From dae3d40f2a0356b6eb90d1dd02a3a9d7bbf41760 Mon Sep 17 00:00:00 2001 From: "shamatulskiy.r" Date: Mon, 16 Jun 2025 14:31:27 +0300 Subject: [PATCH] added 1-6 labs --- .../shamatulskiy_409870/lab1/lab1.docx" | Bin 0 -> 26461 bytes .../shamatulskiy_409870/lab1/lab1.py" | 69 +++++++ .../shamatulskiy_409870/lab2/lab2.py" | 125 ++++++++++++ .../shamatulskiy_409870/lab2/lab2_system.py" | 93 +++++++++ .../shamatulskiy_409870/lab2/report.docx" | Bin 0 -> 680265 bytes .../shamatulskiy_409870/lab3/report.docx" | Bin 0 -> 207368 bytes .../shamatulskiy_409870/lab3/task.py" | 168 ++++++++++++++++ .../shamatulskiy_409870/lab4/report.docx" | Bin 0 -> 548207 bytes .../shamatulskiy_409870/lab4/task.py" | 179 ++++++++++++++++++ .../shamatulskiy_409870/lab5/report.docx" | Bin 0 -> 384234 bytes .../shamatulskiy_409870/lab5/task.py" | 129 +++++++++++++ .../shamatulskiy_409870/lab6/report.docx" | Bin 0 -> 242344 bytes .../shamatulskiy_409870/lab6/task.py" | 129 +++++++++++++ 13 files changed, 892 insertions(+) create mode 100644 "\320\2403212/shamatulskiy_409870/lab1/lab1.docx" create mode 100644 "\320\2403212/shamatulskiy_409870/lab1/lab1.py" create mode 100644 "\320\2403212/shamatulskiy_409870/lab2/lab2.py" create mode 100644 "\320\2403212/shamatulskiy_409870/lab2/lab2_system.py" create mode 100644 "\320\2403212/shamatulskiy_409870/lab2/report.docx" create mode 100644 "\320\2403212/shamatulskiy_409870/lab3/report.docx" create mode 100644 "\320\2403212/shamatulskiy_409870/lab3/task.py" create mode 100644 "\320\2403212/shamatulskiy_409870/lab4/report.docx" create mode 100644 "\320\2403212/shamatulskiy_409870/lab4/task.py" create mode 100644 "\320\2403212/shamatulskiy_409870/lab5/report.docx" create mode 100644 "\320\2403212/shamatulskiy_409870/lab5/task.py" create mode 100644 "\320\2403212/shamatulskiy_409870/lab6/report.docx" create mode 100644 "\320\2403212/shamatulskiy_409870/lab6/task.py" diff --git "a/\320\2403212/shamatulskiy_409870/lab1/lab1.docx" 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+++ "b/\320\2403212/shamatulskiy_409870/lab1/lab1.py" @@ -0,0 +1,69 @@ +import numpy as np + +def gauss_seidel(A, b, tol, max_iter=10000): + n = len(b) + x = np.zeros(n) + for k in range(1, max_iter+1): + x_old = x.copy() + for i in range(n): + s = np.dot(A[i, :i], x[:i]) + np.dot(A[i, i+1:], x_old[i+1:]) + x[i] = (b[i] - s)/A[i, i] + err = np.abs(x - x_old) + if np.all(err < tol): + return x, k, err + return x, k, err + +def make_diagonally_dominant(A, b): + n = A.shape[0] + for i in range(n): + for j in range(i, n): + if abs(A[j, i]) >= np.sum(np.abs(A[j, :])) - abs(A[j, i]): + if j != i: + A[[i, j]], b[[i, j]] = A[[j, i]].copy(), b[[j, i]].copy() + break + for i in range(n): + if abs(A[i, i]) < np.sum(np.abs(A[i, :])) - abs(A[i, i]): + return False + return True + +def read_matrix(): + mode = input("Чтение матрицы из файла? (y/n): ") + if mode.lower() == "y": + fn = input("Имя файла: ") + data = np.loadtxt(fn) + A = data[:, :-1] + b = data[:, -1] + else: + n = int(input("n: ")) + A = np.zeros((n, n)) + b = np.zeros(n) + for i in range(n): + row = list(map(float, input(f"Строка {i+1} (n+1 чисел): ").split())) + A[i, :] = row[:-1] + b[i] = row[-1] + return A, b + +def read_tol(): + mode = input("Чтение точности из файла? (y/n): ") + if mode.lower() == "y": + fn = input("Имя файла с точностью: ") + tol = float(open(fn).read().strip()) + else: + tol = float(input("Точность: ")) + return tol + +def main(): + A, b = read_matrix() + tol = read_tol() + if not make_diagonally_dominant(A, b): + print("Невозможно добиться диагонального преобладания") + return + x, iters, err = gauss_seidel(A, b, tol) + norm = np.linalg.norm(A, ord=np.inf) + print("Норма матрицы (сумма по строкам, бесконечная):", norm) + print("Решение x:", x) + print("Число итераций:", iters) + print("Вектор погрешностей:", err) + +if __name__ == "__main__": + main() diff --git "a/\320\2403212/shamatulskiy_409870/lab2/lab2.py" "b/\320\2403212/shamatulskiy_409870/lab2/lab2.py" new file mode 100644 index 0000000..be62797 --- /dev/null +++ "b/\320\2403212/shamatulskiy_409870/lab2/lab2.py" @@ -0,0 +1,125 @@ +import math +import sys +import matplotlib.pyplot as plt + +def f1(x): + return x**3 - 2*x - 5 + +def f2(x): + return math.sin(x) - 0.5*x + +def f3(x): + return math.exp(-x) - x + +functions = { + '1': ('x^3 - 2x - 5', f1), + '2': ('sin(x) - 0.5x', f2), + '3': ('e^(-x) - x', f3), +} + +def phi3(x): + return math.exp(-x) + +phis = { + '3': phi3, +} + +def bisection(f,a,b,eps): + fa,fb=f(a),f(b) + if fa*fb>0: return None,None,None + n=0 + while (b-a)/2>eps: + c=(a+b)/2 + if f(a)*f(c)<=0: + b=c + else: + a=c + n+=1 + return (a+b)/2,f((a+b)/2),n + +def secant(f,a,b,eps): + fa,fb=f(a),f(b) + if fa==fb: return None,None,None + n=0 + while abs(b-a)>eps: + c=b - fb*(b-a)/(fb-fa) + a,b,fa,fb=b,c,fb,f(c) + n+=1 + return b,f(b),n + +def simple_iter(f,phi,a,b,eps): + mp = max(abs((phi(a+1e-6)-phi(a))/1e-6),abs((phi(b)-phi(b-1e-6))/1e-6)) + if mp>=1: return 'bad',None,None + x0=(a+b)/2 + n=0 + while True: + x1=phi(x0) + n+=1 + if abs(x1-x0)0: + print('На интервале нет единственного корня'); sys.exit() + if method=='1': + root,val,it=bisection(f,a,b,eps) + elif method=='2': + root,val,it=secant(f,a,b,eps) + else: + phi=phis.get(eq) + if not phi: + print('Итерационная форма не задана'); sys.exit() + root,val,it=simple_iter(f,phi,a,b,eps) + if root=='bad': + print('Условие сходимости не выполнено'); sys.exit() + output(root,val,it) + plot(f,a,b) + +if __name__=='__main__': + main() diff --git "a/\320\2403212/shamatulskiy_409870/lab2/lab2_system.py" "b/\320\2403212/shamatulskiy_409870/lab2/lab2_system.py" new file mode 100644 index 0000000..71c5067 --- /dev/null +++ "b/\320\2403212/shamatulskiy_409870/lab2/lab2_system.py" @@ -0,0 +1,93 @@ +import math +import sys +import numpy as np +import matplotlib.pyplot as plt + +def sys1(x): + u,v = x + return np.array([u**2 + v**2 - 4, u*v - 1]) + +def phi1(x): + u,v = x + return np.array([math.sqrt(max(0,4 - v**2)), 1/u]) + +def sys2(x): + u,v = x + return np.array([math.sin(u) + v - 1, u + math.cos(v) - 1]) + +def phi2(x): + u,v = x + return np.array([math.asin(1 - v), math.acos(1 - u)]) + +systems = { + '1': (sys1, phi1), + '2': (sys2, phi2), +} + +def jacobian(phi, x, h=1e-6): + n = len(x) + J = np.zeros((n,n)) + for i in range(n): + dx = np.zeros(n); dx[i] = h + J[:,i] = (phi(x+dx) - phi(x-dx))/(2*h) + return J + +def simple_iter(sysf, phi, x0, eps, maxit=1000): + J = jacobian(phi, x0) + if max(abs(np.linalg.eigvals(J))) >= 1: + print('Условие сходимости не выполнено'); sys.exit() + x_prev = np.array(x0) + errors = [] + for k in range(1, maxit+1): + x_new = phi(x_prev) + err = np.abs(x_new - x_prev) + errors.append(err) + if np.all(err < eps): + return x_new, k, errors + x_prev = x_new + print('Не сошлось за max итераций'); sys.exit() + +def get_input(): + print('Системы:') + print('1: x^2 + y^2 = 4; x*y = 1') + print('2: sin(x) + y = 1; x + cos(y) = 1') + sysn = input('Выберите номер системы: ') + x0 = float(input('x0= ')), float(input('y0= ')) + eps = float(input('ε= ')) + return sysn, x0, eps + +def output(x, it, err): + if input('Вывод в файл? (y/n) ').lower()=='y': + fn = input('Имя файла: ') + with open(fn, 'w') as g: + g.write(f'x = {x}\n') + g.write(f'Итераций = {it}\n') + g.write(f'Ошибка последней итерации = {err}\n') + else: + print('x =', x) + print('Итераций =', it) + print('Ошибка последней итерации =', err) + +def plot(sysf, a=-2, b=2): + xs = np.linspace(a, b, 400) + ys = np.linspace(a, b, 400) + X, Y = np.meshgrid(xs, ys) + Z1 = sysf([X, Y])[0] + Z2 = sysf([X, Y])[1] + plt.contour(X, Y, Z1, levels=[0]) + plt.contour(X, Y, Z2, levels=[0]) + plt.axhline(0, color='black') + plt.axvline(0, color='black') + plt.show() + +def main(): + sysn, x0, eps = get_input() + sysf, phi = systems.get(sysn, (None,None)) + if not sysf: + print('Неизвестная система'); sys.exit() + x, it, errors = simple_iter(sysf, phi, x0, eps) + output(x, it, list(errors[-1])) + plot(sysf) + +if __name__=='__main__': + main() diff --git "a/\320\2403212/shamatulskiy_409870/lab2/report.docx" "b/\320\2403212/shamatulskiy_409870/lab2/report.docx" new 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math +import sys + + +def f1(x): + """f(x) = x^2""" + return x**2 + +def f2(x): + """f(x) = sin(x)""" + return math.sin(x) + +def f3(x): + """f(x) = e^x""" + return math.exp(x) + +def f4(x): + """f(x) = 1/x (при x ≠ 0)""" + return 1.0 / x + +def f5(x): + """f(x) = x^3 + 2x""" + return x**3 + 2*x + +functions = [f1, f2, f3, f4, f5] + +def rectangle_method(f, a, b, n, mode="left"): + h = (b - a) / n + total = 0.0 + + if mode == "left": + for i in range(n): + x = a + i * h + total += f(x) + + elif mode == "right": + for i in range(1, n+1): + x = a + i * h + total += f(x) + + elif mode == "middle": + for i in range(n): + x = a + (i + 0.5) * h + total += f(x) + else: + raise ValueError("Неизвестный режим прямоугольников.") + + return total * h + +def trapezoid_method(f, a, b, n): + h = (b - a) / n + total = 0.5 * (f(a) + f(b)) + for