python - 为什么 Numpy 和 Scipy QR 分解给我不同的值？

Question

我有以下向量。

x = np.array([[ 0.87695113],
              [ 0.3284933 ],
              [-0.35078323]])

当我调用 qr 的 numpy 版本时

from numpy.linalg import qr as qr_numpy
qr_numpy(x)

我得到

(array([[-0.87695113],
        [-0.3284933 ],
        [ 0.35078323]]), array([[-1.]]))

而当我运行 scipy 版本时，我得到了完全不同的东西。

from scipy.linalg import qr as qr_scipy
qr_scipy(x)

带输出

(array([[-0.87695113, -0.3284933 ,  0.35078323],
        [-0.3284933 ,  0.94250897,  0.06139208],
        [ 0.35078323,  0.06139208,  0.93444215]]), array([[-1.],
        [ 0.],
        [ 0.]]))

到底是怎么回事？？

标签： pythonnumpyscipylinear-algebralapack

Answer 1

默认modefornumpy.linalg.qr()是'reduced'而 for scipy.linalg.qr()it 是'full'.

因此，要获得相同的结果，请使用'economic'scipy-qr 或'complete'numpy-qr：

from numpy.linalg import qr as qr_numpy
qr_numpy(x)

(array([[-0.87695113],
        [-0.3284933 ],
        [ 0.35078323]]),
 array([[-1.]]))

与 scipy-qr 的输出匹配：

from scipy.linalg import qr as qr_scipy
qr_scipy(x, mode='economic')

(array([[-0.87695113],
        [-0.3284933 ],
        [ 0.35078323]]),
 array([[-1.]]))

---

并获得两者的“完整”版本：

from numpy.linalg import qr as qr_numpy
qr_numpy(x, mode='complete')

(array([[-0.87695113, -0.3284933 ,  0.35078323],
        [-0.3284933 ,  0.94250897,  0.06139208],
        [ 0.35078323,  0.06139208,  0.93444215]]),
 array([[-1.],
        [ 0.],
        [ 0.]]))

from scipy.linalg import qr as qr_scipy
qr_scipy(x)

(array([[-0.87695113, -0.3284933 ,  0.35078323],
        [-0.3284933 ,  0.94250897,  0.06139208],
        [ 0.35078323,  0.06139208,  0.93444215]]),
 array([[-1.],
        [ 0.],
        [ 0.]]))