def test_Givens_cs2rho_Givens_rho2cs():
'verify consistency'
np.random.seed(11)
a = 10*np.random.rand()
b = 10*np.random.rand()
c, s = qra.Givens_rotation(a=a, b=b)
rho = qra.Givens_cs2rho(c=c, s=s)
c2, s2 = qra.Givens_rho2cs(rho=rho)
aae(c, c2, decimal=10)
aae(s, s2, decimal=10)
def test_QR_Givens_Q_from_QR_Givens_decomposition():
'verify the computed Q and R matrices'
np.random.seed(7)
M = 6
N = 5
A = np.random.rand(M,N)
A2 = A.copy()
qra.QR_Givens(A2)
Q = qra.Q_from_QR_Givens(A=A2)
R = np.triu(A2)
aae(A, np.dot(Q, R), decimal=10)
def test_House_vector_matvec_matmat_reflection():
'verify matrix-matrix product with Householder reflections'
np.random.seed(32)
# real vector
N = 7
a = np.random.rand(N)
v, beta = qra.House_vector(x=a)
P = np.identity(a.size) - beta*np.outer(v,v)
A = np.random.rand(N,N)
PA1 = np.dot(P, A)
PA2 = qra.House_matvec(A=A, v=v, beta=beta, order='PA')
aae(PA1, PA2, decimal=10)
AP1 = np.dot(A, P)
AP2 = qra.House_matvec(A=A, v=v, beta=beta, order='AP')
aae(AP1, AP2, decimal=10)
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)
def test_House_vector_orthogonal_reflection():
'verify that the resulting Householder reflection is orthogonal'
np.random.seed(14)
# real vector
a = np.random.rand(7)
v, beta = qra.House_vector(x=a)
P = np.identity(a.size) - beta*np.outer(v,v)
aae(np.dot(P.T,P), np.dot(P,P.T), decimal=10)
aae(np.dot(P.T,P), np.identity(a.size), decimal=10)
def test_QR_Givens_Cholesky():
'verify that R is the transpose of the Cholesky factor of ATA'
np.random.seed(8)
M = 6
N = 5
A = np.random.rand(M,N)
ATA = np.dot(A.T,A)
qra.QR_Givens(A)
R = np.triu(A)
aae(ATA, np.dot(R.T,R), decimal=10)
def test_House_vector_reflection_property():
'verify that Px = norm_2(x) u_0'
np.random.seed(43)
# real vector
x = np.random.rand(7)
v, beta = qra.House_vector(x=x)
# compute the Householder reflection
P = np.identity(x.size) - beta*np.outer(v,v)
x_norm_2 = np.linalg.norm(x)
u_0 = np.zeros_like(x)
u_0[0] = 1
aae(np.dot(P,x), x_norm_2*u_0, decimal=10)
def test_House_vector_parameter_beta():
'verify that beta = 2/dot(v,v)'
np.random.seed(23)
a = np.random.rand(7)
v, beta = qra.House_vector(x=a)
aae(beta, 2/np.dot(v,v), decimal=10)