i in range(1, n): + x = a + i * h + total += f(x) + return total * h + +def simpson_method(f, a, b, n): + if n % 2 != 0: + n += 1 + + h = (b - a) / n + total = f(a) + f(b) + odd_sum = 0.0 + even_sum = 0.0 + for k in range(1, n): + x = a + k * h + if k % 2 == 0: + even_sum += f(x) + else: + odd_sum += f(x) + total += 4 * odd_sum + 2 * even_sum + return total * h / 3.0 + +def integrate_with_precision(f, a, b, eps, method_name="rectangle", mode="left", n_init=4): + if method_name == "rectangle": + method_func = lambda f,a,b,n: rectangle_method(f,a,b,n,mode=mode) + p = 2 if mode == "middle" else 1 + if p == 2: + runge_factor = 1.0/3.0 + else: + runge_factor = 1.0 # Если p=1 + + elif method_name == "trapezoid": + method_func = trapezoid_method + runge_factor = 1.0/3.0 + + elif method_name == "simpson": + method_func = simpson_method + runge_factor = 1.0/15.0 + + else: + raise ValueError("Неизвестный метод: " + method_name) + + n = n_init + I_old = method_func(f, a, b, n) + while True: + n *= 2 + I_new = method_func(f, a, b, n) + # оценка погрешности по правилу Рунге: + error_est = abs(I_new - I_old) * runge_factor + + if error_est < eps: + return I_new, n + I_old = I_new + +def main(): + print("Выберите функцию для интегрирования:") + for idx, func in enumerate(functions, start=1): + print(f"{idx}) {func.__doc__}") + + choice = int(input("Введите номер функции (1..5): ")) + if choice < 1 or choice > len(functions): + print("Некорректный выбор функции. Завершение.") + return + f = functions[choice - 1] + + a = float(input("Введите левую границу интегрирования a = ")) + b = float(input("Введите правую границу интегрирования b = ")) + if a == b: + print("Границы интегрирования совпадают, интеграл будет 0.") + return + + eps = float(input("Введите требуемую точность (например, 1e-6): ")) + + n_init = int(input("Введите начальное число разбиений (n >= 2), по умолчанию 4: ") or 4) + if n_init < 2: + n_init = 4 + + print("\nВыберите метод интегрирования:") + print("1) Прямоугольники (левые)") + print("2) Прямоугольники (правые)") + print("3) Прямоугольники (средние)") + print("4) Трапеции") + print("5) Симпсон") + + method_choice = int(input("Номер метода: ")) + + if method_choice == 1: + method_name = "rectangle" + mode = "left" + elif method_choice == 2: + method_name = "rectangle" + mode = "right" + elif method_choice == 3: + method_name = "rectangle" + mode = "middle" + elif method_choice == 4: + method_name = "trapezoid" + mode = "" + elif method_choice == 5: + method_name = "simpson" + mode = "" + else: + print("Некорректный выбор метода. Завершение.", file=sys.stderr) + return + + I, N = integrate_with_precision(f, min(a, b), max(a, b), eps, method_name, mode, n_init) + + # вывод рез-тов + print(f"\nРезультаты:") + print(f"Метод: {method_name}, вариант = {mode if mode else '—'}") + print(f"Число разбиений, достигнутое для заданной точности: {N}") + print(f"Приблизительное значение интеграла: {I:.8f}") + print(f"Точность (заданная): {eps}") + +if __name__ == "__main__": + main() diff --git "a/\320\2403212/shamatulskiy_409870/lab4/report.docx" "b/\320\2403212/shamatulskiy_409870/lab4/report.docx" new file mode 100644 index 0000000000000000000000000000000000000000..38bc118c9d91eba69a6a94fefcd286dd4c349e5c GIT binary patch literal 548207 zcmeFYW0xjf&^1`-vTfV8ZQHiZF00G7ZQHhOblL23)irfLGqct^Yvv!klON89JXo10 zW5?bRaZ*tR^cN}+I1nTd5D*a%P$;{6CNL0?2?P)j3J@fSj);T3tC_v4fvT6InTsC1 zr=2Zf;V%%%d?1jY@&Etzzwr(mvYVI;52eZ848X+esEmg&`O!!T>P6r91;ix(jrn^it6!4@5CJrN)DSo;~mhGgh_ zL>xK$uLeeoc*;k%4V?gDQ#6VD*y`K07tX39%`@-avmVsM>CI$FM3{77i~Imb>nNLg 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plt + +# ----------------------------------------------------- +# helpers ------------------------------------------------ + +def read_points_interactive(): + print("Введите пары x y (пустая строка — конец ввода):") + pts = [] + while True: + line = input().strip() + if not line: + break + parts = line.replace(",", ".").split() + if len(parts) != 2: + print("Ожидалось два числа через пробел") + continue + pts.append((float(parts[0]), float(parts[1]))) + return pts + + +def read_points_from_file() -> list: + fname = input("Имя файла с данными: ").strip() + path = Path(fname) + if not path.exists(): + print("Файл не найден"); sys.exit(1) + pts = [] + for line in path.read_text().splitlines(): + line = line.strip() + if not line: + continue + parts = line.replace(",", ".").split() + if len(parts) != 2: + continue + pts.append((float(parts[0]), float(parts[1]))) + return pts + + +def request_points(): + mode = input("Читать точки из файла? (y/n): ").strip().lower() + pts = read_points_from_file() if mode == "y" else read_points_interactive() + if not (8 <= len(pts) <= 12): + print("Нужно 8–12 точек"); sys.exit(1) + xs, ys = map(np.array, zip(*pts)) + return xs, ys + + +# ----------------------------------------------------- +# метрики ------------------------------------------------ + +# Сигма - среднеквадратичное отклонение +def rms(pred, real): + return math.sqrt(np.mean((pred - real) ** 2)) +# R^2 - +def r2_score(pred, real): + ss_res = np.sum((real - pred) ** 2) + ss_tot = np.sum((real - real.mean()) ** 2) + return 1 - ss_res / ss_tot if ss_tot else 0.0 + + +# ----------------------------------------------------- +# модели ------------------------------------------------ + +def model_linear(xs, ys): + c = np.polyfit(xs, ys, 1) + return np.polyval(c, xs), c + +def model_poly2(xs, ys): + c = np.polyfit(xs, ys, 2) + return np.polyval(c, xs), c + +def model_poly3(xs, ys): + c = np.polyfit(xs, ys, 3) + return np.polyval(c, xs), c + +def model_expo(xs, ys): + if np.any(ys <= 0): + return None, None + b, ln_a = np.polyfit(xs, np.log(ys), 1) + a = math.exp(ln_a) + return a * np.exp(b * xs), (a, b) + +def model_log(xs, ys): + if np.any(xs <= 0): + return None, None + b, a = np.polyfit(np.log(xs), ys, 1) + return a + b * np.log(xs), (a, b) + +def model_power(xs, ys): + if np.any(xs <= 0) or np.any(ys <= 0): + return None, None + b, ln_a = np.polyfit(np.log(xs), np.log(ys), 1) + a = math.exp(ln_a) + return a * xs ** b, (a, b) + +MODELS = { + "linear": (model_linear, "y = a₀ + a₁ x"), + "poly2": (model_poly2, "y = a₂ x² + a₁ x + a₀"), + "poly3": (model_poly3, "y = a₃ x³ + a₂ x² + a₁ x + a₀"), + "expo": (model_expo, "y = A · e^{Bx}"), + "log": (model_log, "y = A + B · ln x"), + "power": (model_power, "y = A · x^{B}"), +} + +# ----------------------------------------------------- +# main -------------------------------------- + +def main(): + print("Будут исследоваться следующие формы зависимостей:") + for name, (_, formula) in MODELS.items(): + print(f" {name:6s}: {formula}") + print() + + xs, ys = request_points() + + results = {} + for name, (fn, _) in MODELS.items(): + pred, coeffs = fn(xs, ys) + if pred is None: + continue + results[name] = { + "sigma": rms(pred, ys), + "coeffs": coeffs, + "pred": pred, + "r2": r2_score(pred, ys), + } + + if not results: + print("Не удалось построить ни одной модели"); sys.exit(1) + + best_name = min(results, key=lambda n: results[n]["sigma"]) + + out_mode = input("Сохранить результаты в файл? (y/n): ").strip().lower() + if out_mode == "y": + out_path = Path(input("Имя выходного файла: ").strip()) + with out_path.open("w", encoding="utf-8") as f: + for name, res in results.items(): + f.write(f"{name}: σ={res['sigma']:.3f} R²={res['r2']:.3f} coeffs={res['coeffs']}\n") + f.write(f"Лучшее приближение: {best_name} (σ={results[best_name]['sigma']:.3f})\n") + if "linear" in results: + r = np.corrcoef(xs, ys)[0, 1] + f.write(f"Корреляция Пирсона (linear): r={r:.3f}\n") + print(f"Результаты сохранены в {out_path}") + else: + for name, res in results.items(): + print(f"{name:6s}: σ={res['sigma']:.3f} R²={res['r2']:.3f} coeffs={res['coeffs']}") + print(f"Лучшее приближение => {best_name} (σ={results[best_name]['sigma']:.3f})") + if "linear" in results: + r = np.corrcoef(xs, ys)[0, 1] + print(f"Корреляция Пирсона (linear): r={r:.3f}") + + plt.scatter(xs, ys, label="data", zorder=3) + pu = np.linspace(xs.min(), xs.max(), 400) + for name, res in results.items(): + if name.startswith("poly"): + y_plot = np.polyval(res["coeffs"], pu) + elif name == "linear": + y_plot = res["coeffs"][0] * pu + res["coeffs"][1] + elif name == "expo": + y_plot = res["coeffs"][0] * np.exp(res["coeffs"][1] * pu) + elif name == "log": + y_plot = res["coeffs"][0] + res["coeffs"][1] * np.log(pu) + elif name == "power": + y_plot = res["coeffs"][0] * pu ** res["coeffs"][1] + else: + continue + plt.plot(pu, y_plot, label=name) + plt.title("Аппроксимация точек") + plt.legend() + plt.grid(True) + plt.show() + + +if __name__ == "__main__": + main() diff --git "a/\320\2403212/shamatulskiy_409870/lab5/report.docx" "b/\320\2403212/shamatulskiy_409870/lab5/report.docx" new file mode 100644 index 0000000000000000000000000000000000000000..2ee7d8e56a2eb21d4574e60379d164cda30daf8a GIT binary patch literal 384234 zcmeFX<9B9F@HQGdnb@|SOq|@YGqG)ZCbn(cwr$%J+xA5F$@Bi!IqN-Zo&VrH{b7IT zs=a%4?YgSEs;^%%;1K8_P#`cMARr_lAiRWt&`lvldLGm|8AeNwee*0j$&?