数值分析题目

求方程组
$\left\{ \begin{array}{c} 5x_1+2x_2 + x_3 = -12 \\ -x_1 + 4x_2+2x_3 = 20 \\ 2x_1+-3x_2+10x_3=2 \end{array} \right.$

超松弛迭代其实基于高斯-塞德尔迭代的

可以对比着之前的高斯迭代法对比一下效果
高斯–塞德尔迭代法求方程组的解（Python实现）

代码

import numpy as np

A = np.array([[5.0, 2, 1], [-1, 4, 2], [2, -3, 10]])
B = np.array([-12.0, 20, 3])
x0 = np.array([1.0, 1, 1])
x = np.array([0.0, 0, 0])
w = 1.5
times, MT = 0, 1000

while times < MT:
    tempx = x0.copy()
    for i in range(3):
        temp = 0
        for j in range(3):
            if i != j:
                temp += x0[j] * A[i][j]
        x[i] = (B[i] - temp) / A[i][i]
        x0[i] = x[i]
    x = w * x + (1-w) * tempx
    calTemp = max(abs(x - tempx))
    times += 1
    if calTemp < 1e-4:
        break
print(times)
print(x)

• 结果算出来的X为[-4, 3 , 2]