) + + + def __solution(self, f): + diff = float('inf') + x = self.x0 + self.cnt = 0 + + while(diff > self.e): + x_n = self.__phi(f, x) + self.cnt+=1 + diff = abs(x - x_n) + x = x_n + + self.solution = x + self.f_x = f(self.solution) + + + def __phi(self, f, x): + return x + self.Lambda * f(x) + + def __is_algorithm_convergence(self, fp): + if 1 + self.Lambda * fp(self.x0) >= 1: + print(f"Ошибка: условие сходимости не выполнено на данном интервале") + exit() + + + def __find_x0_and_lambda(self, fp, interval): + if abs(fp(interval[0])) > abs(fp(interval[1])): + self.Lambda = -1/abs(fp(interval[0])) if fp(interval[0]) > 0 else 1/abs(fp(interval[0])) + self.x0 = interval[0] + else: + self.Lambda = -1/abs(fp(interval[1])) if fp(interval[1]) > 0 else 1/abs(fp(interval[1])) + self.x0 = interval[1] + + + def __root_check(self, f, fp, interval): + if(f(interval[0]) * f(interval[1]) > 0): + print(f"Ошибка: на данном интервале нет корней, либо их несколько") + exit() + + else: + linspace = np.linspace(interval[0], interval[1], num=500) + fp_linspace = fp(linspace) + + if not (np.all(fp_linspace[:-1] <= fp_linspace[1:]) or np.all(fp_linspace[:-1] >= fp_linspace[1:])): + print(f"Предупреждение: на данном интервале может быть несколько корней") + + + def __is_correct(self, interval, e): + if interval[0] >= interval[1]: + print(f"Ошибка: некорректный интервал - {interval}") + exit() + + if e >= 1 or e < 0: + print(f"Ошибка: некорректное значение e - {e}") + exit() + + + # ввод всех параметров + def __input_params(self): + + print("Input a: ", end="") + self.interval[0] = self.__get_param("a", float, input) + + print("Input b: ", end="") + self.interval[1] = self.__get_param("b", float, input) + + print("Input e: ", end="") + self.e = self.__get_param("e", float, input) + + print() + + # чтение всех параметов из файла + def __read_params(self, file_name): + try: + with open(file_name) as file: + + self.interval[0] = self.__get_param("a", float, file.readline) + self.interval[1] = self.__get_param("b", float, file.readline) + self.e = self.__get_param("e", float, file.readline) + except: + print(f"Incorrect file name: {file_name}") + exit() + + # получение 1 параметра + def __get_param(self, name, type, func): + try: + return type(func()) + except: + print(f"Incorrect {name}") + exit() \ No newline at end of file diff --git "a/\320\2403208/kaparulin_408737/lab2/non_lin_eq/__init__.py" "b/\320\2403208/kaparulin_408737/lab2/non_lin_eq/__init__.py" new file mode 100644 index 0000000..0e6ff3f --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab2/non_lin_eq/__init__.py" @@ -0,0 +1,3 @@ +from .Main import Main as Lin_Eq + +__all__ = ["Lin_Eq"] \ No newline at end of file diff --git "a/\320\2403208/kaparulin_408737/lab2/non_lin_eq/__pycache__/Main.cpython-312.pyc" "b/\320\2403208/kaparulin_408737/lab2/non_lin_eq/__pycache__/Main.cpython-312.pyc" new file mode 100644 index 0000000..db9e4e4 Binary files /dev/null and "b/\320\2403208/kaparulin_408737/lab2/non_lin_eq/__pycache__/Main.cpython-312.pyc" differ diff --git "a/\320\2403208/kaparulin_408737/lab2/non_lin_eq/__pycache__/PolDel.cpython-312.pyc" "b/\320\2403208/kaparulin_408737/lab2/non_lin_eq/__pycache__/PolDel.cpython-312.pyc" new file mode 100644 index 0000000..a8b8e53 Binary files /dev/null and "b/\320\2403208/kaparulin_408737/lab2/non_lin_eq/__pycache__/PolDel.cpython-312.pyc" differ diff --git "a/\320\2403208/kaparulin_408737/lab2/non_lin_eq/__pycache__/Secant.cpython-312.pyc" 