def test_QR_MGS_Q_orthogonal():
'verify the orthogonality of the computed Q'
np.random.seed(1)
M = 6
N = 5
A = np.random.rand(M,N)
Q1, R1 = qra.QR_MGS(A)
aae(np.identity(N), np.dot(Q1.T,Q1), decimal=10)
def test_QR_MGS_decomposition():
'verify the computed Q and R matrices'
np.random.seed(6)
M = 6
N = 5
A = np.random.rand(M,N)
Q1, R1 = qra.QR_MGS(A)
aae(A, np.dot(Q1, R1), decimal=10)
def test_QR_MCS_Cholesky():
'verify that R is the transpose of the Cholesky factor of ATA'
np.random.seed(98)
M = 6
N = 5
A = np.random.rand(M,N)
ATA = np.dot(A.T,A)
Q1, R1 = qra.QR_MGS(A)
aae(ATA, np.dot(R1.T,R1), decimal=10)