Python矩阵乘法求解Ax <=b

Question

不确定如何用 Python

解决这个问题

有一个 3x3 矩阵，它是 A

3x1 矩阵（未知）是 X

和已知的 b

如何找到 Ax <= b 的所有解？提前致谢

Answer 1

让我们假设 A 是可逆的。然后你可以写条件

Ax <= b

作为

Ax = b + c   |   c <= 0

其中 c 是列向量。由于我们假设 A 是可逆的，我们可以求解 x:

x = A^-1 (b + c)

给定 c，如果 c 的所有分量都是 non-positive，x 将是一个精确的解。

让我们用 numpy 检查一下：

>>> import numpy as np
>>>
# create example A and b
>>> A = np.random.random((3, 3))
>>> b = np.random.random((3, 1))
>>>
# invert A and compute solution for c == 0
>>> invA = np.linalg.inv(A)
>>> x0 = invA @ b
>>> 
# create 10,000 different vectors c
>>> c = np.random.uniform(-10, 10, (3, 10000))
# and mark those which have three non-pos components
>>> all_non_pos = np.all(c <= 0, axis=0)
# compute corresponding vectors x
>>> x = x0 + invA @ c
# and check which are solutions
>>> all_le_b = np.all(A @ x <= b, axis=0)
# finally, check whether solutions correspond to non-neg c 
>>> np.all(all_non_pos == all_le_b)
True

Python矩阵乘法求解Ax <=b

Python Matrix multiplication solve Ax <=b

python

numpy

matrix

matrix-multiplication