From 95c00708408c37fe1fe588a53b8b163bfa16d56c Mon Sep 17 00:00:00 2001 From: Karim Ziyatdinov <408659@niuitmo.ru> Date: Tue, 3 Jun 2025 12:07:00 +0300 Subject: [PATCH] =?UTF-8?q?=D0=94=D0=BE=D0=B1=D0=B0=D0=B2=D0=B8=D0=BB=20?= =?UTF-8?q?=D0=B2=D1=81=D0=B5=20=D0=BB=D0=B0=D0=B1=D1=8B?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- "\320\2403213/ziyatdinov_408659/lab1/main.py" | 157 ++++++++++ ...0\253\320\247\320\234\320\220\320\242.pdf" | Bin 0 -> 564124 bytes .../ziyatdinov_408659/lab2/lab2(1_method).py" | 290 ++++++++++++++++++ ...42_\320\233\320\220\320\221\320\220_2.pdf" | Bin 0 -> 1015137 bytes .../ziyatdinov_408659/lab3/Vich_Mat_3.py" | 263 ++++++++++++++++ ...13\321\207_\320\274\320\260\321\202_3.pdf" | Bin 0 -> 1056907 bytes .../ziyatdinov_408659/lab4/Vich_mat_4.py" | 264 ++++++++++++++++ ...13\321\207_\320\274\320\260\321\202_4.pdf" | Bin 0 -> 808989 bytes .../ziyatdinov_408659/lab5/Vich_mat_5.py" | 223 ++++++++++++++ ...13\321\207_\320\274\320\260\321\202_5.pdf" | Bin 0 -> 792398 bytes .../ziyatdinov_408659/lab6/Vich_mat_6.py" | 155 ++++++++++ ...13\321\207_\320\274\320\260\321\202_6.pdf" | Bin 0 -> 490790 bytes 12 files changed, 1352 insertions(+) create mode 100644 "\320\2403213/ziyatdinov_408659/lab1/main.py" create mode 100644 "\320\2403213/ziyatdinov_408659/lab1/reports/\320\233\320\220\320\221\320\2201_\320\222\320\253\320\247\320\234\320\220\320\242.pdf" create mode 100644 "\320\2403213/ziyatdinov_408659/lab2/lab2(1_method).py" create mode 100644 "\320\2403213/ziyatdinov_408659/lab2/reports/\320\222\320\253\320\247_\320\234\320\220\320\242_\320\233\320\220\320\221\320\220_2.pdf" create mode 100644 "\320\2403213/ziyatdinov_408659/lab3/Vich_Mat_3.py" create mode 100644 "\320\2403213/ziyatdinov_408659/lab3/reports/\320\222\321\213\321\207_\320\274\320\260\321\202_3.pdf" create mode 100644 "\320\2403213/ziyatdinov_408659/lab4/Vich_mat_4.py" create mode 100644 "\320\2403213/ziyatdinov_408659/lab4/reports/\320\222\321\213\321\207_\320\274\320\260\321\202_4.pdf" create mode 100644 "\320\2403213/ziyatdinov_408659/lab5/Vich_mat_5.py" create mode 100644 "\320\2403213/ziyatdinov_408659/lab5/reports/\320\222\321\213\321\207_\320\274\320\260\321\202_5.pdf" create mode 100644 "\320\2403213/ziyatdinov_408659/lab6/Vich_mat_6.py" create mode 100644 "\320\2403213/ziyatdinov_408659/lab6/reports/\320\222\321\213\321\207_\320\274\320\260\321\202_6.pdf" diff --git "a/\320\2403213/ziyatdinov_408659/lab1/main.py" "b/\320\2403213/ziyatdinov_408659/lab1/main.py" new file mode 100644 index 0000000..a250fa4 --- /dev/null +++ "b/\320\2403213/ziyatdinov_408659/lab1/main.py" @@ -0,0 +1,157 @@ +import sys +import numpy as np + +def random_matrix(): + n = int(input("Введите размер (n): ")) + matrix = np.random.randint(-1000, 1000, (n, n)) + d = np.random.randint(-1000, 1000, n) + return n, matrix, d + + +def create_matrix(): + n = int(input("Введите размер (n): ")) + + print("Введите матрицу (n на n):") + matrix = [] + for i in range(n): + while True: + row = input().split() + if len(row) == n: + try: + matrix.append([float(x) for x in row]) + break + except ValueError: + pass + print(f"Введите пж {n} чисел") + print("Введите вектор b:") + while True: + b = input().split() + if len(b) == n: + try: + b = [float(x) for x in b] + break + except ValueError: + pass + print(f"Введите пж {n} чисел") + + return n, matrix, b + + +def perebor_strok(lst): + if len(lst) <= 1: + yield lst + else: + for i in range(len(lst)): + first = lst[i] + rest = lst[:i] + lst[i + 1:] + for p in