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+python-versions = "^3.12" +content-hash = "e6e55ef53b461471a19bbad4c85145e68f9db01e7d74829cbe87453840cd4590" diff --git "a/\320\2403208/kaparulin_408737/lab2/pyproject.toml" "b/\320\2403208/kaparulin_408737/lab2/pyproject.toml" new file mode 100644 index 0000000..034018e --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab2/pyproject.toml" @@ -0,0 +1,21 @@ +[tool.poetry] +name = "lab2" +version = "0.1.0" +description = "" +authors = ["Luhter1 "] +readme = "README.md" + +[tool.poetry.dependencies] +python = "^3.12" +numpy = "^2.2.4" +matplotlib = "^3.10.1" +simpy = "^4.1.1" +sympy = "^1.13.3" + + +[tool.poetry.group.dev.dependencies] +ipykernel = "^6.29.5" + +[build-system] +requires = ["poetry-core"] +build-backend = "poetry.core.masonry.api" diff --git "a/\320\2403208/kaparulin_408737/lab2/reports/Comp_math_2.pdf" "b/\320\2403208/kaparulin_408737/lab2/reports/Comp_math_2.pdf" new file mode 100644 index 0000000..bb90b13 Binary files /dev/null and "b/\320\2403208/kaparulin_408737/lab2/reports/Comp_math_2.pdf" differ diff --git "a/\320\2403208/kaparulin_408737/lab2/sys_non_lin_eq/Newton.py" "b/\320\2403208/kaparulin_408737/lab2/sys_non_lin_eq/Newton.py" new file mode 100644 index 0000000..00b7e2d --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab2/sys_non_lin_eq/Newton.py" @@ -0,0 +1,117 @@ +import numpy as np +import matplotlib.pyplot as plt +from sympy import Matrix + +sys = [ + + [ + [lambda x, y: x + y, lambda x, y: -3*x + 5*y], + [ + [lambda x, y: 1, lambda x, y: 1], + [lambda x, y: -3, lambda x, y: 5] + ] + ], + + [ + [lambda x, y: x**2 + y**2 - 4, lambda x, y: -3*x**2 + y], + [ + [lambda x, y: 2*x, lambda x, y: 2*y], + [lambda x, y: -6*x, lambda x, y: 1] + ] + ], + +] + +class Newton: + def __init__(self, system): + + self.__plot(system[0]) + self.__input_params() + self.__solution(system) + + def __solution(self, system): + f = system[0][0] + g = system[0][1] + dfdx = system[1][0][0] + dfdy = system[1][0][1] + dgdx = system[1][1][0] + dgdy = system[1][1][1] + + def matrix(x, y): + return Matrix([ + [dfdx(x, y), dfdy(x, y)], + [dgdx(x, y), dgdy(x, y)] + ]), Matrix([g(x,y), f(x,y)]) + + + + + diff = float('inf') + cnt = 0 + x = self.x0 + y = self.y0 + + while(diff > self.e): + cnt += 1 + M, ans = matrix(x, y) + + res = M.solve_least_squares(ans) + dx = res[0] + dy = res[1] + + x = x + dx + y = y + dy + + diff = min(abs(dx), abs(dy)) + + print("x1, x2: ", x, y) + print("Number of Iteration: ", cnt) + print("|x_k - x_k+1|: ", diff) + + + def __plot(self, eqs): + # Определяем диапазон значений для x и y + x = np.linspace(-10, 10, 400) + y = np.linspace(-10, 10, 400) + X, Y = np.meshgrid(x, y) + + # Определяем функцию + Z1 = eqs[0](X,Y) + Z2 = eqs[1](X,Y) + + # Строим график + plt.contour(X, Y, Z1, levels=[0], colors='blue') + plt.contour(X, Y, Z2, levels=[0], colors='red') + plt.title('График системы нелинейных уравнений') + plt.xlabel('x') + plt.ylabel('y') + plt.grid(True) + plt.savefig("System_graph.png") + + + # ввод всех параметров + def __input_params(self): + + print("Input x0: ", end="") + self.x0 = self.__get_param("x0", float, input) + + print("Input y0: ", end="") + self.y0 = self.__get_param("y0", float, input) + + print("Input e: ", end="") + self.e = self.