$|o@ 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lagrange(xv,yv,x): + s=0.0 + for i in range(len(xv)): + p=1.0 + for j in range(len(xv)): + if i!=j: + p*=(x-xv[j])/(xv[i]-xv[j]) + s+=yv[i]*p + return s + +def newton_divided(xv,yv,x): + div=yv.copy() + n=len(xv) + for k in range(1,n): + for i in range(n-1,k-1,-1): + div[i]=(div[i]-div[i-1])/(xv[i]-xv[i-k]) + s=div[-1] + for k in range(n-2,-1,-1): + s=s*(x-xv[k])+div[k] + return s + +def newton_fd(xv,yv,x): + h=xv[1]-xv[0] + table=finite_diffs(yv) + n=len(xv) + if abs(x-xv[0])=2: + a,b=map(float,parts[:2]) + xv.append(a) + yv.append(b) + return xv,yv + +def plot_method(title,xv,yv,ys_func): + xs=np.linspace(min(xv),max(xv),400) + plt.figure() + plt.plot(xs,[ys_func(xx) for xx in xs]) + plt.scatter(xv,yv,color="black",zorder=5) + plt.title(title) + plt.xlabel("x") + plt.ylabel("y") + plt.grid(True) + + +def main(): + if input("Ввод из файла? (y/n): ").lower().startswith("y"): + xv,yv=load_file(input("Файл: ")) + else: + if input("Сгенерировать по функции? (y/n): ").lower().startswith("y"): + xv,yv=func_choice() + else: + xv,yv=read_points() + steps=[round(xv[i+1]-xv[i],10) for i in range(len(xv)-1)] + if len(set(steps))!=1: + print("Шаг по x должен быть равномерным") + sys.exit() + table=finite_diffs(yv) + print("Таблица конечных разностей:") + for i in range(len(xv)): + row=[f"{xv[i]:g}",f"{yv[i]:g}"] + for k in range(1,len(table)): + if iv+q zWjd0LTCkKw83fw$3KD&Uz{y_9jj`bnZ6R z1O*^K6nOwZzyAMs`#<;%G^I#O12dw8yaxUljHz*z(}@%*qw}RwY4qDrw=yx}~Zgx!j^?Wv;VBOtrh8J?LWr0$cfZL*?z;qt)Ht|!X3m%_Ns zGlZ77GWGUk%Af%%3S6R7wGPFo;goj-Q!82kw{KGZFb7?*vvx;#&|&GL4;_@I>lU`> z=)3A4Dda92-qLsY5uKz?+{04as=07f8E%^X*EQotMU>t|ib#k-3%bDfV{aK{RcA^i z*t*{MmITt2Xi4>6tJWKUOL0v@e39@aAB5x4jd0h{V&+ZuCkA zGSU8kz(2ph6gxK_g2X&{|7Iuh0dg5V&%!r@9Y`gwxEUk-*7-kfLjKCUjMiCx^Wah_ z%D_HP1~%J}0W3oX&%C}|G&XReO2>`}^MwdM_E*y|vloJI%OIA>|DrTr0#@toFHYrt84LYOX+1|1YbScT|Lp%SmH!Xs@BjAbRf+wU zgA6bNw?SV)^WBOY{TPLE^hR@=80+8=T9Puzn`@S<-(NgyYd|{Z2I7+oi%D~yPFZ3u 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z)fxgt!6^T+AZrQ&hn%ssgRYXRtua9R*D5Y2`F8^N(bx?aK%fjT1K7V2m;sL8-{8~{YSr9;bFhta^<^j4l|8i~q^MT>~18!_>WNqW{yM^#S;p7B6 z)zLsWII#A=rm8=mA?`omCN?$>|06!|^JbwWu*U7!ARuUef`Okx`2N5F9c1J4cv z{{3y~`@IOj;{Ns!ftb+*G@MX_g8T}9Cjlk;2MH@S} zmA|u*f2H!Z@Sjxv(uw|@3I5J_{gpsu(O(JtH#Y2_h56_5{U^2fl0SZtzb)o+lHkA* U@N4{h07(D?0eLP5&K)5C2ayXVvH$=8 literal 0 HcmV?d00001 diff --git "a/\320\2403212/shamatulskiy_409870/lab6/task.py" "b/\320\2403212/shamatulskiy_409870/lab6/task.py" new file mode 100644 index 0000000..ed8adba --- /dev/null +++ "b/\320\2403212/shamatulskiy_409870/lab6/task.py" @@ -0,0 +1,129 @@ +import math +import matplotlib.pyplot as plt + +def eq1_f(x, y): + return x + y + +def eq1_exact(x, x0, y0): + return (y0 + 1) * math.exp(x - x0) - x - 1 + x0 + +def eq2_f(x, y): + return y * x * x + +def eq2_exact(x, x0, y0): + return y0 * math.exp((x**3 - x0**3) / 3) + +def eq3_f(x, y): + return y * x + +def eq3_exact(x, x0, y0): + return y0 * math.exp((x**2 - x0**2) / 