perebor_strok(rest): + yield [first] + p + + +def change_stroke(A, b): + n = len(A) + indices = list(range(n)) + + for el_perebor in perebor_strok(indices): + val = True + for i in range(n): + diagonal = abs(A[el_perebor[i]][i]) + nodiagonal = sum(abs(A[el_perebor[i]][j]) for j in range(n) if j != i) + if diagonal < nodiagonal: + val = False + break + + if val: + new_A = [A[idx] for idx in el_perebor] + new_b = [b[idx] for idx in el_perebor] + print("Найдена перестановка строк для диагонального преобладания:") + print("Перестановка индексов:", el_perebor) + return new_A, new_b + + print("Перестановкой строк добиться диагонального преобладания не удалось.") + return A, b + + +def displaying(matrix): + for row in matrix: + print(*row) + + +def create_C_d(A, b): + n = len(A) + C = [[0.0]*n for _ in range(n)] + d = [0.0]*n + + for i in range(n): + d[i] = b[i] / A[i][i] + for j in range(n): + if i == j: + C[i][j] = 0.0 + else: + C[i][j] = -A[i][j] / A[i][i] + return C, d + + +def matrix_norm(C): + return max(sum(abs(x) for x in row) for row in C) + + +def gaussSeidel(C, d, eps, max_iter): + n = len(C) + old = d + + for k in range(max_iter): + new = old[:] # копируем список + + for i in range(n): + s1 = sum(C[i][j] * new[j] for j in range(i)) + s2 = sum(C[i][j] * old[j] for j in range(i + 1, n)) + new[i] = d[i] + s1 + s2 + + diff = max(abs(new[i] - old[i]) for i in range(n)) + print(f"Погрешность для {k + 1}ой итерации:", diff) + if diff < eps: + return new, k + 1 + + old = new + + return old, max_iter + + +def main(): + mode = input("Выберите источник ввода (1 - консоль, 2 - файл): ") + if mode.strip() == "2": + filename = input("Введите имя файла: ") + sys.stdin = open(filename, "r", encoding="utf-8") + + n, A, b = random_matrix() + displaying(A) + # n, A, b = create_matrix() + + A, b = change_stroke(A, b) + C, d = create_C_d(A, b) + + print("матрица С") + displaying(C) + + norma = matrix_norm(C) + print("Норма матрицы C (по строкам) =", norma) + if norma < 1: + print("Условие ||C|| < 1 выполняется") + else: + print("Условие ||C|| < 1 не выполняется") + + eps_str = input("Введите точность: ") + eps = float(eps_str) + max_iter_str = input("Введите максимальное число итераций: ") + max_iter = int(max_iter_str) + + x, k = gaussSeidel(C, d, eps, max_iter) + + print("Результат решения системы:") + for i in range(n): + print(f"x[{i+1}] =", x[i]) + 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zR7dJ?G-=R4_#@kw;_dqv`VAahY#WHuz(Y>k2Bbk-TZ+PALlQ&$0QkB6wFJ`iZYT$7 UB^GSjW529rqm2gd($?~S0B-0y$N&HU literal 0 HcmV?d00001 diff --git "a/\320\2403213/ziyatdinov_408659/lab2/lab2(1_method).py" "b/\320\2403213/ziyatdinov_408659/lab2/lab2(1_method).py" new file mode 100644 index 0000000..c34dd0d --- /dev/null +++ "b/\320\2403213/ziyatdinov_408659/lab2/lab2(1_method).py" @@ -0,0 +1,290 @@ +import math +import random +import numpy as np +import matplotlib.pyplot as plt + +def r_uravnenie(degree, min_k, max_k): + + koefs = [random.randint(min_k, max_k) for _ in range(degree + 1)] + def uravnenie(x): + return sum(koefs[i] * x**i for i in range(len(koefs))) + return uravnenie, koefs + + +def plot_function_system(f_system, x_min, x_max, y_min, y_max, root, title="График системы с решением"): + def f1(x, y): + return f_system([x, y])[0] + + def f2(x, y): + return f_system([x, y])[1] + + X, Y = np.meshgrid( + np.linspace(x_min, x_max, 300), + np.linspace(y_min, y_max, 300) + ) + + Z1 = np.zeros_like(X) + Z2 = np.zeros_like(X) + + for i in range(X.shape[0]): + for j in