__get_param("e", float, input) + + print() + + + + # получение 1 параметра + def __get_param(self, name, type, func): + try: + return type(func()) + except: + print(f"Incorrect {name}") + exit() + +if __name__ == "__main__": + Newton(sys[1]) \ No newline at end of file diff --git "a/\320\2403208/kaparulin_408737/lab2/test.txt" "b/\320\2403208/kaparulin_408737/lab2/test.txt" new file mode 100644 index 0000000..622cb21 --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab2/test.txt" @@ -0,0 +1,3 @@ +-1 +0 +0.01 \ No newline at end of file diff --git "a/\320\2403208/kaparulin_408737/lab3/Comp_math_3.pdf" "b/\320\2403208/kaparulin_408737/lab3/Comp_math_3.pdf" new file mode 100644 index 0000000..de9faa6 Binary files /dev/null and "b/\320\2403208/kaparulin_408737/lab3/Comp_math_3.pdf" differ diff --git "a/\320\2403208/kaparulin_408737/lab3/main.py" "b/\320\2403208/kaparulin_408737/lab3/main.py" new file mode 100644 index 0000000..1720879 --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab3/main.py" @@ -0,0 +1,85 @@ +import warnings +warnings.simplefilter("error") + +from numpy import sin, cos +import numpy as np + +from methods import TrapezoidRule, RectangleRule, SimpsonRule + + +funcs = [ + lambda x: 3*x**3 + 5*x**2 + 3*x - 6, + lambda x: sin(x) + cos(x), + lambda x: x**2 + 1, + lambda x: np.log(x), +] + +methods = [ + RectangleRule.left_rectangle_rule, + RectangleRule.midpoint_rectangle_rule, + RectangleRule.right_rectangle_rule, + TrapezoidRule.trapezoid_rule, + SimpsonRule.simpson_rule, +] + +class Main: + def __init__(self): + eq_nmb = 0 + method_nmb = 0 + + while(eq_nmb not in [1, 2, 3, 4]): + print("[1] f(x) = 3*x**3 + 5*x**2 + 3*x - 6") + print("[2] f(x) = sin(x) + cos(x)") + print("[3] f(x) = x**2 + 1") + print("[4] f(x) = ln(x)") + print("Введите номер уравнения: ", end="") + eq_nmb = self.__get_param("equation number", int, input) + print() + + while(method_nmb not in [1, 2, 3, 4, 5]): + print("[1] Метод левых прямоугольников") + print("[2] Метод серединных прямоугольников") + print("[3] Метод правых прямоугольников") + print("[4] Метод трапеций ") + print("[5] Метод Симпсона ") + print("Введите номер метода решения уравнения: ", end="") + method_nmb = self.__get_param("method name", int, input) + print() + + print("Input a: ", end="") + a = self.__get_param("a", float, input) + print("Input b: ", end="") + b = self.__get_param("b", float, input) + print("Input e: ", end="") + e = self.__get_param("e", float, input) + print() + + if (eq_nmb == 4 and a<=0) and not (eq_nmb == 4 and a==0 and (method_nmb==2 or method_nmb==3)): + print("Error: Integrated function has discontinuity in current interval") + exit() + elif a>b: + print("Error: incorrect interval") + exit() + elif a==b: + Integ, n = 0, 0 + else: + Integ, n = methods[method_nmb-1](funcs[eq_nmb-1], a, b, e=e) + + + print("Integral value: ", Integ) + + print("Number of splits: ", n) + + + + # получение 1 параметра + def __get_param(self, name, type, func): + try: + return type(func()) + except: + print(f"Incorrect {name}") + exit() + + +if __name__ == "__main__": + Main() \ No newline at end of file diff --git "a/\320\2403208/kaparulin_408737/lab3/methods/Rectangle.py" "b/\320\2403208/kaparulin_408737/lab3/methods/Rectangle.py" new file mode 100644 index 0000000..607166f --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab3/methods/Rectangle.py" @@ -0,0 +1,42 @@ +def _rectangle_rule(func, a, b, n, frac): + """Обобщённое правило прямоугольников.""" + dx = (b - a) / n + ans = 0.0 + xstart = a + frac * dx # 0 <= frac <= 1 задаёт долю смещения