识别以下MATLAB程序，并生成相应Python代码：clc clear close all syms x x0 y0 y1 y2 y3 y4 h real a = [1, x0, x0^2; 1, (x0 + h), (x0 + h)^2; 1, (x0 + 2 * h), (x0 + 2 * h)^2] \ [y0; y1; y2]; %一元二次多项式y(x) = a1 + a2 * x + a3 * x^2的系数 y(x) = a(1) + a(2) * x + a(3) * x^2; dy(x) = diff(y, 1); ddy(x) = diff(y, 2); dy_two_order_central_difference = simplify(dy(x0 + h)) ddy_two_order_central_difference = simplify(ddy(x0 + h)) a = [1, x0, x0^2, x0^3, x0^4; 1, (x0 + h), (x0 + h)^2, (x0 + h)^3, (x0 + h)^4; 1, (x0 + 2 * h), (x0 + 2 * h)^2, (x0 + 2 * h)^3, (x0 + 2 * h)^4; ... 1, (x0 + 3 * h), (x0 + 3 * h)^2, (x0 + 3 * h)^3, (x0 + 3 * h)^4; 1, (x0 + 4 * h), (x0 + 4 * h)^2, (x0 + 4 * h)^3, (x0 + 4 * h)^4] \ [y0; y1; y2; y3; y4]; %一元四次多项式y(x) = a1 + a2 * x + a3 * x^2 + a4 * x^3 + a5 * x^4的系数 y(x) = a(1) + a(2) * x + a(3) * x^2 + a(4) * x^3 + a(5) * x^4; dy(x) = diff(y, 1); ddy(x) = diff(y, 2); dy_four_order_central_difference = simplify(dy(x0 + 2 * h)) ddy_four_order_central_difference = simplify(ddy(x0 + 2 * h)) %% 验证 n = 50; x = linspace(0, 2*pi, n); h = x(2) - x(1); y = sin(x); dy = cos(x); ddy = -sin(x); dy1 = nan * zeros(size(x)); ddy1 = nan * zeros(size(x)); for i = 2 : n - 1 dy1(i) = (y(i + 1) - y(i - 1)) / (2.0 * h); ddy1(i) = (y(i - 1) - 2.0 * y(i) + y(i + 1)) / h^2; end dy2 = nan * zeros(size(x)); ddy2 = nan * zeros(size(x)); for i = 3 : n - 2 dy2(i) = (y(i - 2) - 8.0 * y(i - 1) + 8.0 * y(i + 1) - y(i + 2)) / (12.0 * h); ddy2(i) = -(y(i - 2) - 16.0 * y(i - 1) + 30.0 * y(i) - 16.0 * y(i + 1) + y(i + 2)) / (12.0 * h^2); end max_dy1_err = max(abs(dy1(2 : n - 1) - dy(2 : n - 1))); max_ddy1_err = max(abs(ddy1(2 : n - 1) - ddy(2 : n - 1))); max_dy2_err = max(abs(dy2(3 : n - 2) - dy(3 : n - 2))); max_ddy2_err = max(abs(ddy2(3 : n - 2) - ddy(3 : n - 2))); disp(['一阶导数的二阶和四阶中心差分近似，最大误差分别为:', num2str(max_dy1_err), ',' , num2str(max_dy2_err)]) disp(['二阶导数的二阶和四阶中心差分近似，最大误差分别为:', num2str(max_ddy1_err), ',' , num2str(max_ddy2_err)])

import numpy as np
import sympy as sp

x, x0, y0, y1, y2, y3, y4, h = sp.symbols('x x0 y0 y1 y2 y3 y4 h', real=True)

a = sp.Matrix([[1, x0, x0**2],
[1, (x0 + h), (x0 + h)**2],
[1, (x0 + 2 * h), (x0 + 2 * h)**2]]).inv() * sp.Matrix([y0, y1, y2])

y = a[0] + a[1] * x + a[2] * x**2
dy = sp.diff(y, 1)
ddy = sp.diff(y, 2)
dy_two_order_central_difference = sp.simplify(dy.subs(x, x0 + h))
ddy_two_order_central_difference = sp.simplify(ddy.subs(x, x0 + h))

a = sp.Matrix([[1, x0, x0**2, x0**3, x0**4],
[1, (x0 + h), (x0 + h)**2, (x0 + h)**3, (x0 + h)**4],
[1, (x0 + 2 * h), (x0 + 2 * h)**2, (x0 + 2 * h)**3, (x0 + 2 * h)**4],
[1, (x0 + 3 * h), (x0 + 3 * h)**2, (x0 + 3 * h)**3, (x0 + 3 * h)**4],
[1, (x0 + 4 * h), (x0 + 4 * h)**2, (x0 + 4 * h)**3, (x0 + 4 * h)**4]]).inv() * sp.Matrix([y0, y1, y2, y3, y4])

y = a[0] + a[1] * x + a[2] * x**2 + a[3] * x**3 + a[4] * x**4
dy = sp.diff(y, 1)
ddy = sp.diff(y, 2)
dy_four_order_central_difference = sp.simplify(dy.subs(x, x0 + 2 * h))
ddy_four_order_central_difference = sp.simplify(ddy.subs(x, x0 + 2 * h))

n = 50
x_vals = np.linspace(0, 2 * np.pi, n)
h = x_vals[1] - x_vals[0]
y_vals = np.sin(x_vals)
dy_vals = np.cos(x_vals)
ddy_vals = -np.sin(x_vals)

dy1_vals = np.full_like(x_vals, np.nan)
ddy1_vals = np.full_like(x_vals, np.nan)
for i in range(1, n - 1):
dy1_vals[i] = (y_vals[i + 1] - y_vals[i - 1]) / (2.0 * h)
ddy1_vals[i] = (y_vals[i - 1] - 2.0 * y_vals[i] + y_vals[i + 1]) / h**2

dy2_vals = np.full_like(x_vals, np.nan)
ddy2_vals = np.full_like(x_vals, np.nan)
for i in range(2, n - 2):
dy2_vals[i] = (y_vals[i - 2] - 8.0 * y_vals[i - 1] + 8.0 * y_vals[i + 1] - y_vals[i + 2]) / (12.0 * h)
ddy2_vals[i] = -(y_vals[i - 2] - 16.0 * y_vals[i - 1] + 30.0 * y_vals[i] - 16.0 * y_vals[i + 1] + y_vals[i + 2]) / (12.0 * h**2)

max_dy1_err = np.max(np.abs(dy1_vals[1 : n - 1] - dy_vals[1 : n - 1]))
max_ddy1_err = np.max(np.abs(ddy1_vals[1 : n - 1] - ddy_vals[1 : n - 1]))
max_dy2_err = np.max(np.abs(dy2_vals[2 : n - 2] - dy_vals[2 : n - 2]))
max_ddy2_err = np.max(np.abs(ddy2_vals[2 : n - 2] - ddy_vals[2 : n - 2]))

print(f"一阶导数的二阶和四阶中心差分近似，最大误差分别为: {max_dy1_err}, {max_dy2_err}")
print(f"二阶导数的二阶和四阶中心差分近似，最大误差分别为: {max_ddy1_err}, {max_ddy2_err}")