range(X.shape[1]): + Z1[i, j] = f1(X[i, j], Y[i, j]) + Z2[i, j] = f2(X[i, j], Y[i, j]) + + plt.figure(figsize=(7, 6)) + + plt.contour(X, Y, Z1, levels=[0], colors='red') + plt.contour(X, Y, Z2, levels=[0], colors='blue') + + + x, y = root + plt.axhline(0, color='black', linewidth=0.8) + plt.axvline(0, color='black', linewidth=0.8) + + plt.plot(x, y, 'go', label=f'Решение ({x:.4f}, {y:.4f})') + plt.text(x, y, f"({x:.4f}, {y:.4f})", fontsize=9, ha='left', va='bottom') + + plt.grid(True) + plt.xlim(x_min, x_max) + plt.ylim(y_min, y_max) + plt.title(title) + plt.legend() + plt.show() + + +def p_r_uravnenie(koefs): + + p_koefs = [] + for i in range(1, len(koefs)): + p_koefs.append(i * koefs[i]) + def p_uravnenie(x): + return sum(p_koefs[i] * x**i for i in range(len(p_koefs))) + return p_uravnenie, p_koefs + + + +def f_example(x): + # return math.cos(2*x) - 7 * math.sin(x) - 4 + return x**3 - x + 4 + +def df_example(x): + # return -2 * math.sin(2 * x) - 7 * math.cos(x) + return 3*x**2 - 1 + + + +def f_system(vars_xy): + x, y = vars_xy + # return np.array([ + # math.cos(x - 1) + y - 0.5, + # x - math.cos(y) - 3 + # ]) + return np.array([ + x**2 + y**2 - 4, + y - 3*x**2 + ]) + +def df_system(vars_xy): + x, y = vars_xy + # return np.array([ + # [-math.sin(x - 1), 1], + # [1, math.sin(y)] + # ]) + return np.array([ + [2*x, 2*y], + [-6*x, 1] + ]) + +def plot_function(f, a, b,root, title="График функции"): + + x_vals = np.linspace(a, b, 400) + y_vals = [f(x) for x in x_vals] + plt.figure(figsize=(7, 4)) + plt.plot(x_vals, y_vals, label='f(x)') + plt.axhline(0, color='black', linewidth=0.8) + + y_root = f(root) + plt.plot(root, y_root, 'ro', label=f'Решение: x={root:.4f}') + plt.text(root, y_root, f"({root:.4f}, {y_root:.4f})", fontsize=9, ha='left', va='bottom') + + plt.grid(True) + plt.xlim(a, b) + ymin, ymax = min(y_vals), max(y_vals) + plt.ylim(ymin - abs(ymin)*0.1, ymax + abs(ymax)*0.1) + plt.title(title) + plt.legend() + plt.show() + + +def chord_method(f, a, b, accuracy, m_c): + + if f(a) * f(b) > 0: + raise ValueError("На отрезке [a, b] функция не меняет знак. Корень может отсутствовать или быть не единственным.") + count = 0 + for _ in range(int(m_c)): + count += 1 + x = (a * f(b) - b * f(a)) / (f(b) - f(a)) + if abs(f(x)) <= accuracy: + return x, count, f(x) + if f(a) * f(x) < 0: + b = x + else: + a = x + x = (a * f(b) - b * f(a)) / (f(b) - f(a)) + return x, count, f(x) + + +def secant_method(f, x0, x1, accuracy, m_c): + count = 0 + for _ in range(int(m_c)): + count += 1 + value_in_x0 = f(x0) + value_in_x1 = f(x1) + if abs(value_in_x1 - value_in_x0) < 1e-14: + raise ZeroDivisionError("Разница f(x1) и f(x0) слишком мала, деление невозможно.") + x2 = x1 - value_in_x1 * (x1 - x0) / (value_in_x1 - value_in_x0) + if abs(x2 - x1) < accuracy: + return x2, count, f(x2) + x0, x1 = x1, x2 + return x2, count, f(x2) + + +def iteration_easy_method(fi, x0, accuracy, m_c): + count = 0 + for _ in range(int(m_c)): + count += 1 + x1 = fi(x0) + if abs(x1 - x0) < accuracy: + return x1, count + x0 = x1 + return x0, count + + +def compute_alpha(df, a, b, steps=100): + mas_val = np.linspace(a, b, steps) + max_val = max(abs(df(x)) for x in mas_val) + if (max_val == 0): + max_val = 1 + return 1.0 / max_val + + + +def newton_method_system(fun, dfun, x0, accuracy, count): + x = np.array(x0, dtype=float) + for i in range(count): + F = fun(x) + dF = dfun(x) + try: + delta = np.linalg.solve(dF, F) + except np.linalg.LinAlgError: + return x, i + x_new = x - delta + + + if abs(x_new[0] - x[0]) <= accuracy and abs(x_new[1] - x[1]) <= accuracy: + return