2) + +def euler(f, x0, y0, xn, h): + xs = [] + ys = [] + x = x0 + y = y0 + while x <= xn + 1e-12: + xs.append(x) + ys.append(y) + y = y + h * f(x, y) + x = x + h + return xs, ys + +def rk4(f, x0, y0, xn, h): + xs = [] + ys = [] + x = x0 + y = y0 + while x <= xn + 1e-12: + xs.append(x) + ys.append(y) + k1 = f(x, y) + k2 = f(x + h/2, y + h * k1 / 2) + k3 = f(x + h/2, y + h * k2 / 2) + k4 = f(x + h, y + h * k3) + y = y + h * (k1 + 2*k2 + 2*k3 + k4) / 6 + x = x + h + return xs, ys + +def milne(f, x0, y0, xn, h): + xs_rk, ys_rk = rk4(f, x0, y0, x0 + 3*h, h) + xs = xs_rk[:] + ys = ys_rk[:] + fs = [f(xs[i], ys[i]) for i in range(4)] + i = 3 + while xs[i] + h <= xn + 1e-12: + x_next = xs[i] + h + y_pred = ys[i-3] + (4*h/3) * (2*fs[i-2] - fs[i-1] + 2*fs[i]) + f_pred = f(x_next, y_pred) + y_corr = ys[i-1] + (h/3) * (fs[i-1] + 4*fs[i] + f_pred) + xs.append(x_next) + ys.append(y_corr) + fs.append(f(x_next, y_corr)) + i += 1 + return xs, ys + +def runge_error(method, f, x0, y0, xn, h, p): + _, y_h = method(f, x0, y0, xn, h) + _, y_h2 = method(f, x0, y0, xn, h/2) + y_end_h = y_h[-1] + y_end_h2 = y_h2[-1] + return abs(y_end_h2 - y_end_h) / (2**p - 1) + +def main(): + print("Выберите уравнение:") + print("1) y' = x + y") + print("2) y' = y * x^2") + print("3) y' = y * x") + choice = int(input()) + if choice == 1: + f = eq1_f + exact = eq1_exact + elif choice == 2: + f = eq2_f + exact = eq2_exact + else: + f = eq3_f + exact = eq3_exact + x0 = float(input("x0 = ")) + y0 = float(input("y0 = ")) + xn = float(input("xn = ")) + h = float(input("h = ")) + eps = float(input("epsilon = ")) + xs_e, ys_e = euler(f, x0, y0, xn, h) + xs_rk, ys_rk = rk4(f, x0, y0, xn, h) + xs_m, ys_m = milne(f, x0, y0, xn, h) + print("\n i\t x\t Euler\t\t RK4\t\t Milne\t\t Exact") + N = len(xs_e) + for i in range(N): + x_val = xs_e[i] + y_ex = exact(x_val, x0, y0) + y_e = ys_e[i] if i < len(ys_e) else float('nan') + y_r = ys_rk[i] if i < len(ys_rk) else float('nan') + y_m = ys_m[i] if i < len(ys_m) else float('nan') + print(f"{i:2d}\t{x_val:8.4f}\t{y_e:10.6f}\t{y_r:10.6f}\t{y_m:10.6f}\t{y_ex:10.6f}") + err_euler = runge_error(euler, f, x0, y0, xn, h, 1) + err_rk4 = runge_error(rk4, f, x0, y0, xn, h, 4) + max_err_m = 0.0 + for i, xm in enumerate(xs_m): + ye = exact(xm, x0, y0) + ym = ys_m[i] + max_err_m = max(max_err_m, abs(ye - ym)) + print(f"\nОценка погрешности Эйлера (правило Рунге): {err_euler:.6e}") + print(f"Оценка погрешности RK4 (правило Рунге): {err_rk4:.6e}") + print(f"Максимальная погрешность Милна: {max_err_m:.6e}") + xs_exact = [x0 + i*h/10 for i in range(int((xn - x0)/(h/10)) + 1)] + ys_exact = [exact(x, x0, y0) for x in xs_exact] + plt.plot(xs_exact, ys_exact, label="Exact", color="black") + plt.plot(xs_e, ys_e, label="Euler", linestyle="--") + plt.plot(xs_rk, ys_rk, label="RK4", linestyle="-.") + plt.plot(xs_m, ys_m, label="Milne", linestyle=":") + plt.legend() + plt.xlabel("x") + plt.ylabel("y") + plt.title("Numerical vs Exact Solution") + plt.show() + +if __name__ == "__main__": + main()