точки, + # в которой вычисляется функция, + # от левого края отрезка dx + for i in range(n): + ans += func(xstart + i * dx) + + return ans*dx + +def _rectangle_rule_with_e(func, a, b, n, e, frac): + ans = _rectangle_rule(func, a, b, n, frac) + err_est = max(1, abs(ans)) + + while (err_est > e): + old_ans = ans + ans = _rectangle_rule(func, a, b, 2*n, frac) + + n *= 2 + err_est = abs(ans - old_ans) + + return ans, n + +class RectangleRule: + + @classmethod + def left_rectangle_rule(self, func, a, b, n = 4, e = 1e-4): + """Правило левых прямоугольников""" + return _rectangle_rule_with_e(func, a, b, n, e, 0.0) + + @classmethod + def right_rectangle_rule(self, func, a, b, n = 4, e = 1e-4): + """Правило правых прямоугольников""" + return _rectangle_rule_with_e(func, a, b, n, e, 1.0) + + @classmethod + def midpoint_rectangle_rule(self, func, a, b, n = 4, e = 1e-4): + """Правило прямоугольников со средней точкой""" + return _rectangle_rule_with_e(func, a, b, n, e, 0.5) + \ No newline at end of file diff --git "a/\320\2403208/kaparulin_408737/lab3/methods/Simpson.py" "b/\320\2403208/kaparulin_408737/lab3/methods/Simpson.py" new file mode 100644 index 0000000..22f4203 --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab3/methods/Simpson.py" @@ -0,0 +1,28 @@ +from .Trapezoid import TrapezoidRule + +class SimpsonRule: + + @classmethod + def simpson_rule(self, func, a, b, n = 4, e = 1e-4): + """Интегрирование методом парабол с заданной точностью. + e - относительная точность, + n - число отрезков начального разбиения""" + """ + SS = (4*T(2n) - T(N))/3 + """ + old_trapez_sum, _ = TrapezoidRule.trapezoid_rule(func, a, b, n, e=float('inf')) + new_trapez_sum, _ = TrapezoidRule.trapezoid_rule(func, a, b, 2*n, e=float('inf')) + ans = (4 * new_trapez_sum - old_trapez_sum) / 3 + + err_est = max(1, abs(ans)) + + while (err_est > e): + n*=2 + old_ans = ans + old_trapez_sum = new_trapez_sum + new_trapez_sum, _ = TrapezoidRule.trapezoid_rule(func, a, b, 2*n, e=float('inf')) + + ans = (4 * new_trapez_sum - old_trapez_sum) / 3 + err_est = abs(old_ans - ans) + + return ans, n \ No newline at end of file diff --git "a/\320\2403208/kaparulin_408737/lab3/methods/Trapezoid.py" "b/\320\2403208/kaparulin_408737/lab3/methods/Trapezoid.py" new file mode 100644 index 0000000..6768231 --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab3/methods/Trapezoid.py" @@ -0,0 +1,40 @@ +def _rectangle_rule(func, a, b, n, frac): + """Обобщённое правило прямоугольников.""" + dx = 1.0 * (b - a) / n + sum = 0.0 + xstart = a + frac * dx # 0 <= frac <= 1 задаёт долю смещения точки, + # в которой вычисляется функция, + # от левого края отрезка dx + for i in range(n): + sum += func(xstart + i * dx) + + return sum * dx + +def midpoint_rectangle_rule(func, a, b, n): + """Правило прямоугольников со средней точкой""" + return _rectangle_rule(func, a, b, n, 0.5) + +class TrapezoidRule: + @classmethod + def trapezoid_rule(self, func, a, b, n = 4, e = 1e-4): + """Правило трапеций + e - желаемая относительная точность вычислений + n0 - начальное число отрезков разбиения""" + old_ans = 0.0 + dx = (b - a) / n + ans = 0.5 * (func(a) + func(b)) + + for i in range(1, n): + ans += func(a + i * dx) + + ans *= dx + err_est = max(1, abs(ans)) + while (err_est > e): + old_ans = ans + ans = 0.5 * (ans + midpoint_rectangle_rule(func, a, b, n)) # новые точки для уточнения интеграла + # добавляются ровно в середины предыдущих отрезков + n *= 2 + err_est = abs(ans - old_ans) + + return ans, n + \ No newline at end of file diff --git "a/\320\2403208/kaparulin_408737/lab3/methods/__init__.py" "b/\320\2403208/kaparulin_408737/lab3/methods/__init__.py" new file mode 100644 index 0000000..75ff69c --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab3/methods/__init__.py" @@ -0,0 +1,5 @@ +from .Simpson import SimpsonRule +from .Trapezoid import TrapezoidRule +from .Rectangle import RectangleRule + +__all__ = ["RectangleRule", "TrapezoidRule", "SimpsonRule"] \ No newline at end of file diff --git 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Функции для аппроксимации +def linear(x, a, b): + return a * x + b + +def quadratic(x, a, b, c): + return a * x**2 + b * x + c + +def cubic(x, a, b, c, d): + return a * x**3 + b * x**2 + c * x + d + +def exponential(x, a, b): + return a * np.exp(b * x) + +def logarithmic(x, a, b): + return a * np.log(x + 1e-9) + b # +1e-9 для избежания log(0) + +def power(x, a, b): + return a * (x + 1e-9)**b # +1e-9 для избежания 0^b + +# Метод наименьших квадратов для всех моделей +models = { + "Линейная": (linear, 2), + "Квадратичная": (quadratic, 3), + "Кубическая": (cubic, 4), + "Экспоненциальная": (exponential, 2), + "Логарифмическая": (logarithmic, 2), + "Степенная": (power, 2), +} + +class Main: + def __init__(self): + point_nmb = 0 + + while(point_nmb not in [8, 9, 10, 11, 12]): + print("Введите количество вводимых точек(8-12): ", end="") + point_nmb = self.__get_param("equation number", int, input) + print() + + print("Введите путь до данных(\"\" - ввод в терминале): ", end="") + file = self.__get_param("file name", str, input) + + if file: + with open(file) as f: + for i in f.readlines(): + xi, yi = map(float, i.split(" ")) + X.append(xi) + Y.append(yi) + else: + for i in range(point_nmb): + xi, yi = map(float, input().split(" ")) + X.append(xi) + Y.append(yi) + + + if len(X) != point_nmb: + print("Error: invalid point's number") + exit() + + x = np.array(X) + y = np.array(Y) + + # решение + results = {} + for name, (func, params) in models.items(): + try: + popt, pcov = curve_fit(func, x, y, maxfev=10000) + y_pred = func(x, *popt) + S = np.sum((y_pred - y)**2) + n = len(x) + mse = np.sqrt(S / n) + results[name] = { + "coeffs": popt, + "S": S, + "mse": mse, + "y_pred": y_pred, + } + except Exception as e: + print(f"Ошибка в модели {name}: {str(e)}") + + # Коэффициент корреляции Пирсона (для линейной) + if "Линейная" in results: + r = np.corrcoef(x, y)[0, 1] + results["Линейная"]["r"] = r + R2 = r**2 + results["Линейная"]["R2"] = R2 + + # Вывод результатов + print("Результаты аппроксимации:") + for name, data in results.items(): + print(f"\n--- {name} ---") + print(f"Коэффициенты: {np.round(data['coeffs'], 4)}") + print(f"S = {data['S']:.4f}") + print(f"СКО = {data['mse']:.4f}") + if name == "Линейная": + print(f"Коэф. корреляции Пирсона: {data['r']:.4f}") + print(f"Коэф. детерминации R²: {data['R2']:.4f}") + + # Выбор наилучшей модели + best_model = min(results.items(), key=lambda x: x[1]["mse"]) + print(f"\nНаилучшая модель: {best_model[0]} (СКО = {best_model[1]['mse']:.4f})") + + # Построение графиков + plt.figure(figsize=(12, 8)) + plt.scatter(x, y, label="Исходные данные", color="black") + + x_plot = np.linspace(x.min() - 0.5, x.max() + 0.5, 100) + for name, data in results.items(): + if name == "Линейная": + y_plot = linear(x_plot, *data["coeffs"]) + elif name == "Квадратичная": + y_plot = quadratic(x_plot, *data["coeffs"]) + elif name == "Кубическая": + y_plot = cubic(x_plot, *data["coeffs"]) + elif name == "Экспоненциальная": + 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"a/\320\2403208/kaparulin_408737/lab5/Comp_math_5.pdf" "b/\320\2403208/kaparulin_408737/lab5/Comp_math_5.pdf" new file mode 100644 index 0000000..9e650a0 Binary files /dev/null and "b/\320\2403208/kaparulin_408737/lab5/Comp_math_5.pdf" differ diff --git "a/\320\2403208/kaparulin_408737/lab5/data1.txt" "b/\320\2403208/kaparulin_408737/lab5/data1.txt" new file mode 100644 index 0000000..86b2676 --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab5/data1.txt" @@ -0,0 +1,3 @@ +0.0 0.0 +1.0 1.0 +2.0 2.0 \ No newline at end of file diff --git "a/\320\2403208/kaparulin_408737/lab5/data2.txt" "b/\320\2403208/kaparulin_408737/lab5/data2.txt" new file mode 100644 index 0000000..189b897 --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab5/data2.txt" @@ -0,0 +1,4 @@ +0.0 0.0 +1.0 1.0 +2.0 4.0 +3.0 9.0 \ No newline at end of file diff --git "a/\320\2403208/kaparulin_408737/lab5/data3.txt" "b/\320\2403208/kaparulin_408737/lab5/data3.txt" new file mode 100644 index 0000000..358a2ca --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab5/data3.txt" @@ -0,0 +1,3 @@ +0.0 1.0 +1.0 2.718 +2.0 7.389 \ No newline at end of file diff --git "a/\320\2403208/kaparulin_408737/lab5/graph.png" "b/\320\2403208/kaparulin_408737/lab5/graph.png" new file mode 100644 index 0000000..c7138a8 Binary files /dev/null and "b/\320\2403208/kaparulin_408737/lab5/graph.png" differ diff --git "a/\320\2403208/kaparulin_408737/lab5/main.py" "b/\320\2403208/kaparulin_408737/lab5/main.py" new file mode 100644 index 0000000..d8df696 --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab5/main.py" @@ -0,0 +1,200 @@ +import numpy as np +import matplotlib.pyplot as plt +from scipy.special import factorial +from math import sin, cos, exp, log + +# Построение таблицы конечных разностей +def finite_diff(y): + n = len(y) + table = np.zeros((n, n)) + table[:,0] = y + for j in range(1, n): + for i in range(n - j): + table[i][j] = table[i+1][j-1] - table[i][j-1] + return table + +# 1. Интерполяция многочленом Лагранжа +def lagrange_interpolation(x, y, x_val): + result = 0.0 + n = len(x) + for i in range(n): + term = y[i] + for j in range(n): + if j != i: + term *= (x_val - x[j]) / (x[i] - x[j]) + result += term + return result + +# 2. Интерполяция Ньютона с разделенными разностями +def newton_divided_diff(x, y, x_val): + n = len(x) + coef = np.zeros([n, n]) + coef[:,0] = y + + for j in range(1, n): + for i in range(n - j): + coef[i][j] = (coef[i+1][j-1] - coef[i][j-1]) / (x[i+j] - x[i]) + + result = coef[0][0] + for j in range(1, n): + term = coef[0][j] + for k in range(j): + term *= (x_val - x[k]) + result += term + return result + +# 3. Интерполяция Ньютона с конечными разностями (1 и 2 формулы) +def newton_forward(x, y, x_val): + h = x[1] - x[0] + t = (x_val - x[0]) / h + diff = finite_diff(y) + result = y[0] + for i in range(1, len(x)): + term = 1.0 + for j in range(i): + term *= (t - j) + result += (term / factorial(i)) * diff[0, i] + return result + +def newton_backward(x, y, x_val): + h = x[1] - x[0] + t = (x_val - x[-1]) / h + diff = finite_diff(y) + result = y[-1] + for i in range(1, len(x)): + term = 1.0 + for j in range(i): + term *= (t + j) + result += (term / factorial(i)) * diff[-i-1, i] + return result + +# Функции для выбора в режиме (c) +def