x_new, i + 1 + x = x_new + return x, count + + + + +def main(): + print("Хотите решить ОДНОМЕРНОЕ уравнение или СИСТЕМУ?") + print("1) Одномерное уравнение") + print("2) Система (метод Ньютона)") + choice_global = input("Выберите (1 или 2): ") + + if choice_global == '1': + print("1) Использовать случайное уравнение?") + print("2) Использовать функцию ?") + choice = input("Выберите (1 или 2): ") + + if choice == '1': + deg = int(input("Введите степень случайного уравнения: ")) + poly, coeffs = r_uravnenie(deg, -5, 5) + print(f"Случайно сгенерированное уравнение (коэффициенты): {coeffs[::-1]}") + f = poly + dpoly, dcoeffs = p_r_uravnenie(coeffs) + df = dpoly + else: + f = f_example + df = df_example + + print("Введите границы интервала (a b): ") + inp = input().split() + a = float(inp[0]) + b = float(inp[1]) + + # plot_function(f, a, b, title="График выбранной функции на интервале") + + print("Введите требуемую точность: ") + accuracy = float(input()) + + print("Введите максимальное число итераций: ") + count = int(input()) + + print("Какой метод использовать?") + print("1) Метод хорд") + print("2) Метод секущих") + print("3) Метод простой итерации") + choice_method = input("Выберите (1, 2, 3): ") + + if choice_method == '1': + try: + root, n_iter, f_val = chord_method(f, a, b, accuracy, count) + print(f"\nМетод хорд:\nНайденный корень: {root}\nКоличество итераций: {n_iter}\nf(root) = {f_val}") + plot_function(f, a, b, root, title="График выбранной функции на интервале") + except ValueError as e: + print(f"Ошибка: {e}") + + elif choice_method == '2': + try: + root, n_iter, f_val = secant_method(f, a, b, accuracy, count) + print(f"\nМетод секущих:\nНайденный корень: {root}\nКоличество итераций: {n_iter}\nf(root) = {f_val}") + plot_function(f, a, b, root, title="График выбранной функции на интервале") + except ZeroDivisionError as e: + print(f"Ошибка: {e}") + + elif choice_method == '3': + alpha = compute_alpha(df, a, b) + print(f"Вычислен alpha = {alpha}") + def fi(x): + return x - alpha * f(x) + x0 = (a + b) / 2 + + root, n_iter = iteration_easy_method(fi, x0, accuracy, count) + print(f"\nМетод простой итерации:\nНайденный корень: {root}\nКоличество итераций: {n_iter}\nf(root) = {f(root)}") + plot_function(f, a, b, root, title="График выбранной функции на интервале") + else: + print("Неправильный выбор метода!") + + else: + print("Решаем систему методом Ньютона.") + print("Для демонстрации возьмём систему со слайдов:\n 1) x^2 + y^2 = 4\n 2) y - 3x^2 = 0\n") + + print("Введите начальное приближение (x0, y0): ") + inp = input().split() + x0 = float(inp[0]) + y0 = float(inp[1]) + + print("Введите требуемую точность: ") + accuracy = float(input()) + + print("Введите максимальное число итераций: ") + count = int(input()) + + root, iters = newton_method_system(f_system, df_system, [x0, y0], accuracy, count) + if (root[0] == x0 and root[1] == y0) and (f_system([x0, y0])[0] != 0 or f_system([x0, y0])[1] != 0): + root, iters = newton_method_system(f_system, df_system, [x0 + accuracy, y0 + accuracy], accuracy, count) + print(f_system([0, 0])) + print("\nМетод Ньютона") + print(f"Решения: x = {root[0]:.6f}, y = {root[1]:.6f}") + print(f"Число итераций: {iters}") + print(f"f1(x,y) = {f_system(root)[0]:.6e}") + print(f"f2(x,y) = {f_system(root)[1]:.6e}") + plot_function_system(f_system, root[0] - 3, root[0] + 3, root[1] - 3, root[1] + 3, root, title="График системы с решением") + + +if __name__ == "__main__": + main() diff --git 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