编制 Jacobi 迭代法、Gauss-Seidel 迭代法和超松弛(SOR)迭代法解方程组 Ax=b 的程序

当然可以，这些迭代法通常用于求解线性系统Ax = b，特别是在数值分析和工程计算中。以下是简单的Python示例，演示如何使用Jacobi、Gauss-Seidel和SOR迭代法：

import numpy as np

def jacobi(A, b, iterations, tolerance=1e-6):
    n = A.shape[0]
    x = np.zeros(n)
    delta = np.inf
    
    for _ in range(iterations):
        old_delta = delta
        x_new = (b - A.dot(x)) / A.diagonal()
        delta = np.max(np.abs(x_new - x))
        x = x_new
        
        if delta < tolerance:
            break
            
    return x_new

def gauss_seidel(A, b, iterations, tolerance=1e-6):
    n = A.shape[0]
    x = np.zeros(n)
    
    for _ in range(iterations):
        for i in range(n):
            if i != 0:
                x[i] = (b[i] - np.sum(A[i, :i] * x[:i])) / A[i, i]
            else:
                x[i] = b[i] / A[i, i]
                
            if abs(x[i]) < tolerance:
                break
                
    return x

def sor(A, b, omega, iterations, tolerance=1e-6):
    n = A.shape[0]
    x = np.zeros(n)
    delta = np.inf
    
    for _ in range(iterations):
        x_new = (omega * (x + A @ x) + (1 - omega) * b) / (omega * A.diagonal() + (1 - omega))
        delta = np.max(np.abs(x_new - x))
        x = x_new
        
        if delta < tolerance:
            break
            
    return x_new

# 示例用法
A = np.array([[4, -1], [-1, 5]])
b = np.array([7, 9])
iterations = 100

jacobi_sol = jacobi(A, b, iterations)
gauss_seidel_sol = gauss_seidel(A, b, iterations)
sor_sol = sor(A, b, 1.5, iterations)

print(f"Jacobi 方法结果：{jacobi_sol}")
print(f"Gauss-Seidel 方法结果：{gauss_seidel_sol}")
print(f"SOR 方法结果（ω=1.5）：{sor_sol}")