func_sin(x): + return sin(x) + +def func_cos(x): + return cos(x) + +def func_exp(x): + return exp(x) + +def func_log(x): + return log(x) if x > 0 else float('nan') + +def input_data(): + print("Выберите способ ввода данных:") + print("a) Ввод с клавиатуры") + print("b) Загрузка из файла") + print("c) Генерация на основе функции") + choice = input("Ваш выбор (a/b/c): ").strip().lower() + + if choice == 'a': + # Режим (a): ввод с клавиатуры + n = int(input("Введите количество точек: ")) + x = [] + y = [] + for i in range(n): + xi = float(input(f"x[{i}]: ")) + yi = float(input(f"y[{i}]: ")) + x.append(xi) + y.append(yi) + return x, y + + elif choice == 'b': + # Режим (b): загрузка из файла + filename = input("Введите имя файла: ") + with open(filename, 'r') as f: + lines = f.readlines() + x = [] + y = [] + for line in lines: + xi, yi = map(float, line.strip().split()) + x.append(xi) + y.append(yi) + return x, y + + elif choice == 'c': + # Режим (c): генерация на основе функции + print("Доступные функции:") + print("1. sin(x)") + print("2. cos(x)") + print("3. exp(x)") + print("4. log(x)") + func_choice = int(input("Выберите функцию (1-4): ")) + a = float(input("Начало интервала: ")) + b = float(input("Конец интервала: ")) + n = int(input("Количество точек: ")) + + x = np.linspace(a, b, n) + if func_choice == 1: + y = [func_sin(xi) for xi in x] + elif func_choice == 2: + y = [func_cos(xi) for xi in x] + elif func_choice == 3: + y = [func_exp(xi) for xi in x] + elif func_choice == 4: + y = [func_log(xi) if xi > 0 else float('nan') for xi in x] + else: + raise ValueError("Неверный выбор функции") + + # Удаляем точки, где log(x) не определен + if func_choice == 4: + x = [xi for xi, yi in zip(x, y) if not np.isnan(yi)] + y = [yi for yi in y if not np.isnan(yi)] + + return x, y + + else: + raise ValueError("Неверный выбор способа ввода") + +if __name__ == "__main__": + try: + x, y = input_data() + print("\nВведенные данные:") + for xi, yi in zip(x, y): + print(f"x = {xi:.4f}, y = {yi:.4f}") + + # Запрос точки интерполяции + x_val = float(input("\nВведите значение x для интерполяции: ")) + + # Вычисление интерполяции + print("\nРезультаты интерполяции:") + print(f"Лагранж: y({x_val}) ≈ {lagrange_interpolation(x, y, x_val):.6f}") + print(f"Ньютон (раздел. разности): y({x_val}) ≈ {newton_divided_diff(x, y, x_val):.6f}") + + if len(set(np.diff(x))) == 1: # Проверка равномерности сетки + print(f"Ньютон (вперед): y({x_val}) ≈ {newton_forward(x, y, x_val):.6f}") + print(f"Ньютон (назад): y({x_val}) ≈ {newton_backward(x, y, x_val):.6f}") + print() + print("Таблица конечных разностей:\n") + + diff_table = finite_diff(y) + + for line in diff_table: + for item in line: + print("%.3f\t" % (item), end="") + print() + print() + else: + print("Методы Ньютона с конечными разностями требуют равномерной сетки!") + + + + # Построение графиков + x_new = np.linspace(min(x), max(x), 100) + y_lagrange = [lagrange_interpolation(x, y, xi) for xi in x_new] + y_newton_div = [newton_divided_diff(x, y, xi) for xi in x_new] + + plt.figure(figsize=(10, 6)) + plt.plot(x, y, 'o', label='Узлы интерполяции') + plt.plot(x_new, y_lagrange, label='Лагранж') + plt.plot(x_new, y_newton_div, '--', label='Ньютон (раздел. разности)') + plt.xlabel('x') + plt.ylabel('y') + plt.legend() + plt.grid() + plt.title('Интерполяция функции') + 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"b/\320\2403208/kaparulin_408737/lab6/graph.png" differ diff --git "a/\320\2403208/kaparulin_408737/lab6/main.py" "b/\320\2403208/kaparulin_408737/lab6/main.py" new file mode 100644 index 0000000..c4ff931 --- /dev/null +++ "b/\320\2403208/kaparulin_408737/lab6/main.py" @@ -0,0 +1,124 @@ +import numpy as np +import matplotlib.pyplot as plt + +# Определение ОДУ и их точных решений +def f1(x, y): + return x + y + +def y_exact1(x, x0, y0): + C = (y0 - x0 - 1) * np.exp(-x0) + return C * np.exp(x) + x + 1 + +def f2(x, y): + return y * np.cos(x) + +def y_exact2(x, x0, y0): + return y0 * np.exp(np.sin(x) - np.sin(x0)) + +def f3(x, y): + return x**2 - y + +def y_exact3(x, x0, y0): + return (y0 - x0**2 + 2*x0 - 2) * np.exp(-(x - x0)) + x**2 - 2*x + 2 + +# Методы решения +def euler(f, x0, y0, h, xn): + x = np.arange(x0, xn + h, h) + y = np.zeros(len(x)) + y[0] = y0 + for i in range(1, len(x)): + y[i] = y[i-1] + h * f(x[i-1], y[i-1]) + return x, y + +def improved_euler(f, x0, y0, h, xn): + x = np.arange(x0, xn + h, h) + y = np.zeros(len(x)) + y[0] = y0 + for i in range(1, len(x)): + k1 = h * f(x[i-1], y[i-1]) + k2 = h * f(x[i-1] + h, y[i-1] + k1) + y[i] = y[i-1] + (k1 + k2) / 2 + return x, y + +def milne(f, x0, y0, h, xn, start_method): + x = np.arange(x0, xn + h, h) + n = len(x) + y = np.zeros(n) + # Используем стартовый метод для первых 4 точек + x_start, y_start = start_method(f, x0, y0, h, x0 + 3*h) + y[:4] = y_start[:4] + for i in range(3, n-1): + # Предиктор + y_pred = y[i-3] + 4*h/3 * (2*f(x[i], y[i]) - f(x[i-1], y[i-1]) + 2*f(x[i-2], y[i-2])) + # Корректор + y[i+1] = y[i-1] + h/3 * (f(x[i+1], y_pred) + 4*f(x[i], y[i]) + f(x[i-1], y[i-1])) + return x, y + +if __name__ == "__main__": + # Выбор ОДУ + print("Выберите ОДУ:") + print("1. y' = x + y") + print("2. y' = y * cos(x)") + print("3. y' = x^2 - y") + choice = int(input("Введите номер ОДУ (1-3): ")) + + if choice == 1: + f = f1 + y_exact = lambda x: y_exact1(x, x0, y0) + elif choice == 2: + f = f2 + y_exact = lambda x: y_exact2(x, x0, y0) + elif choice == 3: + f = f3 + y_exact = lambda x: y_exact3(x, x0, y0) + else: + print("Некорректный выбор") + exit() + + x0 = float(input("Введите x0: ")) + y0 = float(input("Введите y0: ")) + xn = float(input("Введите xn: ")) + h = float(input("Введите шаг h: ")) + epsilon = float(input("Введите точность ε: ")) + + if xn <= x0: + print("Некорректный интервал") + exit() + + # Решение методами + x_euler, y_euler = euler(f, x0, y0, h, xn) + x_ie, y_ie = improved_euler(f, x0, y0, h, xn) + x_milne, y_milne = milne(f, x0, y0, h, xn, improved_euler) + + # Точное решение + x_exact = np.linspace(x0, xn, 100) + y_exact_vals = y_exact(x_exact) + + # Оценка точности + # Правило Рунге для метода Эйлера + h2 = h / 2 + x_euler_h2, y_euler_h2 = euler(f, x0, y0, h2, xn) + error_euler = np.abs(y_euler - y_euler_h2[::2][:len(y_euler)]) + + # Ошибка для Милна через точное решение + y_milne_exact = y_exact(x_milne) + error_milne = np.abs(y_milne - y_milne_exact) + + # Вывод таблицы + print("\nТаблица результатов:") + print("x\tЭйлер\tУ.Эйлер\tМилна\tErr_Э\tErr_М") + for i in range(len(x_euler)): + print(f"{x_euler[i]:.2f}\t{y_euler[i]:.3f}\t{y_ie[i]:.3f}\t{y_milne[i]:.3f}\t{error_euler[i]:.4f}\t{error_milne[i]:.4f}") + + # Графики + plt.figure(figsize=(10, 6)) + plt.plot(x_exact, y_exact_vals, label='Точное решение', color='black') + plt.plot(x_euler, y_euler, 'o--', label='Метод Эйлера') + plt.plot(x_ie, y_ie, 's--', label='У. 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