第三章

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3.2 import sympy as sp def solve_difference_equation(): n = sp.symbols('n', integer=True) x = sp.Function('x') eq = sp.Eq(x(n + 2) - x(n + 1) - 2 * x(n), 0) sol = sp.rsolve(eq, x(n), {x(0): -2, x(1): -2}) return sol

solution = solve_difference_equation()
print(solution)
print("3023")


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3.4 import numpy as np from scipy.sparse.linalg import eigs import matplotlib.pyplot as plt

adjacency_matrix = np.array([
[0, 1, 0, 1, 1, 1],
[0, 0, 0, 1, 1, 1],
[1, 1, 0, 1, 0, 0],
[0, 0, 0, 0, 1, 1],
[0, 0, 1, 0, 0, 1],
[0, 0, 1, 0, 0, 0]
])

out_degree = np.sum(adjacency_matrix, axis=1, keepdims=True)

num_nodes = adjacency_matrix.shape[0]

damping_factor = 0.85

transition_matrix = (1 - damping_factor) / num_nodes + damping_factor * adjacency_matrix / out_degree

eigenvalue, eigenvector = eigs(transition_matrix.T, k=1, which='LR') # 'LR' 表示求最大实部特征值

eigenvector = eigenvector.flatten().real / np.sum(eigenvector.flatten().real)

print("最大特征值为：", eigenvalue.real[0])
print("归一化特征向量为：\n", np.round(eigenvector, 4))

plt.bar(range(1, num_nodes + 1), eigenvector, width=0.6)
plt.xlabel('节点')
plt.ylabel('PageRank值')
plt.title('PageRank分布')
plt.show()
print("3023")