np.linalg.inv() 给出意想不到的结果

Question

我正在使用 numpy 对矩阵求逆，但得到了一些意想不到的结果。

运行我的程序我得到 p1 的最终结果为：

>>> p1
array([[1.69133481e+11, 3.74575030e+09, 8.29681977e+07, 1.83800903e+06],
       [3.74575030e+09, 8.29681977e+07, 1.83800903e+06, 4.07236156e+04],
       [8.29681977e+07, 1.83800903e+06, 4.07236156e+04, 9.02416997e+02],
       [1.83800903e+06, 4.07236156e+04, 9.02416997e+02, 2.00000000e+01]])

然后当我尝试使用 np.linalg.inv() 反转 p1 时，我得到：

>>> np.linalg.inv(p1)
array([[ 2.33378273e+00, -3.16566294e+02,  1.43119558e+04,
        -2.15657094e+05],
       [-3.16566293e+02,  4.29411791e+04, -1.94139249e+06,
         2.92538609e+07],
       [ 1.43119557e+04, -1.94139249e+06,  8.77723669e+07,
        -1.32261292e+09],
       [-2.15657092e+05,  2.92538606e+07, -1.32261291e+09,
         1.99302540e+10]])

这显然是不正确的：

>>> (p1 @ np.linalg.inv(p1))
array([[ 9.99968764e-01, -1.28189335e-03, -7.44723976e-01,
         3.60136516e+00],
       [-2.53043124e-06,  9.99986814e-01, -3.83602548e-02,
         1.12390996e-01],
       [-8.71254524e-08,  9.18930849e-06,  9.99317258e-01,
         4.33950341e-03],
       [-6.98491931e-10,  7.45058060e-08, -1.23977661e-05,
         1.00001526e+00]])
>>> (p1 @ np.linalg.inv(p1)).astype(int)
array([[0, 0, 0, 3],
       [0, 0, 0, 0],
       [0, 0, 0, 0],
       [0, 0, 0, 1]])

我的结果显然不是单位矩阵。

然而，这就是事情变得奇怪的地方。如果我在键入“p1”时使用打印到终端的值重新定义 p1 并计算与之前相同的命令，我得到：

>>> p1 = np.array([[1.69133481e+11, 3.74575030e+09, 8.29681977e+07, 1.83800903e+06],
...        [3.74575030e+09, 8.29681977e+07, 1.83800903e+06, 4.07236156e+04],
...        [8.29681977e+07, 1.83800903e+06, 4.07236156e+04, 9.02416997e+02],
...        [1.83800903e+06, 4.07236156e+04, 9.02416997e+02, 2.00000000e+01]])
>>> p1 @ np.linalg.inv(p1)
array([[ 9.99999999e-01, -4.85794104e-07, -2.14803652e-05,
        -4.95130838e-04],
       [ 2.61335278e-10,  9.99999940e-01,  9.92065715e-07,
        -3.11477404e-05],
       [ 1.87780333e-13, -5.48610778e-10,  1.00000001e+00,
        -1.02864912e-07],
       [ 9.94759830e-14, -2.00088834e-11,  5.82076609e-10,
         9.99999998e-01]])
>>> (p1 @ np.linalg.inv(p1)).astype(int)
array([[0, 0, 0, 0],
       [0, 0, 0, 0],
       [0, 0, 1, 0],
       [0, 0, 0, 0]])

在这里，结果与我得到单位矩阵时的预期非常一致。

在重新定义 p1 之前矩阵求逆不正确导致我的代码出现问题。有什么想法吗？

Answer 1

结果正确，您的问题与浮点（in）精度有关：

In [1]: arr = np.array([[1.69133481e+11, 3.74575030e+09, 8.29681977e+07, 1.83800903e+06], 
   ...:        [3.74575030e+09, 8.29681977e+07, 1.83800903e+06, 4.07236156e+04], 
   ...:        [8.29681977e+07, 1.83800903e+06, 4.07236156e+04, 9.02416997e+02], 
   ...:        [1.83800903e+06, 4.07236156e+04, 9.02416997e+02, 2.00000000e+01]])                                     

In [2]: inv = np.linalg.inv(arr)                                                                                      

In [3]: np.isclose(arr @ inv, np.eye(*arr.shape), atol=1e-3)                                                          
Out[3]: 
array([[ True,  True,  True,  True],
       [ True,  True,  True,  True],
       [ True,  True,  True,  True],
       [ True,  True,  True,  True]])

np.linalg.inv() 给出意想不到的结果

np.linalg.inv() giving unexpected results

python

arrays

numpy

matrix

inverse