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# 2. 列主元高斯消去法（Gaussian Elimination with Partial Pivoting）

下面是使用Python实现列主元高斯消去法（Gaussian Elimination with Partial Pivoting）求解线性方程组的代码。该实现包含前向消去和回代两个步骤，并在每一步选择当前列的绝对值最大的元素作为主元。

```python
import numpy as np

def gaussian_elimination_with_partial_pivoting(A, b):
    """
    Solve the system of linear equations Ax = b using Gaussian Elimination with Partial Pivoting.

    Parameters:
    A : 2D list or numpy array
        Coefficient matrix
    b : 1D list or numpy array
        Right-hand side vector

    Returns:
    x : numpy array
        Solution vector
    """
    n = len(b)
    A = A.astype(float)  # Ensure the matrix is of type float
    b = b.astype(float)  # Ensure the vector is of type float
    
    # Forward elimination with partial pivoting
    for i in range(n):
        # Pivoting: find the row with the largest element in the current column
        max_row_index = np.argmax(np.abs(A[i:n, i])) + i
        if A[max_row_index, i] == 0:
            raise ValueError("Matrix is singular or nearly singular")

        # Swap the current row with the max_row_index row
        A[[i, max_row_index]] = A[[max_row_index, i]]
        b[[i, max_row_index]] = b[[max_row_index, i]]

        # Make the diagonal element 1 and eliminate the column below
        pivot = A[i, i]
        A[i] = A[i] / pivot
        b[i] = b[i] / pivot

        for j in range(i + 1, n):
            factor = A[j, i]
            A[j] = A[j] - factor * A[i]
            b[j] = b[j] - factor * b[i]

    # Backward substitution
    x = np.zeros(n)
    for i in range(n - 1, -1, -1):
        x[i] = b[i] - np.dot(A[i, i + 1:], x[i + 1:])
    
    return x

# Example usage
A = np.array([[2, -1, 1],
              [3, 3, 9],
              [3, 3, 5]], dtype=float)
b = np.array([2, -1, 2], dtype=float)

x = gaussian_elimination_with_partial_pivoting(A, b)
print("Solution:", x)
```

#### 代码说明

1. **输入参数**：
   * `A`：系数矩阵，可以是二维列表或NumPy数组。
   * `b`：右端向量，可以是一维列表或NumPy数组。
2. **前向消去（Forward Elimination with Partial Pivoting）**：
   * 对于每一列，从当前行到最后一行中选择绝对值最大的元素作为主元。
   * 交换当前行和主元所在行。
   * 将当前行的主元归一化。
   * 对当前行以下的每一行进行消去操作，使得该列元素变为零。
3. **回代（Backward Substitution）**：
   * 从最后一行开始，逐步向上回代，求解每一个未知数。

#### 示例

给定系数矩阵 (A) 和右端向量 (b)：

```plaintext
A = [[2, -1, 1],
     [3, 3, 9],
     [3, 3, 5]]
b = [2, -1, 2]
```

运行上述代码，求解线性方程组 (Ax = b)，可以得到解向量 `x`。

#### 注意事项

* 该实现假设矩阵 (A) 是非奇异的，即有唯一解。
* 该实现通过交换行操作避免了数值不稳定的问题，适用于更广泛的情况。


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